*282*
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*English*
*Pages 340
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*Year 2020*

- Author / Uploaded
- Sylvain CROVISIER
- Raphaël KRIKORIAN
- Carlos MATHEUS
- Samuel SENTI

- Categories
- Mathematics
- Dynamical Systems

416

ASTÉRISQUE 2020

SOME ASPECTS OF THE THEORY OF DYNAMICAL SYSTEMS: A TRIBUTE TO JEAN-CHRISTOPHE YOCCOZ Volume II Sylvain Crovisier, Raphaël Krikorian, Carlos Matheus, Samuel Senti, éditeurs

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Astérisque est un périodique de la Société Mathématique de France. Numéro 416, 2020

Comité de rédaction Marie-Claude Arnaud Fanny Kassel Christophe Breuil Eric Moulines Damien Calaque Alexandru Oancea Philippe Eyssidieux Nicolas Ressayre Christophe Garban Sylvia Serfaty Colin Guillarmou Nicolas Burq (dir.) Diffusion Maison de la SMF Case 916 - Luminy 13288 Marseille Cedex 9 France [email protected]

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ISSN: 0303-1179 (print) 2492-5926 (electronic) ISBN 978-2-85629-917-3 doi:10.24033/ast.1108 Directeur de la publication : Stéphane Seuret

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ASTÉRISQUE 2020

SOME ASPECTS OF THE THEORY OF DYNAMICAL SYSTEMS: A TRIBUTE TO JEAN-CHRISTOPHE YOCCOZ Volume II Sylvain Crovisier, Raphaël Krikorian, Carlos Matheus, Samuel Senti, éditeurs

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Sylvain Crovisier CNRS - Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, 91405 Orsay Cedex, France. [email protected] Raphaël Krikorian Department of Mathematics, CNRS UMR 8088, Université de Cergy-Pontoise, 2, av. Adolphe Chauvin F-95302 Cergy-Pontoise, France. [email protected] Carlos Matheus CMLS, École polytechnique, CNRS (UMR 7640), 91128, Palaiseau, France. [email protected] Samuel Senti Instituto de Matemética, Universidade Federal do Rio de Janeiro, CP. 68 530, Rio de Janeiro, Brésil. [email protected]

Classification mathématique par sujet (2010). — 32A10, 37A17, 37A25, 37A50, 37C40, 37C75, 37D30, 37E05, 37E30, 37F25, 37F30, 37F50, 37J50, 53D12, 54H20, 60F05, 70H20; 37C86, 37D25, 37F25, 37J12. Mots-clefs. — Accessibilité, application Gevrey, (C 0 -)commutativité au sens de Poisson, centralisateurs, changements de temps, complète integrabilité, conditions diophantiennes, coordonnées de Fatou, decroissance des correlations, déviations des moyennes ergodiques, difféomorphismes analytiques du cercle, dimension centrale, distributions limites, domaines de rotation, dynamique holomorphe, échange d’intervalle, feuilletage invariant, feuilletages, flots nilpotents de Heisenberg, fonctions génératrices, germes holomorphes de C2 , hamiltoniens, hérissons, homéomorphismes symplectiques, hyperbolicité faible, hyperbolicité partielle, instabilité, linéarisation, mélange, nombre de rotation, pétales invariants, petits diviseurs, point fixe elliptique, points fixes indifférents, renormalisation, renormalisation sectorielle, sommes de Birkhoff, sous-variétés lagrangiennes, symplectomorphisme, système dynamique, théorème de translation plane de Brouwer, théorèmes d’Arnol0 d- Liouville, type Roth, variété centrale, vitesses de mélange.

Keywords. — Accessibility, analytic circle diffeomorphisms, Arnol0 d- Liouville theorem, Birkhoff sums, Brower Plane Translation Theorem, (C 0 -)Poisson commutativity, center manifold, central dimension, centralizers, complete integrability, complex dynamics, decay of correlations, deviation of ergodic averages, Diophantine conditions, dynamical system, elliptic fixed point, Fatou coordinates, foliation, generating functions, Gevrey map, Hamiltonian, hedgehogs, Heisenberg nilpotent flows, holomorphic germs in C2 , holomorphic germs in Cn , indifferent fixed points, instability, interval exchange maps, invariant foliation, invariant petals, Lagrangian submanifolds, limit distributions, linearization, measurable Riemann Mapping Theorem, mixing, mixing rates, non-uniform hyperbolicity, partial hyperbolicity, renormalization, rotation domains, rotation number, Roth type, sector renormalization, small divisors, symplectic homeomorphisms, symplectomorphism, time-changes.

SOME ASPECTS OF THE THEORY OF DYNAMICAL SYSTEMS: A TRIBUTE TO JEAN-CHRISTOPHE YOCCOZ Volume II Sylvain Crovisier, Raphaël Krikorian, Carlos Matheus, Samuel Senti, éditeurs

Abstract. — This is the second of two volumes which celebrate the memory of JeanChristophe Yoccoz. These volumes present research articles on various aspects of the theory of dynamical systems and related topics that were dear to him. Résumé (Quelques aspects de la théorie des systèmes dynamiques : un hommage à JeanChristophe Yoccoz) — Voici le deuxième de deux volumes qui célèbrent la mémoire de Jean-Christophe Yoccoz. Ils regroupent des articles de recherche portant sur divers aspects de la théorie des systèmes dynamiques ainsi que sur des sujets connexes qui lui étaient chers.

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TABLE DES MATIÈRES

Marie-Claude Arnaud & Jinxin Xue — A C 1 Arnol′ d-Liouville theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the proofs and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Lipschitz complete integrability determines the Dynamics on each minimal torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Uniform rate for the ﬂow on the invariant tori . . . . . . . . . . . . . . . . . . . 2.2. Uniform rate and completely irrational rotation vector imply uniform Lipschitz conjugacy to a ﬂow of rotations . . . . . . . . . . . . . . . . . . . . 2.3. Lipschitz conjugacy to a completely irrational rotation implies biLipschitz conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Proofs of Theorem 2.1 and Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . 3. The A-non-degeneracy condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The symplectic homeomorphism in the case of C 1 complete integrability 4.1. A generating function for the Lagrangian foliation . . . . . . . . . . . . . . . 4.2. The C 1 property of the conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The symplectic homeomorphism and the proof of Theorem 1.1 . . . 4.4. The smooth approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. C 0 integrability and C 0 Lagrangian submanifolds . . . . . . . . . A.1. Diﬀerent notions of C 0 integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Smooth Hamiltonians that are C 0 completely integrable . . . Appendix C. Smooth Hamiltonians that are Lipschitz completely integrable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Juan Rivera-Letelier — Asymptotic expansion of smooth interval maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Quantifying asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Nonuniformly hyperbolic interval maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Exponentially mixing acip’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 8 8 10 11 14 16 17 18 18 19 22 23 23 24 28 29 30 30

33 34 34 36 38

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1.4. Topological invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. High-temperature phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. Strategy and organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Fatou and Julia sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Topological exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Diﬀerentiable interval maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Interval maps of class C 3 with non-ﬂat critical points . . . . . . . . . . . . 3. Exponential shrinking of components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Quantifying asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Conjugacy to a piecewise aﬃne map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Nonuniform hyperbolicity conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Lyapunov exponents are nonnegative . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefano Marmi & Corinna Ulcigrai & Jean-Christophe Yoccoz — On Roth type conditions, duality and central Birkhoﬀ sums for i.e.m. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Dual Roth type and distributional limit shapes . . . . . . . . . . . . . . . . . . 1.2. Absolute Roth type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Backgound material and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Interval exchange maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Translation surfaces and suspension data . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. A step of Rauzy–Veech algorithm, Rauzy Veech diagrams and Rauzy-Veech matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Iterations of the Rauzy-Veech map and the Zorich algorithm . . . . . 2.5. Dynamics of the continued fraction algorithms . . . . . . . . . . . . . . . . . . . 2.6. Special Birkhoﬀ sums and the extended Kontsevich-Zorich cocycle 2.7. The boundary operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Summary of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interval exchange maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Translation surfaces and suspension data . . . . . . . . . . . . . . . . . . . . . . . . . . The Rauzy-Veech renormalization algorithm . . . . . . . . . . . . . . . . . . . . . . . Functional spaces, special Birkhoﬀ sums and boundary operator . . . 3. Absolute Roth type and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Roth Type i.e.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The cones C (π) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Absolute Roth type i.e.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Two crucial estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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65 66 67 70 70 71 71 72 73 75 76 77 78 79 80 80 81 81 82 82 83 85 86 87

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3.5. Results on the cohomological equation for absolute Roth type i.e.m 3.6. Results on the cohomological equation for translation surfeces in a.e. direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Dual Roth Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Backward Rauzy-Veech induction and backward rotation numbers 4.2. Dual special Birkhoﬀ sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Dual Roth type condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Dual lengths control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Estimates on dual special Birkhoﬀ sums of Hölder functions . . . . . 5. Distributional limit shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Corrected characteristic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (n) 5.2. The functions Ωα (π, τ, χ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (n) (n′ ) 5.3. Comparing the functions Ωα (π, τ, χ) and Ωα (π, τ, χ) . . . . . . . . . . (n) 5.4. Distributional convergence of the sequence Ωα (π, τ, ξ) as n → −∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Homological interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Bases of homology associated to the Rauzy-Veech algorithm . . . . . The classes θα ∈ H1 (M \ Σ, Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The classes ζα ∈ H1 (M, Σ, Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Duality and relation with Ωπ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Change of bases associated to the Rauzy-Veech algorithm . . . . . . . . 6.3. KZ-hyperbolic translation surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Piecewise-aﬃne paths in H1 (M \ Σ, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . (n) 6.5. The functions Ωα (Υ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Piecewise-aﬃne paths in H1 (M, Σ, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7. The functions Ω∗ (n) α (Υ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Completeness of backward rotation numbers . . . . . . . . . . . . . . . A.1. The quantity H(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Switching times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3. The subset A ′ ⊂ A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4. Decompositions of A \ A ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5. Proof of Proposition 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Adaptation of the proofs of results on cohomological equation for absolute Roth type i.e.m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1. Estimates for special Birkhoﬀ sums with C 1 data . . . . . . . . . . . . . . . . B.2. Higher diﬀerentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3. Growth of B(0, nℓ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4. Adaptation of the proof of Theorems 3.10 and 3.11 in [38] . . . . . . . B.4.1. Adaptation of the proof of Lemma 3.15 in [38] . . . . . . . . . . . . . . B.4.2. Space and Time decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4.3. General Hölder estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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90 92 93 94 95 98 100 101 103 104 106 107 108 110 110 110 111 111 111 112 116 117 118 119 120 121 121 123 123 125 125 125 126 126 127 128 128 128

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Artur Avila & Xavier Buff & Arnaud Chéritat — Smooth Siegel disks everywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Conformal radius, wild combs and the general construction. . . . . . . . . . . 1.1. Siegel disks and restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Properties of the conformal radius as a function of the angle . . . . . 1.3. Properties of the linearizing map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. A remark on continuum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Special case: assuming Brjuno’s condition is optimal . . . . . . . . . . . . . 1.6. Smooth Siegel disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction of a sequence θn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of the limit θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. Other regularity classes for the boundary . . . . . . . . . . . . . . . . . . . . . . . . 1.8. General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Proof of the main lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Transfer of the Lipschitz condition to the lifts . . . . . . . . . . . . . . . 2.2. Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Iterations and rescalings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Reminder on continued fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2. Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Proof of Lemma 2 for α ≡ 0 mod Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Improvement through renormalization for maps tending to a nonzero rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Proof of Lemma 2 for α = p/q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9. Proof of lem:main2 for α ∈ R \ Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Analytic degenerate families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. General statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ricardo Pérez-Marco — On quasi-invariant curves . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Analytic circle diﬀeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Real estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Denjoy-Yoccoz Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Hyperbolic Denjoy-Yoccoz Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Flow interpolation in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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133 133 135 135 137 138 139 140 141 142 142 143 143 148 148 148 149 150 155 155 156 156 157 161 164 170 171 175 178 178 181 181 183 183 183 185 186 186

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4.2. Flow interpolation in C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Hyperbolic Denjoy-Yoccoz Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Quasi-invariant curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Reduction to small non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 188 190 191 191

Tanya Firsova & Mikhail Lyubich & Remus Radu & Raluca Tanase — Hedgehogs for neutral dissipative germs of holomorphic diﬀeomorphisms of (C2 , 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Center manifolds of the semi-indiﬀerent fixed point . . . . . . . . . . . . . . . . . . . 3. Semi-parabolic germs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Associated circle homeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Appendix: Alternative approach in dimension one . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 196 196 197 205 205 206 207 210

Mikhail Lyubich & Remus Radu & Raluca Tanase — Hedgehogs in higher dimensions and their applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Holomorphic germs of (C, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. A complex structure on the center manifold . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Quasiconformal conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Quasiconformal conjugacy to an analytic map . . . . . . . . . . . . . . . . . . . . 4.3. Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Dynamical consequences of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-linearizable germ with a Siegel disk . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. A generalization to germs of (Cn , 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of hedgehogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Semi-parabolic germs of (Cn , 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holomorphic Fatou coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

213 213 217 217 219 226 226 226 229 230 231 232 235 240 244 245 249 250

Giovanni Forni & Adam Kanigowski — Time-changes of Heisenberg nilﬂows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253 254 257

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2.1. Heisenberg Nilflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Heisenberg moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. The renormalization flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Stretching of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Construction of the functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Main properties of the functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Limit distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Square mean lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Analyticity of the functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Bounds on the valency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Measure estimates: the bounded-type case . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Measure estimates: the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Correlations. Proof of Theorems 2.2 and 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257 257 258 259 259 260 261 262 267 272 278 283 287 289 291 294 297

Artur Avila & Marcelo Viana — Stable accessibility with 2-dimensional center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Deformations paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. An intersection property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. An approximation property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Connected subgroups of surface diffeomorphisms . . . . . . . . . . . . . . . . . . . . . 6. Density of stable accessibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

301 301 303 306 307 309 315 319

Bassam Fayad & Jean-Pierre Marco & David Sauzin — Attracted by an elliptic fixed point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries and outline of the strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The attraction mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gevrey functions, maps and flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Gevrey functions and Gevrey maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Banach algebra of uniformly Gevrey-(α, L) functions . . . . . . . . . . The Banach space of uniformly Gevrey-(α, L) maps . . . . . . . . . . . . . . . . Composition with close-to-identity Gevrey-(α, L) maps . . . . . . . . . . . . A.2. Estimates for Gevrey flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321 321 323 326 328 334 334 334 334 335 337 339 339

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Un théorème C 1 d’Arnol ′d-Liouville Marie-Claude Arnaud & Jinxin Xue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Dans cet article, nous montrons une version du théorème d’Arnol′ d-Liouville pour des hamiltoniens de classe C 2 qui ont assez de hamiltoniens de classe C 1 commutant avec eux. Nous montrons que le caractère Lipschitz du feuilletage en tores lagrangiens invariants est crucial pour déterminer la dynamique sur chaque tore invariant et que la régularité C 1 du feuilletage est cruciale pour montrer la continuité des coordonnées d’Arnol′ d-Liouville. Nous explorons aussi diﬀérentes notions d’intégrabilité au sens C 0 ou Lipschitz. Expansion asymptotique des applications lisses d’intervalle Juan Rivera-Letelier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

On associe à chaque application lisse et non dégénérée de l’intervalle un nombre measurant son expansion asymptotique globale. On montre que ce nombre peut être calculé de plusieurs façons distinctes. En conséquence, plusieurs notions d’hyperbolicité faible coïncident. De cette façon on obtient une extension aux applications de l’intervalle avec une nombre arbitraire de points critiques du fameux résultat de Nowicki et Sands caractérisant la condition de Collet-Eckmann pour les applications unimodales. Ceci résout aussi une conjecture de Luzzatto en dimensión 1. En combinaison avec un résultat de Nowicki et Przytycki, ces considérations entraînent que plusieurs notions d’hyperbolicité faible sont invariantes par conjugaison topologique. Une autre conséquence est pour le formalisme thermodynamique : une application lisse et non dégénérée de l’intervalle possède une transition de phase de haute temperature si et seulement si elle n’est pas Lyapunov hyperbolique. Sur les conditions de type Roth, la dualité et les sommes de Birkhoﬀ centrées pour les échange d’intervalles Stefano Marmi & Corinna Ulcigrai & Jean-Christophe Yoccoz 65 Nous introduisons deux conditions diophantiennes pour les nombres de rotation des transformations d’échange d’intervalles (i.e.m.) et des surfaces de translation : la condition absolue de type Roth est un aﬀaiblissement de la notion i.e.m

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de type Roth, tandis que la condition duale de type Roth est une condition sur le nombre de rotation en arrière d’une surface de translation. Nous montrons que les résultats sur l’équation cohomologique prouvés précédemment dans [38] pour les i.e.m. de type Roth restreint (sur la solvabilité en supposant un nombre fini d’obstructions et la régularité des solutions) peuvent être étendues aux i.e.m. de type Roth absolu restreint. Sous la condition duale de type Roth, nous associons des formes limites (limit shapes) distributionnelles à une classe de fonctions avec des déviations sous-polynomiales des moyennes ergodiques (correspondantes aux classes d’homologie relatives), qui sont construites de manière similaire aux formes limites des sommes de Birkhoﬀ associées dans [36] aux fonctions qui correspondent aux exposants de Lyapunov positifs. De l’ubiquité des disques de Siegel à bord lisse Artur Avila & Xavier Buff & Arnaud Chéritat . . . . . . . . . . . . . . . . .

133

Nous démontrons l’existence de disques de Siegel à bord lisse dans la plupart des familles de fonctions holomorphes fixant l’origine. La méthode peut également donner d’autres types de régularité pour le bord. On demande à la famille d’avoir un point fixe indiﬀérent en 0, d’être paramétrisée par le nombre de rotation α, de dépendre d’α de façon Lipschitz-continue et d’être non-dégénérée. Une famille est dite dégénérée si l’ensemble de ses applications non-linéarisables n’est pas dense. Nous donnons une caractérisation des familles dégénérées, qui prouve qu’elles sont assez exceptionnelles. Sur les courbes quasi-invariantes Ricardo Pérez-Marco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181

Les courbes quasi-invariantes sont un outil fondamental dans l’étude de la dynamique des hérissons. Le lemme de Denjoy-Yoccoz est le premier pas dans la théorie de renormalisation de Yoccoz des diﬀéomorphismes analytiques du cercle et l’étude de sa linéarisation. On donne une nouvelle version du lemme de Denjoy-Yoccoz en termes de métrique hyperbolique, ce qui fournit une nouvelle construction directe des courbes quasi-invariantes sans utiliser la renormalisation comme dans la construction originelle.

Hérissons pour les germes dissipatifs neutres des diﬀéomorphismes holomorphes de (C2 , 0) Tanya Firsova & Mikhail Lyubich & Remus Radu & Raluca Tanase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Nous montrons l’existence de hérissons pour les germes de diﬀéomorphismes holomorphes de (C2 , 0) ayant un point fixe semi-neutre à l’origine, en utilisant uniquement des techniques topologiques. Cette approche donne également une preuve alternative d’un théorème de Pérez-Marco sur l’existence de hérissons pour les germes de diﬀéomorphismes holomorphes de (C, 0) ayant un point fixe neutre.

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Hérissons en dimension supérieure et leurs applications Mikhail Lyubich & Remus Radu & Raluca Tanase . . . . . . . . . . . . . . . 213 Dans cet article, on étudie la dynamique des germes de difféomorphismes holomorphes de (Cn , 0) ayant un point fixe à l’origine avec exactement une valeur propre neutre. Nous prouvons que la fonction sur n’importe quelle variété centrale locale de 0 est quasiconformément conjuguée à une fonction holomorphe et utilisons ce théorème pour adapter des résultats en dimension une complexe aux dimensions supérieures. Changements de temps des flots nilpotents d’Heisenberg Giovanni Forni & Adam Kanigowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Nous étudions les flots nilpotents de Heisenberg en dimension trois. Sous une condition Diophantienne de mesure pleine sur le générateur du flot, nous montrons l’existence de fonctionnelles de Bufetov, qui sont asymptotiques aux intégrales ergodiques pour toutes les fonctions suffisamment différentiables, qui ont une propriété modulaire, et satisfont une identité de changement d’échelle sous la dynamique de renormalisation. De la propriété asymptotique, nous dérivons des résultats sur les distributions limites des moyennes ergodiques, qui généralisent les travaux de Griffin et Marklof [17], et Cellarosi et Marklof [8]. Ensuite nous montrons une propriété d’analyticité des fonctionnelles dans les directions transverses au flot. Comme conséquence de cette propriété d’analyticité, nous dérivons l’existence d’un ensemble de mesure pleine de flots nilpotents dont les changements de temps génériques (non-triviaux) sont mélangeant, et de plus ont une vitesse de mélange « polynomiale étirée » pour toutes les fonctions suffisamment différentiables (cela améliore un résultat de Avila, Forni, et Ulcigrai [2]). De plus, nous montrons qu’il existe un ensemble de dimension de Hausdorff maximale de flots nilpotents tels que les changements de temps génériques non-triviaux ont une vitesse de mélange polynomiale. Accessibilité stable de dimension centrale 2 Artur Avila & Marcelo Viana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 1 L’accessibilité est une propriété C -stable parmi les difféomorphismes partiellement hyperboliques à dimension centrale 2. De plus, l’accessibilité (stable) est une propriété C 1 -dense dans le domaine des produits gauches satisfaisant la condition de regroupement central (‘center bunching’). Attiré par un point fixe elliptique Bassam Fayad & Jean-Pierre Marco & David Sauzin . . . . . . . . . . . . 321 Nous donnons des exemples de difféomorphismes symplectiques de R6 pour lesquels l’origine est un point fixe elliptique non résonant qui attire une orbite.

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ABSTRACTS

A C 1 Arnol ′d-Liouville theorem Marie-Claude Arnaud & Jinxin Xue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

In this paper, we prove a version of Arnol d-Liouville theorem for C Hamiltonians having enough C 1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to determine the Dynamics on each Lagrangian torus and that the C 1 regularity of the foliation by invariant Lagrangian tori is crucial to prove the continuity of Arnol′ d-Liouville coordinates. We also explore various notions of C 0 and Lipschitz integrability. ′

Asymptotic expansion of smooth interval maps Juan Rivera-Letelier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

We associate to each non-degenerate smooth interval map a number measuring its global asymptotic expansion. We show that this number can be calculated in various diﬀerent ways. A consequence is that several natural notions of nonuniform hyperbolicity coincide. In this way we obtain an extension to interval maps with an arbitrary number of critical points of the remarkable result of Nowicki and Sands characterizing the Collet-Eckmann condition for unimodal maps. This also solves a conjecture of Luzzatto in dimension 1. Combined with a result of Nowicki and Przytycki, these considerations imply that several natural nonuniform hyperbolicity conditions are invariant under topological conjugacy. Another consequence is for the thermodynamic formalism: A non-degenerate smooth map has a high-temperature phase transition if and only if it is not Lyapunov hyperbolic. On Roth type conditions, duality and central Birkhoﬀ sums for i.e.m. Stefano Marmi & Corinna Ulcigrai & Jean-Christophe Yoccoz

65

We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m.) and translation surfaces: the absolute Roth type condition is a weakening of the notion of Roth type i.e.m., while the dual Roth type condition is a condition on the backward rotation number of a translation surface. We show that results on the cohomological equation previously proved in [38]

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for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted absolute Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with subpolynomial deviations of ergodic averages (corresponding to relative homology classes) distributional limit shapes, which are constructed in a similar way to the limit shapes of Birkhoﬀ sums associated in [36] to functions which correspond to positive Lyapunov exponents. Smooth Siegel disks everywhere Artur Avila & Xavier Buff & Arnaud Chéritat . . . . . . . . . . . . . . . . .

133

We prove the existence of Siegel disks with smooth boundaries in most families of holomorphic maps fixing the origin. The method can also yield other types of regularity conditions for the boundary. The family is required to have an indiﬀerent fixed point at 0, to be parameterized by the rotation number α, to depend on α in a Lipschitz-continuous way, and to be non-degenerate. A degenerate family is one for which the set of non-linearizable maps is not dense. We give a characterization of degenerate families, which proves that they are quite exceptional. On quasi-invariant curves Ricardo Pérez-Marco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181

Quasi-invariant curves are the fundamental tool for the study of hedgehog’s dynamics. The Denjoy-Yoccoz lemma is the preliminary step for Yoccoz’s complex renormalization techniques for the study of linearization of analytic circle diﬀeomorphisms. We give a new geometric interpretation of the Denjoy-Yoccoz lemma using the hyperbolic metric that gives a new direct construction of quasiinvariant curves without renormalization theory as in the original construction. Hedgehogs for neutral dissipative germs of holomorphic diﬀeomorphisms of (C2 , 0) Tanya Firsova & Mikhail Lyubich & Remus Radu & Raluca Tanase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 We prove the existence of hedgehogs for germs of complex analytic diﬀeomorphisms of (C2 , 0) with a semi-neutral fixed point at the origin, using topological techniques. This approach also provides an alternative proof of a theorem of Pérez-Marco on the existence of hedgehogs for germs of univalent holomorphic maps of (C, 0) with a neutral fixed point. Hedgehogs in higher dimensions and their applications Mikhail Lyubich & Remus Radu & Raluca Tanase . . . . . . . . . . . . . . .

213

In this paper we study the dynamics of germs of holomorphic diﬀeomorphisms of (Cn , 0) with a fixed point at the origin with exactly one neutral eigenvalue.

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We prove that the map on any local center manifold of 0 is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher dimensions. Time-changes of Heisenberg nilﬂows Giovanni Forni & Adam Kanigowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 We consider the three dimensional Heisenberg nilﬂows. Under a full measure set Diophantine condition on the generator of the ﬂow we construct Bufetov functionals which are asymptotic to ergodic integrals for suﬃciently smooth functions, have a modular property and scale exactly under the renormalization dynamics. By the asymptotic property we derive results on limit distributions, which generalize earlier work of Griﬃn and Marklof [17] and Cellarosi and Marklof [8]. We then prove analyticity of the functionals in the transverse directions to the ﬂow. As a consequence of this analyticity property we derive that there exists a full measure set of nilﬂows such that generic (non-trivial) time-changes are mixing and moreover have a “stretched polynomial” decay of correlations for suﬃciently smooth functions (this strengthens a result of Avila, Forni, and Ulcigrai [2]). Moreover we also prove that there exists a full Hausdorﬀ dimension set of nilﬂows such that generic non-trivial time-changes have polynomial decay of correlations. Stable accessibility with 2-dimensional center Artur Avila & Marcelo Viana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 For partially hyperbolic diﬀeomorphisms with 2-dimensional center, accessibility is C 1 -stable. Moreover, for center bunched skew-products (stable) accessibility is C ∞ -dense. Attracted by an elliptic fixed point Bassam Fayad & Jean-Pierre Marco & David Sauzin . . . . . . . . . . . . 319 6 We give examples of symplectic diﬀeomorphisms of R for which the origin is a non-resonant elliptic fixed point which attracts an orbit.

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Astérisque 416, 2020, p. 1–31 doi:10.24033/ast.1109

A C 1 ARNOL′ D-LIOUVILLE THEOREM by Marie-Claude Arnaud & Jinxin Xue

Dedicated to Jean-Christophe Yoccoz A generous man

Abstract. — In this paper, we prove a version of Arnol′ d-Liouville theorem for C 2 Hamiltonians having enough C 1 commuting Hamiltonians. We show that the Lipschitz regularity of the foliation by invariant Lagrangian tori is crucial to determine the Dynamics on each Lagrangian torus and that the C 1 regularity of the foliation by invariant Lagrangian tori is crucial to prove the continuity of Arnol′ d-Liouville coordinates. We also explore various notions of C 0 and Lipschitz integrability. Résumé (Un théorème C 1 d’Arnol′d-Liouville). — Dans cet article, nous montrons une version du théorème d’Arnol′ d-Liouville pour des hamiltoniens de classe C 2 qui ont assez de hamiltoniens de classe C 1 commutant avec eux. Nous montrons que le caractère Lipschitz du feuilletage en tores lagrangiens invariants est crucial pour déterminer la dynamique sur chaque tore invariant et que la régularité C 1 du feuilletage est cruciale pour montrer la continuité des coordonnées d’Arnol′ dLiouville. Nous explorons aussi diﬀérentes notions d’intégrabilité au sens C 0 ou Lipschitz.

1. Introduction and Main Results This article elaborates on the following question. Question. — If a C 2 Hamiltonian system has enough commuting integrals (1), can we precisely describe the Hamiltonian Dynamics, even in the case of non C 2 integrals? 2010 Mathematics Subject Classification. — 37J50, 70H20, 53D12. Key words and phrases. — (C 0 -)Poisson commutativity, Hamiltonian, Arnol′ d- Liouville theorem, foliation, Lagrangian submanifolds, generating functions, symplectic homeomorphisms, complete integrability. The first author is member of the Institut universitaire de France. The project is supported by ANR-12-BLAN-WKBHJ. (1) These notions are precisely described in the rest of the introduction.

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When the integrals are C 2 , Arnol′ d-Liouville Theorem (see [4]) implies that the compact level sets are tori and gives a precise description of the Dynamics. The proof in [4] relies on the Abelian group action generated by the commuting Hamiltonian ﬂows. Unfortunately, the result is not valid for C 1 integrals : in this case, there is a priori no Hamiltonian ﬂow that we can associate to the C 1 integrals, the Abelian group action does not exist and so the proof in [4] does not work. Note that in some case, a C 0 -integrability can be shown without knowing if the integrals can be chosen smooth: the case of Tonelli Hamiltonians (2)with no conjugate points on Tn ×Rn (see Theorem B.1). C 0 -integrability for such Hamiltonians is proved in [1] and some partial results concerning the Dynamics on the invariant graphs are given, but no result similar to Arnol′ d-Liouville Theorem is proved. The only case where a more accurate result is obtained is when the Tonelli Hamiltonian gives rise to a Riemannian metric after Legendre transform. Burago & Ivanov proved in [5] that a Riemannian metric with no conjugate points is smoothly integrable, but this is specific to the Riemannian case and cannot be adapted to the general Tonelli case. In these statements and all our results, the topology of the invariant leaves is prescribed: they are tori. Indeed, we cannot use the Abelian group action defined by the commuting Hamiltonian ﬂows as in classical Arnol′ d-Liouville Theorem to recover this topology. And our use of Herman’s theory requires the level set to be Tn , so we work with T ∗ Tn with the topology prescribed. In this article, we consider the case of Lipschitz and C 1 -integrability, that are intermediary between C 0 and C 2 integrability. For a C 2 Hamiltonian that satisfies a certain non-degeneracy condition called A-non-degeneracy condition (3) and that is C 1 -integrable, we will prove — we can define global continuous Arnol′ d-Liouville coordinates, which are defined by using a symplectic homeomorphism; — the Dynamics restricted to every invariant torus is C 1 -conjugate to a rotation; — we can even define a ﬂow for the continuous Hamiltonian vectorfields that are associated to the C 1 integrals (see Proposition 4.2). For a Tonelli Hamiltonian that is Lipschitz integrable, we will prove that the Dynamics restricted to every invariant torus is Lipschitz conjugate to a rotation, but we will obtain no information concerning the transverse dependence to the conjugacy. Let us add that the wide class of Hamiltonians that are defined on the cotangent bundle of the n-dimensional torus and strictly convex in the fiber direction is a part of the set of A-non-degenerate Hamiltonians. (4) In particular, Tonelli Hamiltonian are A-non-degenerate. In order to state our results, let us now introduce some definitions. The definitions can be divided into three classes: the integrability conditions (Definition 1.2, 1.3, (2) (3) (4)

See Definition 1.1. See Definition 1.4. This will be proved in Proposition 3.2.

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1.6, 1.7, 1.8), the nondegeneracy conditions (Definition 1.1, 1.4) and the symplectic coordinates (Definition 1.5). Definition 1.1. — A C 2 Hamiltonian H : T ∗ N → R for a compact Riemannian manifold N is Tonelli if the following two assumptions are satisfied. — H has super-linear growth, i.e., H(q,p) kpk → ∞ as kpk → ∞. — H is convex in the fiber, i.e.,

∂2H ∂p2 (q, p)

is positive definite for all q, p.

For example, a mechanical Hamiltonian H : + V (q) is Tonelli.

T ∗ Tn → R given by H(q, p) =

1 2 2 kpk

Notations. — If H is a C 1 Hamiltonian defined on a symplectic manifold (M (2n) , ω), we denote by XH the Hamiltonian vectorfield, that is defined by ∀ x ∈ M, ∀ v ∈ Tx M,

ω(XH (x), v) = dH(x) · v.

If moreover H is C 2 , the Hamiltonian ﬂow associated to H, that is the ﬂow of XH , is denoted by (ϕH t ). — If H and K are two C 1 Hamiltonians that are defined on M , their Poisson bracket is {H, K}(x) = DH(x) · XK (x) = ω(XH (x), XK (x)). Definition 1.2. — Let k ≥ 1 be an integer. Let U ⊂ M (2n) be an open subset and let H : M → R be a C sup{2,k} Hamiltonian. Then H is C k completely integrable in U if — U is invariant by the Hamiltonian ﬂow of H; — there exist n C k functions H1 , H2 , . . . , Hn : U → R so that — at every x ∈ U , the family dH1 (x), . . . , dHn (x) is independent; — for every i, j, we have {Hi , Hj } = 0 and {Hi , H} = 0. Remarks. — 1. We cannot always take H1 = H. At the critical points of H, dH(x) = 0 and a Tonelli Hamiltonian has always critical points. However, if we consider only the part of phase space without critical points, we can indeed take H = H1 . 2. Observe that when k = 1, the C 1 Hamiltonians H1 , . . . , Hn don’t necessarily define a ﬂow because the corresponding vector field is just continuous. Hence the proof of Arnol′ d-Liouville theorem (see for example [4]) cannot be used to determine what the Dynamics is on the invariant Lagrangian submanifold {H1 = c1 , . . . , Hn = cn }. That is why the results we give in Theorem 1.1 and 1.2 below are non-trivial. In fact, in the setting of next definition for k = 1 and when the Hamiltonian is Tonelli, we will prove in Proposition 4.2 a posteriori that each Hi surprisingly defines a ﬂow. We will sometimes need the following narrower definition of C k integrability.

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Definition 1.3. — A C 2 Hamiltonian H : V → R that is defined on some open subset V ⊂ T ∗ N is called G-C k completely integrable on some open subset U ⊂ V if — it is C k completely integrable and — with the same notations as in the definition of C k complete integrability, every Lagrangian submanifold {H1 = c1 , . . . , Hn = cn } is the graph of a C k map. Remarks. — 1. Observe that for any k ≥ 1, G-C k integrability is equivalent to the existence of an invariant C k foliation into Lagrangian graphs. The direct implication is a consequence of the fact that the Hi are commuting in the Poisson sense (i.e., {Hi , Hj } = 0). For the reverse implication, denote the invariant C k foliation by (ηa )a∈U where a is in some open subset of Rn . Then we define a C k map A = (A1 , . . . , An ) by A(q, p) = a ⇐⇒ p = ηa (q). Observe that each Lagrangian graph Ta of ηa is in the energy level {Ai = ai }. Hence XAi (q, p) ∈ T(q,p) TA(q,p) and thus {Ai , Aj }(q, p) = ω(XAi (q, p), XAj (q, p)) = 0 because all the Ta are Lagrangian. In a similar way, {H, Ai } = 0. 2. When k ≥ 2, U = T ∗ N and H is a Tonelli Hamiltonian, C k -integrability implies G-C k integrability and that N = Tn . Let us give brieﬂy the arguments: in this case the set of fixed points of the Hamiltonian ﬂow N = { ∂H ∂p = 0} is an invariant submanifold that is a graph. Hence N is one of the invariant tori given by Arnol′ d-Liouville theorem. Therefore N = Tn and all the invariant tori of the foliation are also graphs : this is true for those that are close to N , and in this case there is a uniform Lipschitz constant because the Hamiltonian is Tonelli. Using this Lipschitz constant, we can extend the neighborhood where they are graphs to the whole T ∗ Tn . We next introduce a non-degeneracy condition called A-non-degeneracy. We will prove the following proposition in Section 4.2. Proposition 1.1. — Assume that (q, a) ∈ Tn × U 7→ (q, ηa (q)) ∈ U is a C 1 foliation of an open subset U of Tn × Rn into Lagrangian graphs. Then the map c : U → Rn defined by Z ηa (q)dq

c(a) =

Tn

is a C 1 -diﬀeomorphism from U onto its image.

Notations. — Let H : U ⊂ T ∗ Tn → R be a C 1 integrable Hamiltonian. — We will denote by (Ta )a∈U = ({(q, ηa (q)) : q ∈ Tn })a∈U R the invariant foliation and define the function c : U → Rn by c(a) = Tn ηa (q)dq. — The A-function AH : c(U ) → R is defined by AH (c) = H(0, ηc−1 (c) (0)).

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Definition 1.4. — Let H : U → R, U ⊂ T ∗ Tn , be a G-C 1 completely integrable Hamiltonian. We say that H is A-non-degenerate if for every non-empty open subset V ⊂ c(U ), the set ∇AH (V ) is contained in no resonant hyperplane of Rn k1 x1 + · · · + kn xn = 0 n

with (k1 , . . . , kn ) ∈ Z \{(0, . . . , 0)}. In the classical Arnol′ d-Liouville theorem for C k commuting Hamiltonians with k ≥ 2, the invariant foliation is C k but the change of coordinates is only C k−1 transversally to the invariant leaves. Hence we can guess that under the C 1 integrability assumption, the Arnol′ d-Liouville coordinates transform can only be C 0 and we need a notion of C 0 symplectic maps. It was a fundamental result of Gromov [11] and Eliashberg [7] in symplectic geometry that the group of symplectomorphisms on a symplectic manifold is C 0 -closed in the group of diﬀeomorphisms. Definition 1.5. — Following [19], we call a homeomorphism a symplectic homeomorphism if its restriction to every relatively compact open subset is a uniform limit for the C 0 -topology of a sequence of symplectic C ∞ diﬀeomorphisms. Our main theorem is as follows. Theorem 1.1. — Let H : U → R be G-C 1 completely integrable and A-non-degenerate in some open set U ⊂ T ∗ Tn . Then there exist a neighborhood U of 0 in Rn and a symplectic homeomorphism φ : Tn × U → U that is C 1 in the direction of Tn such that n n — ∀ c ∈ U , φ−1 ◦ ϕH t ◦ φ(T × {c}) = T × {c}; −1 H = Rtρ(c) ; — ∀ c ∈ U , φ ◦ ϕt ◦ φ| n T ×{c} n where ρ : U → R is a homeomorphism onto ρ(U ), and Rtρ(c) : Tn → Tn is given by Rtρ(c) (x) := x + tρ(c) mod Zn . This gives some symplectic Arnol′ d-Liouville coordinates in the C 0 sense (see Chapter 10 of [4]) and describes precisely the Dynamics on the leaves of the foliation. Observe that the conjugacy φ that we obtain has the same regularity as the foliation in the direction of the leaves but is just C 0 in the transverse direction. If we replace the C 1 -integrability by a Lipschitz integrability, we loose any transverse regularity and we just obtain some results along the leaves. Let us explain this. Definition 1.6. — A C 2 Hamiltonian H : V → R that is defined on some open subset V ⊂ T ∗ N is called G-Lipschitz completely integrable on some open subset U ⊂ V if U admits a Lipschitz foliation by invariant Lipschitz Lagrangian graphs. Let us recall that a Lipschitz graph L admits Lebesgue almost everywhere a tangent subspace by Rademacher theorem (see [8], Theorem 2, page 81). Such a graph is Lagrangian if and only if these tangent subspaces are Lagrangian. This is equivalent to asking that L is the graph of c+du where c is a closed smooth 1-form and u : N → R is C 1,1 .

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Theorem 1.2. — Suppose that the Hamiltonian H : T ∗ Tn → R is Tonelli and is G-Lipschitz completely integrable. Then restricted to each leaf, the Hamiltonian ﬂow has a unique well-defined rotation vector, and is bi-Lipschitz conjugate to a translation ﬂow by the rotation vector on Tn . Moreover, all the leaves are in fact C 1 . The proof of this result is given in Appendix C. Remark. — Observe that we do not know if the conjugacies are C 1 . The assumption of A-nondegeneracy can be verified in at least two settings: Tonelli Hamiltonian (Definition 1.1) and positive torsion (Definition 1.8). See Corollary 1.1 and 1.2. When H is a Tonelli Hamiltonian, the A function is exactly the α-function of Mather (see [17]). Corollary 1.1. — Suppose that H : T ∗ Tn → R is a Tonelli Hamiltonian that is G-C 1 completely integrable in some open set U ⊂ T ∗ Tn . Then there exist a neighborhood U of 0 in Rn and a symplectic homeomorphism φ : Tn × U → U that is C 1 in the direction of Tn such that n n — ∀ c ∈ U , φ−1 ◦ ϕH t ◦ φ(T × {c}) = T × {c}; −1 H — ∀ c ∈ U , φ ◦ ϕt ◦ φ| n = Rtρ(c) ; T ×{c}

where ρ : U → Rn is a homeomorphism onto ρ(U ), and Rtρ(c) : Tn → Tn is given by Rtρ(c) (x) := x + tρ(c) mod Zn . Theorem 1.1 is global but requires a little more that C 1 integrability. If we have just C 1 integrability (instead of G-C 1 integrability), we obtain a local result. Definition 1.7. — Let H : U → R, U ⊂ M (2n) , be a C 2 Hamiltonian and let T be an invariant C k , (k ≥ 1) Lagrangian torus contained in U . We say that H is locally C k completely integrable at T if there exists a neighborhood U ⊂ U of T such that — H is C k completely integrable in U ; — T is one leaf of the foliation given by level sets of the n integrals. Definition 1.8. — Let T ⊂ M be a C 1 Lagrangian torus in a symplectic manifold (M (2n) , ω) and let H : M → R be a C 2 Hamiltonian. We say that H has positive torsion along T if there exist — a neighborhood U of T in M ; — a neighborhood V of the zero section in T ∗ Tn ; — a C 2 symplectic diﬀeomorphism φ : U → V such that φ(T ) is the graph of a C 1 map and ∀ (q, p) ∈ V ;

∂ 2 (H ◦ φ−1 ) (q, p) ∂p2

is positive definite.

Remark. — It is proved in [21], Extension Theorem in Lecture 5, as well as the proof of Theorem in Lecture 6 (see also Theorem 3.33 of [18]), that a small neighborhood

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of a Lagrangian C k , k ≥ 1, submanifold T is always C k symplectomorphic to a neighborhood of the zero section in T ∗ T . So the two important things in the definition are that — we can choose C 2 coordinates even if T is just C 1 : it is possible just by perturbing a C 1 symplectic diﬀeomorphism into a C 2 one; — in the new coordinates, H has to be strictly convex in the fiber direction. Corollary 1.2. — Let H : M → R be a C 2 Hamiltonian that has an invariant Lagrangian torus T . Suppose H has positive torsion along T and is locally C 1 completely integrable at T . Then there exist a neighborhood U of T in M , an open set U containing 0 in Rn and a symplectic homeomorphism φ : Tn × U → U that is C 1 in the direction of Tn and such that: n n — ∀ c ∈ U , φ−1 ◦ ϕH t ◦ φ(T × {c}) = T × {c}; −1 H = Rtρ(c) ; — ∀ c ∈ U , φ ◦ ϕt ◦ φ| n T ×{c}

where ρ : U → Rn is a homeomorphism onto ρ(U ), and Rtρ(c) : Tn → Tn is given by Rtρ(c) (x) := x + tρ(c) mod Zn . Finally, in both the Tonelli case and the positive torsion case, we have the following approximation results. Corollary 1.3. — Any Tonelli Hamiltonian on T ∗ Tn that is G-C 1 completely integrable lies in the C 1 closure of the set of smooth completely integrable (in the usual Arnol ′d-Liouville sense) Hamiltonians. More precisely, if H : T ∗ Tn → R is Tonelli and G-C 1 completely integrable on some set {H < K} or the whole T ∗ Tn , then for any compact subset K of {H < K} or of the whole T ∗ Tn , there exist a neighborhood U of K and a sequence (Hi ) of C ∞ Hamiltonians Hi : U → R that are G-C ∞ completely integrable on some set containing K such that — the sequence Hi uniformly converges on U to H for the C 1 topology; — the invariant foliation for the Hi uniformly converges to the invariant foliation for H for the C 1 topology; H i — the Hamiltonian ﬂows (ϕH t ), t ∈ [−1, 1] uniformly converge to (ϕt ) on U for 0 the C topology. Remarks. — 1. By the continuous dependence on parameter of solutions of ODEs, the last point is a consequence of the first one : if a sequence of Lipschitz vectorfields (Xi ) converges in C 0 topology to a Lipschitz vectorfield X, then the sequence of ﬂows of the Xi converges in C 0 topology to the ﬂow of X. 2. We do not know if we can choose the Hi being convex in the fiber direction, we are just able to prove that they are symplectically smoothly conjugate to such Hamiltonians. Corollary 1.4. — Let H : M → R be a C 2 Hamiltonian that has an invariant Lagrangian torus T . Suppose H has positive torsion along T and is locally C 1 completely integrable at T . Then there exist a compact neighborhood K of T in M , an

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open set U containing K and a sequence (Hi ) of C ∞ Hi : U → R of Hamiltonians that are C ∞ completely integrable on a set containing K such that — the sequence Hi uniformly converges on U to H for the C 1 topology; — the invariant foliation for the Hi uniformly converges on K to the invariant foliation for H for the C 1 topology; H i — the Hamiltonian ﬂows (ϕH t ), t ∈ [−1, 1] uniformly converge to (ϕt ) on U for 0 the C topology. Structure of the proofs and comments — In Section 2, we will use Herman’s results concerning the conjugacy of torus homeomorphisms to rotations (see [13]) and even extend some of them to describe the Dynamics on the tori that carry a minimal Dynamics (i.e., all the orbits are dense) in the case of Lipschitz or C 1 complete integrability; — in Section 3, we will show that the A-non degeneracy condition implies the density of the union of the minimal tori and that this condition is satisfied by Tonelli Hamiltonians and Hamiltonians with positive torsion. It would be nice to have non-trivial other examples of Hamiltonians that satisfy this condition; — in Section 4, we will prove that C 1 complete integrability and A-non degeneracy imply the existence of Arnol′ d-Liouville coordinates; this proves Theorem 1.1, Corollary 1.1 and 1.2; this is done by using generating functions and HamiltonJacobi equations; in Section 4.4, we will prove Corollary 1.3 and 1.4. Finally we include three appendices exploring possible relaxations of the assumption of the C 1 integrability in the main body of the paper. — Appendix A is devoted to the study of diﬀerent possible definitions of C 0 complete integrability. — Appendix B recalls some known results concerning C 0 completely integrable Tonelli Hamiltonians. — Appendix C contains the proof of Theorem 1.2.

2. Lipschitz complete integrability determines the Dynamics on each minimal torus The goal of this section is to prove the following Theorem 2.1 that tells us that if H is G-Lipschitz integrable: — we can define on every invariant Lagrangian torus of the foliation a rotation vector; — on every such torus with a rotation vector that is completely irrational, the Dynamics is bi-Lipschitz conjugate to a minimal ﬂow of rotations with a Lipschitz constant that is uniform. Theorem 2.1. — Let V ⊂ T ∗ Tn be an open set. Assume that H : V → R is G-Lipschitz completely integrable. We denote a Lipschitz constant of the foliation by K,

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9

i.e., 1 ka − a′ k ≤ kηa (q) − ηa′ (q)k ≤ Kka′ − ak. K where we denote by Ta = {(H1 , . . . , Hn ) = (a1 , . . . , an )} = {(q, ηa (q)); q ∈ Tn } the tori of the invariant foliation.Then the ﬂow (ϕH t ) restricted-projected to every Lagrangian torus Ta , which is denoted by (fta ) and is defined on Tn by ϕH t (q, ηa (q)) = (fta (q), ηa (fta (q))) satisfies ∀ q ∈ Tn , ∀ a, a′ ,

F a (x)−x

1. if (Fta ) is the lift of (fta ) to Rn , then t t uniformly converges with respect to (a, x) to a constant ρ(a) ∈ Rn as t → ∞. Therefore the rotation vector ρ(a) ∈ Rn is well-defined and continuously depends on a; 2. if Rρ(a) is minimal, there exists a homeomorphism ha : Tn → Tn such that ha ◦ fta = Rtρ(a) ◦ ha ; 3. ha is K n -bi-Lipschitz. Remark. — Observe that we do not require in this section that the Hamiltonian is Tonelli or A non-degenerate: the results are valid even if the Hamiltonian is very degenerate. But observe that when H is constant (the very degenerate case), then there is no torus where the Dynamics is minimal and so in this case Theorem 2.1 is almost empty. This theorem will be useful when we are sure that such tori exist. Using the following proposition, we could easily deduce an analogue of Theorem 2.1 in a local C 1 -integrable setting. Proposition 2.1. — Let H : M → R be a C 2 Hamiltonian that has an invariant Lagrangian torus T . Then, if H is locally C 1 integrable at T , there exist an invariant open neighborhood U of T in M , an open subset U of Rn and a symplectic C 2 diffeomorphism ψ : Tn × U → U such that H ◦ ψ is G-C 1 integrable. In the remaining part of this section, we will give and prove three propositions and a corollary that will imply Theorem 2.1. Then we will prove Proposition 2.1. Notations. — For a vector v ∈ Rn , we use kvk to denote its Euclidean norm. For a matrix M ∈ Rn×n , we define its norm by kM k = sup kM vk, kvk=1

and its conorm by m(M ) := inf kvk=1 kM vk. For a vector α ∈ Rn , we denote by Rα the rigid rotation by α on Tn , i.e., Rα : T n → T n ,

Rα (x) = x + α, mod Zn .

For a point (q, p) ∈ Tn × Rn , we introduce two projections π1 (q, p) = q ∈ Tn and π2 (q, p) = p ∈ Rn .

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2.1. Uniform rate for the flow on the invariant tori. — In this section, we first show that the G-Lipschitz completely integrability condition implies a uniform estimate of the derivative Dfta , which is the key ingredient for us to apply Herman theory to obtain a conjugacy. Proposition 2.2. — Assume that H is G-Lipschitz completely integrable on some open subset U ⊂ T ∗ Tn . Then there exists a constant K > 0 such that, restricted on each Lagrangian torus Ta = {(H1 , . . . , Hn ) = (a1 , . . . , an )}, the ﬂow (fta ) satisfies the following Lipschitz estimate at Lebesgue almost every point q ∈ Tn (1)

∀ v ∈ Rn , ∀ t ∈ R,

kvk ≤ kDfta (q)vk ≤ Kkvk. K

Proof. — We denote by ηa : Tn → Rn the map such that the graph of ηa is the invariant submanifold Ta . Then N : (q, a) 7→ (q, ηa (q)) is a bi-Lipschitz-homeomorphism, because (ηa ) defines a Lipschitz foliation. Because of Rademacher Theorem, the set D(N ) where N is diﬀerentiable has full Lebesgue measure. Moreover, N (D(N )) is invariant by (ϕH t ) because the foliation is invariant. Using the notation of Theorem 2.1 we have a a ϕH t (q, ηa (q)) = (ft (q), ηa (ft (q))) .

Diﬀerentiating this equation, we obtain for every (q, a) ∈ D(N ): ! ! ! ∂fta 0 0 H ∂a (q) a + ∂ηa a Dϕt (q, ηa (q)) ∂ηa (2) = ∂ηa a , ∂ft ∂q (ft (q)) ∂a (q) ∂a (q) ∂a (ft (q)) ! ! ∂fta 1 H ∂q (q) Dϕt (q, ηa (q)) ∂ηa = ∂ηa a (3) . ∂fta ∂q (q) ∂q (ft (q)) ∂q (q) Then we use along every orbit in N (D(N )) a symplectic change of bases with matrix in the usual coordinates at (q, ηa (q)): ! 1 0 P(q,a) = ∂ηa ∂q (q) 1 We deduce from (2) and (3) that the matrix of DϕH t (q, ηa (q)) in the new coordinates is ! ∂fta −1 H ∂q (q) bt (q, a) (4) P(f a (q),a) Dϕt (q, ηa (q))P(q,a) = , t 0 dt (q, a) ∂ηa a a where dt (q, a) ∂η ∂a (q) = ∂a (ft (q)). Because the foliation is biLipschitz, there exists a constant K such that, for every (q, a) ∈ D(N ), we have

−1

∂η

√

∂ηa √

a

K and (q) ≤ (q)

≤ K.

∂a

∂a

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As dt (q, a) =

∂ηa a ∂a (ft (q))

−1

∂ηa ∂a (q)

11

, we deduce that

kdt (q, a)k ≤ K

and kdt (q, a)−1 k ≤ K.

The matrix in (4) being symplectic, we deduce that

a

∂ft

−t

≤K

∂q (q) = dt (q, a)

and

∂f a −1

t (q)

= dt (q, a)t ≤ K.

∂q

The proof is not completely finished because the result that we obtain is valid only for (a, q) ∈ D(N ). As D(N ) has total Lebesgue measure, by Fubini theorem, the set of a for which N is diﬀerentiable Lebesgue almost everywhere in {a} × Tn has full Lebesgue measure in Rn . Hence, we obtain local Lipschitz estimates along a dense set of graphs. In other words, there exists a dense set B of parameters a such that ∀ t ∈ R, ∀ a ∈ B, ∀ q, q ′ ∈ Tn ,

1 d(q, q ′ ) ≤ d(fta (q), fta (q ′ )) ≤ Kd(q, q ′ ). K

Approximating any parameter by a sequence of parameters in B and taking the limit, we obtain the same estimates for any parameter and then the wanted result (1) by diﬀerentiation. Corollary 2.1. — Assume that H is G-C 1 completely integrable on some open subset U ⊂ T ∗ Tn . Then for every compact subset K ⊂ U , there exists a constant K > 0 such that, restricted on each Lagrangian torus Ta = {(H1 , . . . , Hn ) = (a1 , . . . , an )} with Ta ∩ K 6= ∅, the ﬂow fta satisfies the following Lipschitz estimate (1)

∀ v ∈ Rn , ∀ t ∈ R, ∀q ∈ Ta

kvk ≤ kDfta (q)vk ≤ Kkvk. K

Proof. — We explain how to deduce Corollary 2.1 from Proposition 2.2. We assume that H is G-C 1 -integrable on some open subset U ⊂ T ∗ Tn and that K ⊂ U is compact and connected. Observe that K0 = {a; Ta ∩ K 6= ∅} is compact. Hence the map (a, q) ∈ K0 × Tn 7→ (a, ηa (q)) restricted to K0 , which is C 1 , is bi-Lipschitz when restricted to the compact K0 × Tn and then Corollary 2.1 can be deduced from Proposition 2.2. 2.2. Uniform rate and completely irrational rotation vector imply uniform Lipschitz conjugacy to a flow of rotations Next, we restrict our attention to one Lagrangian torus Ta and the associated restricted-projected ﬂow fta . Let (Fta ) be the ﬂow that is a lift of (fta ) to Rn . Proposition 2.3. — Assume Equation (1) for the ﬂow fta . Then

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F a (q)−q

(A) The family t t uniformly converges with respect to (a, q) to a constant ρ(a) ∈ Rn when t → +∞. Therefore the rotation vector ρ(a) ∈ Rn is well-defined and continuously depends on a; (B) if Rρ(a) : Tn → Tn is minimal, there exists a homeomorphism ha : Tn → Tn such that ha ◦ fta = Rtρ(a) ◦ ha . (C) The conjugacy ha is K-Lipschitz. Proof. — We first prove part (A). The proof is similar to Proposition 3.1, ch 13 in [13] with small modifications adapted to the ﬂow instead of the map. We consider the following family of functions {Fta (q) − q − Fta (0) | t ∈ R}. It is known that this family is equicontinuous by Inequality (1). Next, for each t, the function Fta (q) − q − Fta (0) is zero at q = 0, and is Zn -periodic. Again by (1), using R1 Fta (q) − Fta (0) = 0 DFta (sq)q ds, we get that sup max kFta (q) − q − Fta (0)k ≤ K + 1.

(5)

q

t

We next pick any invariant Borel probability measure µ of f1a and pick a µ-generic point q ∗ for which the Birkhoﬀ ergodic theorem holds, we get PN −1 a Z (Fi+1 (q ∗ ) − Fia (q ∗ )) FNa (q ∗ ) − q ∗ (F1a (q) − q) dµ(q). = i=0 → N N n T

Next, for each t ∈ R, we have Fta (q ∗ ) − q ∗ = t

a F[t] (q ∗ ) − q ∗

[t]

+

a a (Ft−[t] − IdRn ) ◦ F[t] (q ∗ )

[t]

!

[t] . t

This shows that Fta (q ∗ ) − q ∗ → t

Z

Tn

(F1a (q) − q) dµ(q) as t → ∞,

a (q) − q) is uniformly bounded for all q, t. By (5), we get that the conversince (Ft−[t] gence Z Fta (q) − q (F1a (q) − q) dµ(q) → ρ(a) = t Tn is uniform on Tn . We now apply Equation (5) and Proposition XIII 1.6. page176 of [13] that we recall.

Proposition 2.4 (Proposition 1.6, [13, p. 176, Ch. 13]). — Suppose f ∈ Homeo(Tn ) satisfies sup kf k − id − f k (0)kC 0 = a < ∞. k∈N

Then: 1. if k → +∞,

f k −id k

converges uniformly to an element ρ(f ) ∈ Rn ;

2. supk kf k − id − kρ(f )kC 0 ≤ 2a.

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13

We deduce sup max kFta (q) − q − tρ(a))k ≤ 2(K + 1).

(6)

q

t

This implies that the convergence to ρ(a) is uniform in (q, a). We next work on part (B). We assume that the rotation vector ρ(a) is completely irrational. Let us recall Proposition 3.1, page 181, in ch 13 of [13]. Proposition 2.5. — [Proposition 3.1, page 181, in ch 13 of [13]] Let f ∈ Homeo(Tn ) be k such that {f k | k ∈ Z} is equi-continuous, then as k → +∞, f −Id converges unik formly to ρ(f ) = α ∈ Rn . If the translation Rα on Tn is ergodic, then f is C 0 conjugate to Rα . We apply this proposition to the map f1a : Tn → Tn to get that there is a homeomorphism ha : Tn → Tn such that ha ◦ f1a (q) = ha (q) + ρ(a). Iterating this formula, we get ha ◦ fna (q) = ha (q) + nρ(a) for all n ∈ Z. It remains to show that ha ◦ fta (q) = ha (q) + tρ(a) for all t ∈ R. Equation (6) tells us that kFta (q) − q − tρ(a)k ≤ 2(K + 1) uniformly for all q and t. In the expression Z t 1 a (Fs (q) − sρ(a)) ds , t 0 we have just established the uniform boundedness for all t ∈ R \ {0} and q in a fundamental domain, and the equicontinuity in the q-variable follows from (1). By Arzela-Ascoli, we can extract uniformly convergent subsequence as tk → ∞. Denote by g(q) the limit Z tk 1 g(q) = lim (Fsa (q) − sρ(a)) ds . k tk 0 Next, we consider

g(Fta (q))

1 = lim k tk

tk

(Fsa (Fta (q))

0

Z

tk

− sρ(a)) ds

a (Fs+t (q) − (s + t)ρ(a)) ds + tρ(a) 0 Z tk 1 (Fsa (q) − sρ(a)) ds = lim k tk 0 Z tk +t Z t 1 a + lim (Fs (q) − sρ(a)) ds + tρ(a) − k tk 0 tk = g(q) + tρ(a). 1 = lim k tk

(7)

Z

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Choosing t = 1, we get g(F1a (q)) = g(q) + ρ(a). Then taking diﬀerence with the equation ha (F1a (q)) = ha (q)+ρ(a), we get g(F1a (q))−ha (F1a (q)) = g(q)−ha (q). By assumption we have that Rρ(a) is minimal, so is f1a , therefore we get that g(q) − ha (q) = const. This shows that g is a homeomorphism as ha is. Moreover, we apply the following result that is contained in the proof of Proposition 3.1, page 181, in ch 13 of [13]: ha can be chosen as a limit point of the sequence Pk−1 F a (q)−jρ(a) q 7→ q + j=0 j k . Hence ha is K-Lipschitz because all the Fja are K-Lipschitz. 2.3. Lipschitz conjugacy to a completely irrational rotation implies bi-Lipschitz conjugacy The main goal of this section is to prove: Proposition 2.6. — Let g : Tn → Tn be a bi-Lipschitz homeomorphism such that the family (g k )k∈Z is K-equi-Lipschitz. If the rotation vector α of g is such that Rα is ergodic, then g is bi-Lipschitz-conjugated to Rα by some conjugacy h in Homeo(Tn ). Moreover, if h ◦ g = Rα ◦ h, then the conjugacy h is K-Lipschitz and its inverse h−1 is K n -Lipschitz. Remarks. — Observe that in the proof of Proposition 3.2, page 182, in ch 13 of [13], M. Herman raised the question of the Lipschitzian property of such h−1 and that we give here a positive answer. — Observe too that with the notations of Section 2.2, Proposition 2.6 implies that n every h−1 a is K -Lipschitz. Proof. — We know from Proposition 2.5 that if the rotation vector α of g is such that Rα is minimal, then there exists a homotopic to identity homeomorphism h of Tn such that (8)

h ◦ g = Rα ◦ h,

Let G : Rn → Rn be a lift of g. We know from the proof of Proposition 3.1, page 181, in ch 13 of [13] that h can be chosen as a limit point of the sequence Pk−1 k q 7→ q + j=0 G (q)−jα . Hence h is K-Lipschitz because all the Gj are K-Lipschitz. k −1 Let us prove that h is K n -Lipschitz. Lemma 2.1. — Let g : Tn → Tn be a bi-Lipschitz homeomorphism so that supk∈Z kDg k kC 0 = K < +∞. Then g has an invariant Borel probability measure µ so that for all Borel subset A ⊂ Tn , we have 1 Leb(A) ≤ µ(A) ≤ K n Leb(A). Kn In particular, µ is equivalent to Leb.

ASTÉRISQUE 416

15

INTEGRABLE HAMILTONIANS

Proof. — We apply Krylov-Bogolyubov process (see [15]) to the Lebesgue measure. In other words, for every k ≥ 1, we define µk =

k−1 1X k g Leb. k j=0 ∗

Observe that for every open set A, we have Z Z 1 n |detDg k |dLeb ∈ d(g∗k Leb) = g∗k Leb(A) = Leb(A), K Leb(A) . Kn A A We deduce that µk (A) ∈ K1n Leb(A), K n Leb(A) . Let now µ be a limit point of the sequence (µk ). Then µ is invariant by g and satisfies 1 Leb(A) ≤ µ(A) ≤ K n Leb(A). Kn By Rademacher theorem, g is Lebesgue almost everywhere diﬀerentiable. Because g is bi-Lipschitz, we deduce that the set Z of the points q ∈ Tn such that g is diﬀerentiable at every g k (q) for k ∈ Z has full Lebesgue measure. We will use the following lemma that is easy to prove by using the definition of the diﬀerential and the Lipschitz property. Lemma 2.2. — Let G : Tn → Tn a bi-Lipschitz homeomorphism. Assume that G is diﬀerentiable at some q0 ∈ Tn . Then G−1 is diﬀerentiable at G(q0 ) and −1

DG−1 (G(q0 )) = (DG(q0 ))

.

By Rademacher theorem, h is Lebesgue almost everywhere diﬀerentiable and we have kDhk ≤ K. As we have h = Rα ◦ h ◦ g −1 and Lemma 2.2, the set E of the points of Z where h is diﬀerentiable is invariant under g. Moreover, we know that Leb(E) = 1, i.e., Leb(Tn \E) = 0. By Lemma 2.1, this implies that µ(Tn \E) = 0 and then µ(E) = 1. Moreover, we have on E: (Dh) ◦ g = Dh · (Dg)−1 . Let us now prove that for µ-almost every x ∈ E and every unit vector v ∈ Tx Tn , we have kDh(x)vk ≥ K1n kvk. If not, there exist ε > 0 and a Borel measurable subset E ′ ⊂ E with non-zero µ-measure such that for every x ∈ E ′ , there is a unit vector 1 v ∈ Tx Tn such that kDh(x)vk ≤ 1−ε K n kvk < K n kvk. Because of Equation (8), we have for every k ∈ N Dh(g k x) · Dg k (x) = Dh(x). Hence if we denote by wk ∈ Tgk x Tn the unit vector wk = deduce from k(Dg k )−1 k ≤ K that kDh(g k x)wk k =

1 Dg k (x)v, kDg k (x)vk

then we

1 kwk k kwk k kDh(x)vk ≤ (1 − ε) n ≤ (1 − ε) n−1 . kDg k (x)vk K kDg k (x)vk K

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If now we compute |det(Dh(g k x))| in some orthonormal basis with the first vector equal to wk , we obtain because kDhk ≤ K that |det(Dh(g k x))| ≤

1 − ε n−1 .K = 1 − ε. K n−1

Hence we have proved that |det(Dh)| ≤ 1 − ε on the set [ E = g k (E ′ ). k∈N

As µ is ergodic and µ(E ′ ) 6= 0, we have µ(E ) = 1 and then Leb(E ) = 1. Integrating and using a change of variables, we finally obtain Z Z |detDh(x)| dLeb(x) ≤ 1 − ε. dLeb = 1= Tn

Tn

Hence we have proved that for Lebesgue almost every x ∈ Tn , we have kDh−1 (x)k ≤ K n . This implies in particular that h−1 is K n -Lipschitz. 2.4. Proofs of Theorem 2.1 and Proposition 2.1 Proof of Theorem 2.1. — By Proposition 2.1, the G-Lipschitz completely integrability condition implies a uniform estimate (1) of the derivative Dfta . Next, by Proposition 2.3, the estimate (1) implies the existence of rotation vector and in the minimal case the existence of a Lipschitz conjugacy. This proves the first two items of the theorem. Finally, by Proposition 2.6, the Lipschitz conjugacy is bi-Lipschitz. This completes the proof of the theorem. Proof of Proposition 2.1. — We assume that H is locally C 1 completely integrable at T . Hence there exists an invariant open neighborhood U of T and n C 1 functions H1 , H2 , . . . , Hn : U → R so that — — — —

the Hi are constant on T ; at every x ∈ U , the family dH1 (x), . . . , dHn (x) is independent; for every i, j, we have {Hi , Hj } = 0 and {Hi , H} = 0; T = {H1 = a1 , . . . , Hn = an }.

Then if we take a smaller U , H = (H1 , . . . , Hn ) : U → Rn is a C 1 submersion that gives an invariant by (ϕH t ) Lagrangian foliation and (see [21], Extension Theorem in Lecture 5, as well as the proof of Theorem in Lecture 6) there exists a C 1 symplectic embedding φ : U → Tn × Rn that maps T onto Tn × {0} and then the foliation onto a foliation into C 1 graphs. Perturbing φ, we can assume that φ is smooth and symplectic and that the foliation is into graphs (but of course φ(T ), which is a graph, cannot be the zero section). The diﬀeomorphism ψ of Proposition 2.1 is then given by ψ = φ−1 .

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INTEGRABLE HAMILTONIANS

17

3. The A-non-degeneracy condition In Section 2, we succeeded in describing the Dynamics on the invariant tori having a completely irrational rotation vector. If we want to extend this to all the tori, we need to know that the union of these tori is dense in the set where the Hamiltonian is completely integrable. That is why we introduced our notion of A-non-degeneracy (see Definition 1.4). Let us recall the notations that we gave in the introduction. Notations. — Let H : U ⊂ T ∗ Tn → R be a C 1 integrable Hamiltonian. — We will denote by (Ta )a∈U = ({(q, ηa (q)) :

q ∈ Tn })

R the invariant foliation and define the function c : U → Rn by c(a) = Tn ηa (q)dq. — The A-function AH : c(U ) → R is defined by AH (c) = H(0, ηc−1 (c) (0)). Remark. — Observe that we have ∀ q ∈ Tn , ∀ c ∈ c(U ),

H(q, ηc−1 (c) (q)) = AH (c).

Moreover, the A function is C 1 . The main implication of the A-nondegeneracy condition is the following result. Proposition 3.1. — Assume that H is G-C 1 completely integrable and A-nondegenerate and denote by (Ta )a∈U = ({(q, ηa (q)), q ∈ Tn }) the invariant foliation. Then for every non-empty open subset V ⊂ c(U ), the set ∇AH (V ) contains a completely irrational α = (α1 , . . . , αn ) i.e., α that doesn’t belong to any resonant hyperplane. Proof. — Assume that V ⊂ c(U ) is a non-empty open subset of c(U ) such that ∇AH (V ) contains no completely irrational number. Then V is contained in the countable union of the backward images by the continuous map ∇AH of all the closed resonant hyperplanes. As V is Baire, there exists a non-empty open subset W of V such that ∇AH (W ) is contained in some resonant hyperplane. This contradicts the definition of A-non-degeneracy. The next result shows that the A-non-degeneracy condition is implied by either the Tonelli condition or the positive torsion condition (see Definition 1.1 and 1.8). Hence we deduce Corollary 1.1 and 1.2 from Theorem 1.1. Proposition 3.2. — Assume that H is G-C 1 completely integrable and strictly convex in the momentum variable, i.e., ∀ (q, p) 6= (q, p′ ) ∈ U , ∀ λ ∈ (0, 1),

(q, λp + (1 − λ)p′ ) ∈ U

⇒ H(q, λp + (1 − λ)p′ ) < λH(q, p) + (1 − λ)H(q, p′ ). Then H is A-non-degenerate and even ∇AH is a homeomorphism onto ∇AH (c(U )).

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Proof. — We recall that if F : V → R is a C 1 function that is strictly convex, then DF is an homeomorphism from V onto its image (see Theorem 1.4.5. in [9]). Hence we just have to prove that AH is strictly convex. Let us fix two distinct c1 , c1 in c(U ) and λ ∈ (0, 1). Because H is strictly convex in the fiber direction, we have for every q ∈ Tn , H(q, ληc−1 (c1 ) (q) + (1 − λ)ηc−1 (c2 ) (q)) < λH(q, ηc−1 (c1 ) (q)) + (1 − λ)H(q, ηc−1 (c2 ) (q)). As H(q, ηc−1 (c) (q)) = AH (c), we have (9)

H(q, ληc−1 (c1 ) (q) + (1 − λ)ηc−1 (c2 ) (q)) < λAH (c1 ) + (1 − λ)AH (c2 ).

Observe that λc1 + (1 − λ)c2 is equal to Z Z (ληc−1 (c1 ) (q) + (1 − λ)ηc−1 (c2 ) (q))dq =

ηc−1 (λc1 +(1−λ)c2 ) (q)dq.

Tn

Tn

Hence the 1-form ληc−1 (c1 ) + (1 − λ)ηc−1 (c2 ) − ηc−1 (λc1 +(1−λ)c2 ) is exact and so there exists a q0 ∈ Tn so that ληc−1 (c1 ) (q0 ) + (1 − λ)ηc−1 (c2 ) (q0 ) = ηc−1 (λc1 +(1−λ)c2 ) (q0 ). Replacing in Equation (9), we obtain H(q0 , ηc−1 (λc1 +(1−λ)c2 ) (q0 )) < λAH (c1 ) + (1 − λ)AH (c2 ), i.e., AH (λc1 + (1 − λ)c2 ) < λAH (c1 ) + (1 − λ)AH (c2 ). 4. The symplectic homeomorphism in the case of C 1 complete integrability The goal of this section is to prove Theorem 1.1, which, joint with Proposition 3.2, implies Corollary 1.1 and 1.2. 4.1. A generating function for the Lagrangian foliation Proposition 4.1. — Assume that (q, a) ∈ Tn × U → (q, ηa (q)) ∈ U is a C 1 foliation of an open subset U of Tn × Rn into Lagrangian graphs. Then there exists a C 1 map S : U → R such that — ∂S ∂q (q, a) = ηa (q) − c(a); ∂S — ∂q is C 1 in the two variables (q, a); 1 — ∂S ∂a is C in the variable q. Proof. — As the graph of ηa is Lagrangian, there exist a C 2 function ua : Tn → R such that ∂ua . ηa = c(a) + ∂q Observe that ua is unique up to the addition of some constant. We then define: S(q, a) = ua (q) − ua (0).

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Then S has a derivative with respect to q that is ∂S ∂q (q, a) = ηa (q) − c(a) = 1 which is C in the two variables (q, a). Moreover, Z 1 Z 1 ∂S S(q, a) = (ηa (q) − c(a)) γ˙ q (t)dt, (γq (t), a)γ˙ q (t)dt = 0 ∂q 0

19

∂ua ∂q (q),

where γq : [0, 1] → Tn is any C 1 arc joining 0 to q. This implies that S is C 1 in the variable a and that Z 1 2 Z 1 ∂ S ∂S ∂ηa ∂c (10) (q, a) = (γq (t), a)γ˙ q (t)dt = (γq (t), a) − (a) γ˙ q (t)dt. ∂a ∂a ∂a 0 ∂a∂q 0

We deduce that S has partial derivatives with respect to a and q that continuously depend on (q, a). This implies that S is in fact C 1 . Moreover, we deduce from Equation (10) that for every ν ∈ R and δq ∈ Rn , we have Z ν ∂ηa ∂S ∂c ∂S (q + νδq, a) − (q, a) = (q + tδq)δqdt − ν (a)δq. ∂a ∂a ∂a ∂a 0 This implies that is:

∂S ∂a

has a partial derivative with respect to q in the direction δq that

∂ηa ∂c ∂2S (q, a) · δq = (q) − (a) δq. ∂q∂a ∂a ∂a 1 Because the foliation is C , then all these partial derivatives continuously depend 1 on q. This implies that ∂S ∂a is C in the q variable. 4.2. The C 1 property of the conjugacy. — Now we will prove Proposition 1.1 that we rewrite by using the generating function. Proposition. — 1.1 Assume that (q, a) ∈ Tn × U 7→ (q, c(a) + ∂S ∂q (q, a)) ∈ U is a C 1 foliation of an open subset U of Tn × Rn into Lagrangian graphs. The function c : U → Rn is a C 1 diﬀeomorphism from U onto its image. Proof. — Observe that c is injective. If a 6= a′ , then Ta ∩ Ta′ = ∅ because we have a foliation. Assume that c(a) = c(a′ ). Let q0 be a critical point of S(·, a) − S(·, a′ ). ∂S ′ Then we have ∂S ∂q (q0 , a) − ∂q (q0 , a ) = 0 and c(a) +

∂S ∂S (q0 , a) = c(a′ ) + (q0 , a′ ), ∂q ∂q

which contradicts that Ta ∩ Ta′ = ∅. Hence c is C 1 and injective. To prove the lemma, we just have to prove that Dc(a) is invertible at every point of U . If not, let us choose a ∈ U and a non-zero vector v ∈ Rn such that Dc(a)v = 0. Now, we choose q0 ∈ Tn such that q 7→ ∂S ∂a (q, a)v attains its maximum at q0 . Then we have ∂ηa ∂2S (q0 )v = Dc(a)v + (q0 , a)v = 0. ∂a ∂q∂a

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This contradicts the fact that (ηa ) defines a C 1 -foliation. Let us recall the notation that we introduced in the statement of Theorem 2.1: Notation. — The ﬂow (ϕH t ) restricted-projected to every Lagrangian torus Ta , which a a is denoted by (fta ), is defined on Tn by ϕH t (q, ηa (q)) = (ft (q), ηa (ft (q))). Theorem 4.1. — Suppose H : U ⊂ T ∗ Tn → R is a G-C 1 completely integrable Hamiltonian that is A-non-degenerate and let (q, a) ∈ Tn × U → (q, c(a) + ∂S ∂q (q, a)) ∈ U be the invariant C 1 foliation. Then ga (q) := q + Dc(a)−1

∂S (q, a) ∂a

is a C 1 diﬀeomorphism conjugating (fta ) to Rtρ(a) where ρ was defined in Proposition 2.3 and in fact ρ = (∇AH ) ◦ c. Hence, when ∇AH (c(a)) is completely irrational, we have ga = ha up to an additive constant where ha is the conjugacy associated to the torus Ta given by Theorem 2.1. Proof. — Write ηa (q) = c(a) + ∂S ∂q (q, a). We have ∂S (11) H q, c(a) + (q, a) = AH (c(a)), ∂q R where c(a) = Tn ηa (q)dq C 1 depends on a and AH is the A-function. Diﬀerentiating Equation (11) with respect to a, we deduce that if t 7→ (q(t), p(t)) = ϕH t (q, p) is contained in Ta , we have Dc(a) · q(t) ˙ +

∂2S (q(t), a)q(t) ˙ = Dc(a) · ∇AH (c(a)) ∂q∂a

and then Dc(a) (q(t) − q(0) − t∇AH (c(a))) +

∂S ∂S (q(t), a) − (q(0), a) = 0. ∂a ∂a

We deduce fta (q) + Dc(a)−1

∂S a ∂S (ft (q), a) = q + Dc(a)−1 (q, a) + t∇AH (c(a)). ∂a ∂a

If we define ga (q) = q + Dc(a)−1 ∂S ∂a (q, a), we have proved that ga (fta (q)) = ga (q) + t∇AH (c(a)). Hence ga is a semi-conjugacy between (fta ) and (Rt∇AH (c(a)) ). Lemma 4.1. — Let f : Tn → Tn be a homeomorphism. Assume that g, h ∈ C 0 (Tn , Tn ) are homotopic to identity and such that g ◦ f = g + α and h ◦ f = h + β for some α, β ∈ Tn . Then β = α.

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21

Proof. — From g ◦ f = g + α and h ◦ f = h + β, we deduce that (12)

(g − h) ◦ f = g − h + α − β.

Because g and h are homotopic to identity, there exists a continuous map G : Tn → Rn such that the class of G(q) modulo Zn is (g − h)(q). Then Equation (12) and the continuity of G implies that G ◦ f − G is a constant B ∈ Rn whose class modulo Zn is α − β. If µ is a Borel probability measure that is invariant by f , we then obtain by integrating Z Z G dµ =

G dµ + B.

Hence B = 0 and α = β mod Zn .

We denote by A the set of a ∈ U such that ρ(a) is completely irrational. We deduce from Lemma 4.1 that for a ∈ A , we have ρ(a) = ∇AH (c(a)) and then ga (fta (q)) = ga (q)+tρ(a). We deduce from ga (fta (q)) = ga (q)+tρ(a) and ha (fta (q)) = ha (q)+tρ(a) that (ga − ha ) ◦ fta = (ga − ha ). Hence ga − ha is constant Lebesgue almost everywhere and then by continuity constant. Hence ga itself is a conjugacy between (fta ) and (Rtρ(a) ). As H is G-C 1 completely integrable and A-non-degenerate, we deduce from Proposition 3.1 that A is dense in U . If now a ∈ U , let K be a compact neighborhood of Ta in U . We deduce from Theorem 2.1 that there exists a constant K such that for every a ∈ A , if Ta ∩ K 6= ∅, then ha is K-bi-Lipschitz. Then we choose a sequence (ai ) in A such that — every Tai meets K ; — (ai ) converges to a. Every gan is a K-bi-Lipschitz homeomorphism and the sequence (gan ) C 0 converges to ga . We deduce that ga is a K-bi-Lipschitz homeomorphism too and then a C 0 conjugacy between (fta ) and (Rt∇AH (c(a)) ). This implies that ∇AH (c(a)) = ρ(a). Moreover, all the ga s are C 1 . Observe that a C 1 homeomorphism that is bi-Lipschitz is a C 1 diﬀeomorphism. Hence all the ga s are C 1 diﬀeomorphisms. Proposition 4.2. — Each continuous vectorfield XHi generates a ﬂow. Proof. — Note that if we define for every a ∈ U and every i ∈ {1, . . . , n} γia by γia = Dga (q)

∂Hi (q, ηa (q)), ∂p

then the ﬂow defined on Tn × U by −1 −1 i ϕH t (q, ηa (q)) = (ga ◦ Rtγia ◦ ga (q), ηa (ga ◦ Rtγia ◦ ga (q)))

preserves the foliation in ηa , is continuous and C 1 along the Ta and such that i ∂ϕH t (x) = XHi (x). ∂t |t=0

Remark. — We don’t aﬃrm that the vector fields XHi are uniquely integrable.

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4.3. The symplectic homeomorphism and the proof of Theorem 1.1 Proof of Theorem 1.1. — With the notations used in the previous part, we define φ0 (q, a) = (ga−1 (q), ηa (ga−1 (q))). By Proposition 4.1 and Theorem 4.1, we get that φ0 satisfies all the conclusions of Theorem 1.1 except the fact that it is not symplectic. Let C : Tn × U → Tn × c(U ) be the C 1 diﬀeomorphism that is defined by C (q, a) = (q, c(a)). Then we define the homeomorphism φ : Tn × c(U ) → U by φ = φ0 ◦ C −1 . Then φ is a homeomorphism that is C 1 in the direction of Tn and such that: n n — ∀c ∈ c(U ), φ−1 ◦ ϕH t ◦ φ(T × {c}) = T × {c}; −1 H — ∀c ∈ c(U ), φ ◦ ϕt ◦ φ| n = Rtρ◦c−1 (c) . T ×{c}

To see that φ is a homeomorphism, we note that φ = f0 ◦ ψ0−1 is a composition of a C 1 diﬀeomorphism given by the foliation f0 : (q, c) 7→ (q, c +

∂S (q, c−1 (c))), ∂q

and the inverse function of the conjugacy preserving the standard foliation (by the Tn × {c}) ψ0 : (q, c) 7→ (gc−1 (c) (q), c). Note that ψ0 is C 1 in the q direction but not C 1 in the c direction. Observe that if S(q, a) is the function that we introduced in Proposition 4.1, the function (q, c) 7→ S (q, c) = S(q, c−1 (c)) is a generating function for φ in the sense that if φ(x, c) = (q, p), then we have ( p = c + ∂∂qS (q, c), x=q+

∂S ∂c

(q, c).

Now we prove that φ can be approximated in C 0 topology by smooth symplectomorphisms φǫ as ǫ → 0. We introduce Sǫ (q, c)

= S (q, c) ∗ ϕǫ (c) ∗ ϕǫ (q)

where ϕǫ is a C approximating Dirac-δ function as ǫ → 0 and ∗ is the convolution. We get that Sǫ (q, c) is C ∞ in both variables and is a well-defined function on T ∗ Tn (Zn periodic in q). Moreover, when ǫ tends to 0, (Sǫ ) tends to S in C 1 topology Sǫ uniformly on compact subsets and ( ∂∂q ) tends to ∂∂qS in C 1 topology uniformly on compact subsets. Sǫ (q, c)) is Hence, if V is relatively compact, for ǫ small enough, fǫ (q, p) = (q, c + ∂∂q 1 a smooth map that is C close to the diﬀeomorphism that gives the initial foliation (q, c) 7→ (q, c + ∂∂qS (q, c)) and thus is a smooth diﬀeomorphism when restricted to V . Observe too that the approximating foliations are smooth and C 1 converge to the initial one. ∞

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Sǫ (q, c), c). Let W be a compact subset of Tn ×Rn . We now define ψǫ : (q, c) 7→ (q+ ∂∂c ∂S 1 Let us recall that for every c, the map gc−1 (c) : q 7→ q + ∂c (q, c) is a C diﬀeomorphism 2

∂ Sǫ Sǫ (q, c) and q 7→ 1+ ∂q∂c (q, c) uniformly converge of Tn . Moreover, the maps q 7→ q + ∂∂c 2

∂ S (q, c) respectively. We deduce that the on W to q 7→ q + ∂∂cS (q, c) and q 7→ 1 + ∂q∂c restriction of ψǫ to W is also a smooth diﬀeomorphism for ǫ small enough. If W is relatively compact, we can now define φǫ by φǫ = fǫ ◦ ψǫ−1 on W for ǫ small enough. Then Sǫ (q, c) is a generating function of the smooth symplectic diﬀeomorphim φǫ (x, c) = (q, p) i.e., ( Sǫ p = c + ∂∂q (q, c), ∂ Sǫ ∂c (q, c). f0 ◦ ψ0−1 in

x=q+

Hence φǫ is symplectic and φǫ tends to φ = set when ε tends to 0.

C 0 topology on every compact

4.4. The smooth approximation. — The goal of this section is to prove Corollaries 1.3 and 1.4. We will use some notations from Section 4.3. The Hamilton-Jacobi equation H(q, c + ∂∂qS (q, c)) = A(c) can be rewritten as H(q, p) = A ◦ π2 ◦ f0−1 (q, p) where f0 : (q, c) 7→ (q, c + ∂∂qS (q, c)) is the C 1 map that defines the foliation. The corollaries then follow by approximating A by a sequence of C ∞ functions {Aǫ = A ∗ ϕǫ } of only c ∈ Rn and approximating f0 by the sequence of C ∞ diﬀeomorphisms {fǫ } constructed in the proof of Theorems 1.1 and 1.2. We denote Hǫ (q, p) = Aǫ ◦ π2 ◦ fǫ−1 (q, p). By the construction of Aǫ , we get that kAǫ − AkK ,C 1 → 0 on every compact subset K and that every Aǫ has positive definite Hessian. In the proof of Theorems 1.1 and 1.2, we proved that on every compact subset K , we have kfǫ − f0 kK ,C 1 → 0. This implies that kHǫ − HkK ,C 1 → 0. This proves the first and third bullet points in Corollaries 1.3 and 1.4. The second bullet point follows from kfǫ − f0 kK ,C 1 → 0. Appendix A C integrability and C 0 Lagrangian submanifolds 0

In this appendix, we study various notions of C 0 integrability and prove a result on the relations between two notions of C 0 integrability.

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A.1. Different notions of C 0 integrability. — There are diﬀerent notions of C 0 integrability existing in literature. We give a list here. The following definition of C 0 Lagrangian submanifold was first introduced in [12] by Herman. Definition A.1 (C 0 Lagrangian submanifold #1). — Suppose that a n-dimensional C 0 submanifold of T ∗ N is the graph of a one-form p(q)dq. We say that this manifold is C 0 Lagrangian in the sense #1 if the one-form p(q) dq is closed in the distribution sense. Remarks. — 1. It can be proved that p(q) dq is closed in the distribution sense if and only if its integral along every closed homotopic to a point loop is zero (see Proposition A.1). 2. Observe that it is proved in [9] (Theorem 4.11.5) that every C 0 -Lagrangian graph in the sense #1 that is invariant by a Tonelli ﬂow is in fact a Lipschitz Lagrangian graph. Hence it has a tangent space Lebesgue almost everywhere and this tangent space is Lagrangian. Proposition A.1. — Suppose N = Tn . If the graph T of p(q) dq is C 0 Lagrangian in the sense #1, then there exists a unique c ∈ Rn and a unique C 1 function u : Tn → R such that p = c + du. Then c is called the cohomology class of T . Proof. — We define the cohomology class for the torus T . Suppose T = {(q, p(q)), q ∈ Tn }. Then we pick cycles γj ⊂ Tn with homology class ej := (0, . . . , 0, 1, 0, . . . , 0) ∈ H1 (Tn , Z) where 1 is in the j-th entry. Then the cohomology class c ∈ H 1 (Tn , R) of T is a vector in Rn determined by the equation Z (13) hc, ej i = p(q) dq. γj

To see that the cohomology class is well-defined, we pick another cycle ηj homologous to γj , so the diﬀerence γj − ηj of two cycles can be realized as the boundary of a piece of surface Sj . We next smoothen p(q) to pǫ (q) by convoluting each of the components of p(q) = (p1 , . . . , pn ) by the same C ∞ approximating Dirac-δ function. The components of pǫ (q) satisfy ∂qj pi,ǫ = ∂qi pj,ǫ since pdq is closed in the distribution sense. This implies that pǫ dq is a smooth closed one-form. So we get Z Z Z p(q) dq = lim pǫ (q) dq = lim d(pǫ (q) dq) = 0. γj −ηj

ǫ→0

∂Sj

ǫ→0

Sj

Next, we get that (p(q) − c) dq is exact. To see this, we pick any C ∞ curve R1 γq : [0, 1] → Tn with γq (0) = 0 and γq (1) = q and define uc (q) = 0 (p(γq ) − c) · γ˙ q dt. This function uc is well-defined and C 1 by the closedness of pdq and the definition of the cohomology class c.

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25

The following definition of C 0 Lagrangian submanifold was introduced in [14] motivated by the C 0 closedness of the symplectomorphism group of Gromov-Eliashberg. Definition A.2 (C 0 Lagrangian submanifold #2). — We say a C 0 submanifold of a symplectic manifold is C 0 Lagrangian in the sense #2, if it is symplectically homeomorphic to a smooth Lagrangian submanifold via a symplectic homeomorphism (see Definition 1.5). It is proved in Proposition 26 of [14] that a C 0 Lagrangian graph in the sense #1 is necessary a C 0 Lagrangian submanifold in the sense #2. However, the other direction of implication for a submanifold that is a graph is not clear. Using the above two notions of C 0 Lagrangian submanifold, we get two notions of C 0 integrability. Definition A.3 (C 0 integrability #1). — We say that a Tonelli Hamiltonian H : T ∗ Tn → R is C 0 integrable in the sense #1 if there exists a continuous foliation of T ∗ Tn by invariant and C 0 Lagrangian graphs in the sense #1. This definition of C 0 integrability was given by Arcostanzo-Arnaud-BolleZavidovique in [1]. Similarly, we have the following. Definition A.4 (C 0 integrability #2). — We say a Tonelli Hamiltonian H : T ∗ Tn → R is C 0 integrable in the sense #2 if there exists a continuous foliation of T ∗ Tn by invariant and C 0 Lagrangian graphs in the sense #2. Our next notion of C 0 integrability is based on the following remarkable C 0 rigidity result of Poisson bracket discovered by Cardin-Viterbo in [6]: if M is a symplectic manifold and Fn , Gn , F, G are in C ∞ (M ), if (Fn , Gn ) C 0 converges to (F, G) and lim k{Fn , Gn }kC 0 = 0,

n→∞

then {F, G} = 0. With this result, it is natural to define the Poisson commutativity for C 0 Hamiltonians (this definition with a diﬀerent name was introduced in in [6], see also Chapter 2.1 of [20]). Definition A.5 (C 0 Poisson commutativity). — For two C 0 functions F and G on a symplectic manifold M , we say that F and G Poisson commute, i.e., {F, G} = 0, if there exist two sequences {Fn }, {Gn } with Fn , Gn ∈ C ∞ (M ) satC0

isfying (Fn , Gn ) −−→ (F, G) and lim k{Fn , Gn }kC 0 = 0.

n→∞

Next, we define the third version of C 0 -integrability, which can be considered as a C version of Arnol′ d-Liouville integrability. 0

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Definition A.6 (C 0 -integrability #3). — Suppose that U ⊂ T ∗ Tn is open. We say that a C 0 Hamiltonian H : T ∗ Tn → R is C 0 integrable in U in the sense #3, if there is a sequence of C 2 n-uples Hi (q, p) := (H1,i (q, p), . . . , Hn,i (q, p)) : U → Rn , satisfying (1) (Poisson commutativity) Hi → H := (H1 , . . . , Hn ), in the C 0 norm as i → ∞ and k{Hj,i , Hk,i }kC 0 → 0 for all j, k, as i → ∞. We assume that H1 = H. (2) (Non-degeneracy 1)The matrix Mi formed by ∂p Hj,i , j = 1, . . . , n as rows is invertible and the inverse Mi−1 is uniformly bounded on U and in i; (3) (Non-degeneracy 2) the limiting tuple H(q, .) := (H1 , . . . , Hn ) : U → Rn , for each fixed q ∈ Tn as a function of p is an homeomorphism onto its image and this image U does not depend on q. Remark. — In item (1) of Definition A.6, the Poisson commutativity of the Hamiltonians Hj , j = 1, . . . , n for n > 2 is narrower than applying Definition A.5 directly to each pair. Indeed, by Definition A.5, the sequence of {Hi,n } defining {Hi , Hj } = 0 might be diﬀerent from that used to defining {Hi , Hk } = 0 for j 6= k. We stick to the narrower definition here for simplicity. It is not clear to us how to recover Theorem A.1 below using the broader definition. Since we do not need to talk about ﬂow invariant objects in Definition A.6, different from Definition A.3 and A.4, we assume the Hamiltonian to be C 0 only, instead of C 2 as in Definition A.3 and A.4. We need the non-degeneracy 1 hypothesis to prove Theorem A.1. This hypothesis tells us that the map H(q, .)−1 is uniformly Lipschitz with respect to the variable p. It can be slightly weakened to allow kMi−1 k2C 0 to blow up to infinity slowly as i → ∞ provided the following limit holds (14)

max k{Hj,i , Hk,i }kC 0 kMi−1 k2C 0 → 0, as i → ∞. j,k

Observe that the non-degeneracy 1 hypothesis implies that the function H defined in point (3) of Definition A.6 is a local bi-Lipschitz homeomorphism for each fixed q, which gives a part of the non-degeneracy 2 hypothesis. In our C 1 Arnol′ d-Liouville theorem, we rely crucially on the C 1 or Lipschitz assumption to conjugate the Dynamics on each Lagrangian torus Ta

:= {(q, p) ∈ U | (H1 , . . . , Hn ) = a ∈ Rn }

to a translation on Tn by ρ(a), from both the side of Herman theory and the side of generating functions. When we drop the C 1 or Lipschitz assumption, we lose control of the Dynamics and we do not know how to get a conjugacy. However, we still have the following theorem on the topological structure of the phase space.

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27

Theorem A.1. — Suppose H is a C 0 -integrable Hamiltonian defined on U in the sense #3, then the level sets Ta := {(q, p) ∈ U | (H1 , . . . , Hn ) = a} for a ∈ U define a C 0 foliation of U by Lagrangian graphs in the sense #1 (i.e., in the sense of distribution). Proof. — Hypothesis of non-degeneracy 2 implies that every level set Ta for a ∈ U is a continuous graph and that the foliation (Ta ) is continuous. Hypothesis of non-degeneracy 1 and the usual implicit function theorem imply that if a ∈ U , then for i large enough the set Ti,a

:= {(q, p) ∈ U | (H1,i , . . . , Hn,i ) = a}

is the graph of some C 2 function pi : Tn → U . The C 0 convergence of Hi to H implies that pi C 0 -converges to p if Ta is the graph of p. ∂H ∂H We now think each vector ∂qj,i and ∂pj,i as a row vector. We compute ∂Hj,i ∂pi ∂Hj,i + · = 0. ∂q ∂p ∂q This gives t t ∂Hj,i ∂pi ∂Hk,i ∂Hk,i ∂Hj,i + = 0. · · · ∂q ∂p ∂p ∂q ∂p Permuting the subscripts k and j, we get t t ∂Hj,i ∂Hj,i ∂Hk,i ∂Hk,i ∂pi · · · + = 0. ∂q ∂p ∂p ∂q ∂p

Taking diﬀerence, this implies that

∂Hj,i {Hj,i , Hk,i } + · ∂p

∂pi − ∂q

∂pi ∂q

t ! t ∂Hk,i · = 0. ∂p

We denote by Pi the matrix ({Hj,i , Hk,i })j,k and Mi the matrix with j = 1, . . . , n as rows. We get the following abbreviation t ! ∂pi ∂pi Mit = −Pi . − Mi ∂q ∂q

∂Hj,i ∂p ,

By assumption Mi is non-degenerate and kMi−1 k is uniformly bounded in i and Pi → 0 in the C 0 topology. So we get in the C 0 topology t ! ∂pi ∂pi = −Mi−1 Pi Mi−t → 0. − ∂q ∂q By integrating against a C ∞ test function ψ : Tn → R, we see that the limiting one-form p(q)dq is closed in the distribution sense. The proof also goes through if we assume (14) instead.

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Remark. — If H is a Tonelli Hamiltonian, Theorem A.1 tells us that C 0 complete integrability in the sense #3 implies C 0 complete integrability in the sense #1. Open questions remain that we list now. Problem A.1. — Is a Tonelli Hamiltonian that is C 0 completely integrable in the sense #1 or #2 C 0 completely integrable in the sense #3? Problem A.2. — Is a graph that is C 0 Lagrangian in the sense #2 necessarily C 0 Lagrangian in the sense #1? Problem A.3. — Does there exist any Tonelli Hamiltonian that is C 0 integrable (in the sense #1 or #2) but not C 1 integrable? not Lipschitz integrable? When H is Tonelli and C 0 integrable, a lot of questions concerning the Dynamics restricted to the leaves remain open. We will explain in next section what is known and what is unknown. Appendix B Smooth Hamiltonian That are C 0 completely integrable The Hamiltonian that we use in Definition A.6 and Theorem A.1 is assumed to be C 0 only, thus there is a priori no Dynamics. In this section, we discuss existing dynamical results if we assume more regularity for the Hamiltonian. More precisely, we assume that the Hamiltonian H : T ∗ Tn → R is Tonelli and C 0 integrable in the sense #1. The following results are proved in [3],[2] and [1]. Theorem B.1. — For a Tonelli Hamiltonian H : T ∗ Tn → R, the following assertions are equivalent — H has no conjugate points; — H is C 0 integrable in the sense #1; — H admits a continuous foliation into Lipschitz invariant Lagrangian graphs. In this case, a lot of leaves are in fact C 1 . Theorem B.2. — Suppose that H : T ∗ N → R is Tonelli, where N is a compact manifold, and is C 0 integrable in the sense #1. Then there exists a ﬂow invariant Gδ subset G of T ∗ N such that each leaf in G is C 1 . If we ask a little more regularity of the Hamiltonian, it can be shown that the Dynamics is non-hyperbolic. Theorem B.3. — Suppose that H : T ∗ Tn → R is a C 3 Tonelli Hamiltonian and is C 0 integrable in the sense #1, then all the Lyapunov exponents of the Hamiltonian ﬂow are zero, with respect to any invariant Borel probability measure.

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If H is C ∞ and is C 0 integrable in the sense #1, some K.A.M. theorems can be proved close to the completely periodic tori (see [1]) that have a lot of nice consequences that we give now. Theorem B.4. — Suppose that H : T ∗ Tn → R is a C ∞ Tonelli Hamiltonian and is C 0 integrable in the sense #1, then 1. there exists a dense subset of T ∗ Tn with Lebesgue positive measure that is foliated by smooth invariant Lagrangian graphs on which the Dynamics is conjugate to a Diophantine rotation; 2. there exists an dense subset of T ∗ Tn that is foliated by smooth invariant Lagrangian graphs on which the Dynamics is conjugate to a rational rotation; 3. there exists a dense Gδ subset of T ∗ Tn that is foliated by invariant Lagrangian graphs on which the Dynamics is strictly ergodic. We recall that a dynamical system is strictly ergodic if it is uniquely ergodic and if the unique invariant Borel probability measure has full support. The following problem was posed in [1] for the first case of C 0 integrability. Problem B.1. — Suppose H is Tonelli and C 0 integrable in the sense #1, #2 or #3. Can an invariant torus of the Hamiltonian ﬂow carry two invariant measures with diﬀerent rotation vectors?

Appendix C Smooth Hamiltonian that are Lipschitz completely integrable In this section, we assume further that H is a G-Lipschitz completely integrable Hamiltonian. We assume here more regularity, in particular in the transverse direction, than the C 0 integrabilities in Appendix A or B, but slightly less regularity than the C 1 integrability. We begin by proving Theorem 1.2 that we recall. Theorem C.1. — Suppose that the Hamiltonian H : T ∗ Tn → R is Tonelli and is G-Lipschitz completely integrable. Then restricted to each leaf, the Hamiltonian ﬂow has a unique well-defined rotation vector, and is bi-Lipschitz conjugate to a translation ﬂow by the rotation vector. Moreover, all the leaves are in fact C 1 . Proof. — First, we deduce from Propositions 2.2, 2.3 and 2.6 that restricted to each leaf Ta , the Hamiltonian ﬂow has a unique well-defined rotation vector ρ(a), and if the rotation vector is completely irrational, then there exists a bi-Lipschitz conjugacy conjugating the ﬂow on the leaf Ta to the rigid translation by tρ(a). Observe too that ρ(a) continuously depends on a: this is a consequence of point(A) of Proposition 2.3.

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Let us prove that ρ is injective. We will use some results due to J. N. Mather concerning the minimizing measures (see [16] and [17]). We assume that a 6= a′ satisfy ρ(a) = ρ(a′ ). Observe that any minimizing measure with cohomology class c(a) (resp. c(a′ ), see Proposition 1.1 for the definition of c) is supported in Ta (resp Ta′ ). Moreover, every minimizing measure supported in Ta (resp Ta′ ) has ρ(a) (resp. ρ(a′ )) for rotation vector. We deduce that every minimizing measure with rotation vector ρ(a) = ρ(a′ ) is minimizing with cohomology class c(a) and c(a′ ), and then its support is in Ta ∩ Ta′ = ∅. We obtain then a contradiction. Hence a 7→ ρ(a) is a continuous and injective map. By the invariance of domain theorem (see for example [10], Theorem 18.9, page 110), ρ is a homeomorphism and then for a dense set A of a, ρ(a) is completely irrational. Hence every a can be approximated by a sequence (an ) such that every ρ(an ) is completely irrational. Then there exists a K bi-Lipschitz conjugacy hk : Tn → Tn such that ftak = h−1 k ◦ Rtρ(ak ) ◦ hk . By Arzela-Ascoli, we can extract from (hk ) a converging subsequence to some h : Tn → Tn that is K-bi-Lipschitz and such that fta = h−1 ◦ Rtα ◦ h for some α. Then necessarily α = ρ(a) and we obtained the wanted bi-Lipschitz conjugacy. To conclude that the leaves are C 1 , we use the following result. Theorem C.2 (Theorem 2 of [2]). — Suppose H : T ∗ Tn → R is Tonelli, and that G is a C 0 Lagrangian graph in the sense #1 which is invariant under the Hamiltonian ﬂow. Suppose the time-1 map of the Hamiltonian ﬂow on G is bi-Lipschitz conjugate to a rotation, then the graph G is C 1 . So in this case, Problem B.1 is answered negatively. Acknowledgment. — M.-C. A. is supported by the Institut Universitaire de France and by the ANR-12-BLAN-WKBHJ. J.X. was supported by NSF grant DMS-1500897 and is supported by NNSF of China (No.11790273) and One Thousand Young Talents Program in China. References [1] M. Arcostanzo, M.-C. Arnaud, P. Bolle & M. Zavidovique – “Tonelli Hamiltonians without conjugate points and C 0 integrability”, Math. Z. 280 (2015), p. 165–194. [2] M.-C. Arnaud – “Fibrés de Green et régularité des graphes C 0 -lagrangiens invariants par un ﬂot de Tonelli”, Ann. Henri Poincaré 9 (2008), p. 881–926. [3]

, “Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures”, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), p. 989– 1007.

[4] V. Arnol′ d – Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976. [5] D. Burago & S. Ivanov – “Riemannian tori without conjugate points are ﬂat”, Geom. Funct. Anal. 4 (1994), p. 259–269.

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[6] F. Cardin & C. Viterbo – “Commuting Hamiltonians and Hamilton-Jacobi multitime equations”, Duke Math. J. 144 (2008), p. 235–284. [7] Y. M. Eliashberg – “A theorem on the structure of wave fronts and its application in symplectic topology”, Funktsional. Anal. i Prilozhen. 21 (1987), p. 65–72, 96. [8] L. C. Evans & R. F. Gariepy – Measure theory and fine properties of functions, Studies in Advanced Math., CRC Press, Boca Raton, FL, 1992. [9] A. Fathi – “Weak KAM theorem in Lagrangian dynamics”, preprint, tenth preliminary version. [10] M. J. Greenberg & J. R. Harper – Algebraic topology, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings Publishing Co., Advanced Book Program, Reading, Mass., 1981. [11] M. Gromov – Partial diﬀerential relations, Ergebn. Math. und ihrer Grenzg., vol. 9, Springer, 1986. [12] M.-R. Herman – “Inégalités “a priori” pour des tores lagrangiens invariants par des diﬀéomorphismes symplectiques”, Inst. Hautes Études Sci. Publ. Math. 70 (1989), p. 47– 101. [13] M.-R. Herman – “Sur la conjugaison diﬀérentiable des diﬀéomorphismes du cercle à des rotations”, Inst. Hautes Études Sci. Publ. Math. 49 (1979), p. 5–233. [14] V. Humilière, R. Leclercq & S. Seyfaddini – “Coisotropic rigidity and C 0 -symplectic geometry”, Duke Math. J. 164 (2015), p. 767–799. [15] N. Kryloff & N. Bogoliouboff – “La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques de la mécanique non linéaire”, Ann. of Math. 38 (1937), p. 65–113. [16] D. Massart & A. Sorrentino – “Diﬀerentiability of Mather’s average action and integrability on closed surfaces”, Nonlinearity 24 (2011), p. 1777–1793. [17] J. N. Mather – “Action minimizing invariant measures for positive definite Lagrangian systems”, Math. Z. 207 (1991), p. 169–207. [18] D. McDuff & D. Salamon – Introduction to symplectic topology, second ed., Oxford Mathematical Monographs, The Clarendon Press Univ. Press, 1998. [19] Y.-G. Oh & S. Müller – “The group of Hamiltonian homeomorphisms and C 0 -symplectic topology”, J. Symplectic Geom. 5 (2007), p. 167–219. [20] L. Polterovich & D. Rosen – Function theory on symplectic manifolds, CRM Monograph Series, vol. 34, Amer. Math. Soc., 2014. [21] A. Weinstein – Lectures on symplectic manifolds, CBMS Regional Conference Series in Mathematics, vol. 29, Amer. Math. Soc., 1979.

M.-C. Arnaud, Université de Paris, Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75013 Paris, France E-mail : [email protected] J. Xue, Department of mathematics & Yau Mathematical Sciences Center, Tsinghua University, Haidian district, Beijing, China, 100084 • E-mail : [email protected]

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Astérisque 416, 2020, p. 33–63 doi:10.24033/ast.1110

ASYMPTOTIC EXPANSION OF SMOOTH INTERVAL MAPS by Juan Rivera-Letelier

Dédié à la mémoire de Jean-Christophe Yoccoz

Abstract. — We associate to each non-degenerate smooth interval map a number measuring its global asymptotic expansion. We show that this number can be calculated in various diﬀerent ways. A consequence is that several natural notions of nonuniform hyperbolicity coincide. In this way we obtain an extension to interval maps with an arbitrary number of critical points of the remarkable result of Nowicki and Sands characterizing the Collet-Eckmann condition for unimodal maps. This also solves a conjecture of Luzzatto in dimension 1. Combined with a result of Nowicki and Przytycki, these considerations imply that several natural nonuniform hyperbolicity conditions are invariant under topological conjugacy. Another consequence is for the thermodynamic formalism: A nondegenerate smooth map has a high-temperature phase transition if and only if it is not Lyapunov hyperbolic. Résumé (Expansion asymptotique des applications lisses d’intervalle). — On associe à chaque application lisse et non dégénérée de l’intervalle un nombre measurant son expansion asymptotique globale. On montre que ce nombre peut être calculé de plusieurs façons distinctes. En conséquence, plusieurs notions d’hyperbolicité faible coïncident. De cette façon on obtient une extension aux applications de l’intervalle avec une nombre arbitraire de points critiques du fameux résultat de Nowicki et Sands caractérisant la condition de Collet-Eckmann pour les applications unimodales. Ceci résout aussi une conjecture de Luzzatto en dimensión 1. En combinaison avec un résultat de Nowicki et Przytycki, ces considérations entraînent que plusieurs notions d’hyperbolicité faible sont invariantes par conjugaison topologique. Une autre conséquence est pour le formalisme thermodynamique : une application lisse et non dégénérée de l’intervalle possède une transition de phase de haute temperature si et seulement si elle n’est pas Lyapunov hyperbolique.

2010 Mathematics Subject Classification. — 37E05; 37D25, 37C40. Key words and phrases. — Non-uniform hyperbolicity, mixing rates.

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1. Introduction In the last few decades, the statistical and stochastic properties of nonuniformly hyperbolic maps have been extensively studied in the one-dimensional setting, see for example [6, 12, 16, 37, 39, 45] and references therein. These maps are known to be abundant, see for example [3, 5, 15, 10, 21, 42, 44] for interval maps and [2, 34, 40, 14] for complex rational maps. In this paper we associate to each non-degenerate smooth interval map a number measuring its global asymptotic expansion. Our main result is that this number can be calculated in various diﬀerent ways. For example, it can be calculated using the Lyapunov exponents of periodic points or the Lyapunov exponents of invariant measures, and it can also be calculated using the exponential contraction rate of preimages of a small ball. This implies that several natural notions of nonuniform hyperbolicity coincide, including the existence of an absolutely continuous invariant probability (acip) that is exponentially mixing. In this way we obtain an extension to interval maps with an arbitrary number of critical points of the remarkable result of Nowicki and Sands characterizing the Collet-Eckmann condition for unicritical maps, see [28]. Moreover, this solves in the aﬃrmative a conjecture of Luzzatto in dimension 1, see [19, Conjecture 1]. Combined with a result of Nowicki and Przytycki, we obtain that several natural notions of nonuniform hyperbolicity are invariant under topological conjugacy, see [27]. In particular, for non-degenerate smooth interval maps the existence of an exponentially mixing acip is invariant under topological conjugacy. Combined with [11, 22, 23, 43, 46], these considerations imply that an arbitrary exponentially mixing acip satisfies strong statistical properties, such as the local central limit theorem and the vector-valued almost sure invariant principle. On the other hand, by [37] it follows that for some p > 1 the density of such a measure is in the space Lp (Leb). Our main result provides an important step in the study of the thermodynamic formalism of non-degenerate smooth interval maps in [32]. (1) Combining our main result with [32, Theorem A], we obtain a characterization of those maps having a high-temperature phase transition. We proceed to describe our results more precisely. To simplify the exposition, below we state our results in a more restricted setting than what we are able to handle. For general versions, see §4 and the remarks in §6. 1.1. Quantifying asymptotic expansion. — Let I be a compact interval and f : I → I a smooth map. A critical point of f is a point of I at which the derivative of f vanishes. The map f is non-degenerate if it is non-injective, if the number of its critical points is (1)

The proof of our Main Theorem applies without change to the more general class of maps considered in [32], see Theorem C of that paper. Note however that, although the proof in [32] follows the proof of our Main Theorem, it has a part that is diﬀerent. This modified proof only gives a qualitative version of our Main Theorem, similar to Corollary A.

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finite, and if at each critical point of f some higher order derivative of f is nonzero. A non-degenerate smooth interval map is unicritical if it has a unique critical point. (2) Let f : I → I be a non-degenerate smooth map. For an integer n ≥ 1, a periodic point p of f of period n is hyperbolic repelling if |Df n (p)| > 1. In this case, denote by 1 ln |Df n (p)| n the Lyapunov exponent of p. Similarly, for a Borel probability measure ν on I that is invariant by f denote by Z χp (f ) :=

χν (f ) :=

ln |Df | dν

its Lyapunov exponent. The following is our main result. A non-degenerate smooth map f : I → I is topologically exact, if for every open subset U of I there is an integer n ≥ 1 such that f n (U ) = I. Main Theorem. — For a non-degenerate smooth map f : I → I, the number χper (f ) := inf {χp (f ) : p hyperbolic repelling periodic point of f } is equal to χinf (f ) := inf {χν (f ) : ν invariant probability measure of f } . If in addition f is topologically exact, then there is δ > 0 such that for every interval J contained in I that satisfies |J| ≤ δ, we have 1 ln max |W | : W connected component of f −n (J) = −χinf (f ). lim n→+∞ n Moreover, for each point x0 in I we have 1 (1.1) lim sup ln min |Df n (x)| : x ∈ f −n (x0 ) ≤ χinf (f ), n n→+∞

and there is a subset E of I of zero Hausdorﬀ dimension such that for each point x0 in I \ E the lim sup above is a limit and the inequality an equality. Except for the equality χinf (f ) = χper (f ), the hypothesis that f is topologically exact is necessary, see §1.6. The result above suggests that for a non-degenerate smooth map f the number χper (f ) (equal to χinf (f )) is a natural measure of the asymptotic expansion of f . In fact, χinf (f ) gives a lower bound for the (lower) Lyapunov exponent of every point in a set of total probability. This motivates the following definition. Definition 1.1. — A non-degenerate smooth map f is Lyapunov hyperbolic if χinf (f ) > 0. In this case, we call χinf (f ) the total Lyapunov exponent of f . (2)

Note that every unicritical map is unimodal, but not conversely.

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Lyapunov hyperbolicity can be regarded as a strong form of nonuniform hyperbolicity in the sense of Pesin. A consequence of the Main Theorem is that Lyapunov hyperbolicity coincides with several natural nonuniform hyperbolicity conditions, see §1.2. When restricted to the case where f is unicritical, the Main Theorem gives a quantified version of the fundamental part of [28, Theorem A]. In [28, Theorem A], property (1.1) was only considered in the case where x0 is the critical point of f ; so the assertions concerning (1.1) in the Main Theorem are new, even when restricted to the case where f is unicritical. The proof of [28, Theorem A] relies heavily on delicate combinatorial arguments that are specific to unicritical maps. As is, it does not extend to interval maps with several critical points. When restricted to unicritical maps, our argument is substantially simpler than that of [28]. When f is a complex rational map, the Main Theorem is the essence of [33, Main Theorem]. The proof in [33, Main Theorem] does not extend to interval maps, because at a key point it relies on the fact that a complex rational map is open as a map of the Riemann sphere to itself. Our argument allows us to deal with the fact that a nondegenerate smooth interval map is not an open map in general, see §1.7 for further details. 1.2. Nonuniformly hyperbolic interval maps. — We introduce some terminology to state a consequence of the Main Theorem about the equivalence of various nonuniform hyperbolicity conditions. Let (X, dist) be a compact metric space, T : X → X a continuous map and ν a Borel probability measure that is invariant by T . Then ν is exponentially mixing or has exponential decay of correlations, if there are constants C > 0 and ρ in (0, 1) such that for every continuous function ϕ : X → R and every Lipschitz continuous function ψ : X → R we have for every integer n ≥ 1 Z Z Z ϕ ◦ f n · ψ dν − ϕ dν ψ dν ≤ C sup |ϕ| kψkLip ρn , X X

X

X

′ )| supx,x′ ∈X,x6=x′ |ψ(x)−ψ(x dist(x,x′ ) .

where kψkLip := We denote by Leb the Lebesgue measure on R. For a non-degenerate smooth map f : I → I, we use acip to refer to a Borel probability measure on I that is absolutely continuous with respect Leb and that is invariant by f . A non-degenerate smooth map f : I → I has Uniform Hyperbolicity on Periodic Orbits, if χper (f ) > 0. Moreover, f satisfies the: — Collet-Eckmann condition, if all the periodic points of f are hyperbolic repelling and if for every critical value v of f we have 1 ln |Df n (v)| > 0. n — Backward or Second Collet-Eckmann condition at a point x of I, if there are constants C > 0 and λ > 1, such that for every integer n ≥ 1 and every point y of f −n (x) we have |Df n (y)| ≥ Cλn . lim inf

n→+∞

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— Backward or Second Collet-Eckmann condition, if f satisfies the Backward Collet-Eckmann condition at each of its critical points. — Exponential Shrinking of Components condition, if there are constants δ > 0 and λ > 1 such that for every interval J contained in I that satisfies |J| ≤ δ, the following holds: For every integer n ≥ 1 and every connected component W of f −n (J) we have |W | ≤ λ−n . In the statement of the following corollary we use the following fact: Every nondegenerate smooth map that is topologically exact has strictly positive topological entropy and a unique measure of maximal entropy, see for example [4, §3]. Finally, a measure ρ on I has a power-law lower bound, if there are constants C > 0 and α > 0 such that for every interval J contained in I we have ρ(J) ≥ C|J|α . Corollary A. — For a non-degenerate smooth map f : I → I that is topologically exact, the following properties are equivalent: 1. Lyapunov hyperbolicity (χinf (f ) > 0). 2. Uniform Hyperbolicity on Periodic Orbits (χper (f ) > 0). 3. Existence of an exponentially mixing acip for f . 4. The map f is conjugated to a piecewise aﬃne and expanding multimodal map by a bi-Hölder continuous function. 5. The map f satisfies the Exponential Shrinking of Components condition. 6. The map f satisfies the Backward Collet-Eckmann condition at some point of I. 7. The maximal entropy measure of f has a power-law lower bound. Furthermore, these equivalent conditions are satisfied when f satisfies the ColletEckmann or the Backward Collet-Eckmann condition. The equivalence 1 ⇔ 3 solves [19, Conjecture 1] in dimension 1. When f is unicritical, the equivalence of conditions 1–5 was proved by Nowicki and Sands in [28, Theorem A]. They also showed, still in the case where f is unicritical, that the Collet-Eckmann and the Backward Collet-Eckmann conditions are equivalent and that each of these conditions is equivalent to conditions 1–5. In contrast, for maps with several critical points the Collet-Eckmann and the Backward ColletEckmann conditions are not equivalent and neither of these conditions is equivalent to conditions 1–7, see [33, §6]. When f is a complex rational map, a statement analog to Corollary A was shown by Przytycki, Smirnov, and the author in [33, Main Theorem], (3) [31, Corollary 1.1] and [35, Theorem B]. Even when restricted to the case where f is unicritical, the implication 6 ⇒ 5 of Corollary A is new. It is the main new ingredient of the proof, which is provided by Main Theorem. The implication 5 ⇒ 4 is also new. The rest of the implications are known, or can be easily adapted from known properties of unicritical interval maps or complex rational maps, see §6 for references. (3)

In [33] condition 4 was interpreted as the existence of a “Hölder coding tree.”

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1.3. Exponentially mixing acip’s. — Let f : I → I be a non-degenerate smooth map that is topologically exact and that is Lyapunov hyperbolic. Such a map has a unique exponentially mixing acip. In [31, Theorem C], this measure is constructed using the general method of Young in [46]. (4) When a measure ν on I can be obtained in this way, we say ν can be obtained through a Young tower with an exponential tail estimate. Such a measure has several statistical properties, including the “local central limit theorem” and the “vector-valued almost sure invariant principle,” see [23, 46] for these results and for precisions, and [11, 22, 43] for other statistical properties satisfied by such a measure. On the other hand, for f as above there is p(f ) > 1 with the following property: For p ≥ 1 the density of the unique exponentially mixing acip of f is in the space Lp (Leb) if 1 ≤ p < p(f ), and it is not in Lp (Leb) if p > p(f ). See [37, Corollary 2.19], where a geometric characterization of p(f ) is also given. (5) In view of the results above, the following corollary is a direct consequence of Corollary A and of general properties of non-degenerate smooth interval maps. Corollary B. — Let f be a non-degenerate smooth interval map having an exponentially mixing acip ν. Then there is p > 1 such that the density of ν with respect to Leb is in the space Lp (Leb). Moreover, ν can be obtained through a Young tower with an exponential tail estimate. In particular, ν satisfies the local central limit theorem and the vector-valued almost sure invariant principle. Alves, Freitas, Luzzatto, and Vaienti showed under mild assumptions that in any dimension each polynomially mixing or stretch exponentially mixing acip can be obtained through a Young tower with the corresponding tail estimates, see [1, Theorem C]. In contrast with this last result, in Corollary B the existence of p > 1 for which the density of ν is in Lp (Leb) is obtained as a consequence, and not as a hypothesis. So the following question arises naturally. Question 1.2. — Let f be a non-degenerate smooth interval map having an acip ν. Does there exist p > 1 such that the density of ν with respect to Leb is in the space Lp (Leb)? 1.4. Topological invariance. — A direct consequence of Corollary A and a result of Nowicki and Przytycki in [27], is that each of the conditions 1–7 of Corollary A is invariant under topological conjugacy for maps having all of its periodic points hyperbolic repelling. To state this result more precisely, we recall the definition of the “Topological Collet-Eckmann condition” introduced in [27]. Let f : I → I be a nondegenerate smooth map that is topologically exact and fix r > 0. Given an integer n ≥ 1, the criticality of f n at a point x of I is the number of those j in {0, . . . , n − 1} (4) The proof of [31, Theorem C] is written for complex rational maps and applies without change to topologically exact non-degenerate smooth interval maps. See [37, Corollary 2.19] for a proof written for interval maps. (5) If f is unicritical and we denote its critical point by c, then p(f ) = ℓc /(ℓc − 1).

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such that the connected component of f −(n−j) (B(f n (x), r)) containing f j (x) contains a critical point of f . Then f satisfies the Topological Collet-Eckmann (TCE) condition, if for some choice of r > 0 there are constants D ≥ 1 and θ in (0, 1), such that the following property holds: For each point x in I the set Gx of all those integers m ≥ 1 for which the criticality of f m at x is less than or equal to D, satisfies 1 # (Gx ∩ {1, . . . , n}) ≥ θ. n One of the main features of the TCE condition, which is readily seen from its definition, is that it is invariant under topological conjugacy preserving critical points: If f : I → I is a non-degenerate smooth map satisfying the TCE condition and fe: Ie → Ie is a non-degenerate smooth map that is topologically conjugated to f by a map preserving critical points, then fe also satisfies the TCE condition. Nowicki and Przytycki showed in [27] that for a non-degenerate smooth interval map f , condition 5 of Corollary A implies the TCE condition. They also proved that if in addition all the periodic points of f are hyperbolic repelling, then the TCE condition implies condition 2 of Corollary A. Thus, the following is a direct consequence of Corollary A and [27]. lim inf

n→+∞

Corollary C. — For a non-degenerate smooth interval map that is topologically exact and that only has hyperbolic repelling periodic points, the Topological Collet-Eckmann condition is equivalent to each of the conditions 1–7 of Corollary A. In particular, each of the conditions 1–7 of Corollary A is invariant under topological conjugacy preserving critical points, for maps having only hyperbolic repelling periodic points. Combining [27] and [28, Theorem A], it follows that for unicritical maps having only hyperbolic repelling periodic points the Collet-Eckmann and the Backward ColletEckmann conditions are both invariant under topological conjugacy preserving critical points. This is not the case for maps with several critical points, see [33, Appendix C]. The following is for maps that are not necessarily topologically exact. It is obtained by combining Corollary C with general properties of non-degenerate smooth interval maps, see §6 for the proof. Corollary D. — For non-degenerate smooth interval maps having only hyperbolic repelling periodic points, the property that an iterate has an exponentially mixing acip is invariant under topological conjugacy preserving critical points. 1.5. High-temperature phase transitions. — Corollary A has a very useful application to the thermodynamic formalism of interval maps, that we proceed to describe. Let f : I → I be a non-degenerate smooth interval map that is topologically exact. Denote by M (I, f ) the space of Borel probability measures on I that are invariant by f . For a measure ν in M (I, f ), denote by hν (f ) the measure-theoretic entropy of f with respect to ν and for each real number t put P (t) := sup {hν (f ) − tχν (f ) : ν ∈ M (I, f )} .

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Combining Ruelle’s inequality in [38] with the fact that the Lyapunov exponent of every measure in M (I, f ) is nonnegative, see [30, Theorem B] or Proposition A.1, it follows that the number above is finite and that the function P : R → R so defined is convex and nonincreasing. Moreover, P has at least one zero and that its first zero is in (0, 1]. The function P is called the geometric pressure function of f , and it is related to various multifractal spectra and large deviation rate functions associated to f . Following the usual terminology in statistical mechanics, for a real number t∗ we say f has a phase transition at t∗ , if P is not real analytic at t = t∗ . In accordance with the usual interpretation of t > 0 as the inverse of the temperature in statistical mechanics, if in addition t∗ > 0 and t∗ is less than or equal to the first zero of P , then we say that f has a high-temperature phase transition. The following is an easy consequence of Corollary A and [32, Theorem A], see §6 for the proof. Corollary E. — For a non-degenerate smooth interval map f that is topologically exact, the following properties are equivalent: 1. The map f has a high-temperature phase transition. 2. If we denote by t0 the first zero of P , then for every t ≥ t0 we have P (t) = 0. 3. The function P is nonnegative. 4. The map f is not Lyapunov hyperbolic. When f is a complex rational map, the equivalence of conditions 2–4 is part of [33, Main Theorem]. (6) 1.6. Notes and references. — If the map f is not topologically exact, then by the Main Theorem we have χinf (f ) = χper (f ), but the remaining assertions of the Main Theorem do not hold in general. For an example, consider the logistic map with the Feigenbaum combinatorics, f0 . For this map we have χinf (f0 ) = 0. However, if J is a small closed interval that is disjoint from the post-critical set of f0 , then the limit in the Main Theorem is strictly negative. Similarly, for every point x0 that is not in the post-critical set of f0 , the lim sup in the Main Theorem is strictly positive. This also shows that the implication 6 ⇒ 1 of Corollary A does not hold for f0 . Note also that an infinitely renormalizable map f cannot satisfy any of the conditions 1–5 of Corollary A. See [25] for further examples illustrating the diﬀerence between the Collet-Eckmann condition and conditions 1–7 of Corollary A for maps with at least 2 critical points. Li [17] and Luzzatto and Wang [20] showed that the Collet-Eckmann condition together with a slow recurrence condition is invariant under topological conjugacy preserving critical points. See also [18] for a recent related result. (6)

It is unclear to us if condition 1 is equivalent to 2–4 in the complex setting.

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See [8, 9] and references therein for results on low-temperature phase transitions; that is, phase transitions that occur after the first zero of the geometric pressure function. 1.7. Strategy and organization. — To prove the Main Theorem and Corollary A we follow the structure of the proof of the analog result for complex rational maps in [33, Main Theorem]. The main diﬃculty is the proof that χper (f ) > 0 implies the last statement of the Main Theorem, which is essentially the implication 2 ⇒ 5 of Corollary A. The proof of this fact in [33] relies in an essential way on the fact that a nonconstant complex rational maps is open as a map from the Riemann sphere to itself. The argument provided here allows us to deal with the fact that a multimodal map is not an open map in general. Ultimately, it relies on the fact that the boundary of a bounded interval in R is reduced to 2 points. To prove implication 2 ⇒ 5 of Corollary A we first remark that the proof of the implication 2 ⇒ 6 for rational maps in [33] applies without change to interval maps. Our main technical result is a quantified version of the implication 6 ⇒ 5 for interval maps. This is stated as Proposition 3.1, after some preliminary considerations in §2. Its proof occupies all of §3. In §4 we formulate a strengthened version of the Main Theorem, stated as the Main Theorem′ , and we deduce it from Proposition 3.1 and known results. In the proof we use that the Lyapunov exponent of every invariant measure supported on the Julia set is nonnegative [30, Theorem B]. We provide a simple proof of this fact (Proposition A.1 in Appendix A), which holds for a general continuously diﬀerentiable interval map. This result is used again in the proof of Corollary E. The proofs of Corollaries A, D, and E are given in §6, after we prove the implication 5 ⇒ 4 of Corollary A in §5. Acknowledgments. — I would like to thank the referee for several valuable comments. This article was completed while the author was visiting Brown University and the Institute for Computational and Experimental Research in Mathematics (ICERM). The author thanks both of these institutions for the optimal working conditions provided, and acknowledges partial support from FONDECYT grant 1100922, Chile, and NSF grant DMS-1700291, U.S.A. 2. Preliminaries Throughout the rest of this paper I denotes a compact interval of R. We endow I with the distance dist induced by the absolute value | · | on R. For x in I and r > 0, we denote by B(x, r) the open ball of I centered at x and of radius r. For an interval J contained in I, we denote by |J| its length and for η > 0 we denote by ηJ the open interval of R of length η|J| that has the same middle point as J. Given a map f : I → I, a subset J of I is forward invariant if f (J) = J and it is completely invariant if f −1 (J) = J.

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2.1. Fatou and Julia sets. — Following [24], in this section we introduce the Fatou and Julia sets of a multimodal map and gather some of their basic properties. A non-injective continuous map f : I → I is multimodal, if there is a finite partition of I into intervals on each of which f is injective. A turning point of a multimodal map f : I → I is a point in I at which f is not locally injective. Fix a multimodal map f : I → I. The Fatou set F (f ) of f is the largest open subset of I on which the iterates of f form a normal family. A connected component of F (f ) is called Fatou component of f . A Fatou component U of f is periodic if for some integer p ≥ 1 we have f p (U ) ⊂ U . In this case the least integer p with this property is the period of U . The Julia set J(f ) of f is the complement of F (f ) in I. By definition we have f −1 (F (f )) ⊂ F (f ) and therefore f (J(f )) ⊂ J(f ). In contrast with the complex setting, the Julia set of f might be empty, reduced to a single point, or might not be completely invariant. If the Julia set of f is not completely invariant, then it is possible to make an arbitrarily small smooth perturbation of f outside a neighborhood of J(f ), so that the Julia set of the perturbed map is completely invariant and coincides with J(f ). 2.2. Topological exactness. — Fix a multimodal map f : I → I. We say that f is boundary anchored if f (∂I) ⊂ ∂I and that f is topologically exact on J(f ), if J(f ) is not reduced to a point and if for every open subset U of I intersecting J(f ) an iterate of f | maps U ∩ J(f ) onto J(f ). J(f ) Since it is too restrictive for our applications to assume that a multimodal map is at the same time boundary anchored and topologically exact on its Julia set, we introduce the following terminology. We say that a multimodal map f is essentially topologically exact on J(f ), if there is a compact interval I0 contained in I that contains all the critical points of f and such that the following properties hold: f (I0 ) ⊂ I0 , S+∞ the multimodal map f | : I0 → I0 is topologically exact on J(f | ), and n=0 f −n (I0 ) I0 I0 contains an interval whose closure contains J(f ). 2.3. Differentiable interval maps. — Fix a diﬀerentiable map f : I → I. A critical point of f is a point at which the derivative of f vanishes. A critical value of f is the image by f of a critical point. We denote by Crit(f ) the set of critical points of f . If f is in addition a multimodal map, then we put Crit′ (f ) := Crit(f ) ∩ J(f ). Let J be an interval contained in I and let n ≥ 1 be an integer. Then each connected component of f −n (J) is a pull-back of J of order n, or just a pull-back of J. If in addition f n : W → J is a diﬀeomorphism, then the pull-back W is diﬀeomorphic. Note that if f is boundary anchored and W is a pull-back of J of order n, then f n (∂W ) ⊂ ∂J.

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Let J be an interval contained in I, let n ≥ 1 be an integer, and let W be a pullback of J by f n . We say W is a child of J, (7) if W contains a unique critical point c of f in J(f ) and if there is s in {0, . . . , n − 1} such that f s (c) belongs to Crit(f ) and such that the following properties hold: 1. Either s = n − 1 or the pull-back of J by f n−s−1 containing f s+1 (c) is diﬀeomorphic. ′

′

2. For each s′ in {0, . . . , s} the pull-back of J by f n−s containing f s (c) is either ′ ′ disjoint from Crit(f ) or f s (c) belongs to Crit(f ) and then f s (c) is the unique critical point of f contained in this set. 2.4. Interval maps of class C 3 with non-flat critical points. — A diﬀerentiable interval map f : I → I is of class C 3 with non-ﬂat critical points, if: — The set Crit(f ) is finite and f is of class C 3 outside Crit(f ). — For each critical point c of f there exists a number ℓc > 1 and diﬀeomorphisms φ and ψ of R of class C 3 , such that φ(c) = ψ(f (c)) = 0 and such that on a neighborhood of c on I we have, |ψ ◦ f | = |φ|ℓc . The number ℓc is the order of f at c. Denote by A the collection of non-injective interval maps of class C 3 with non-ﬂat critical points, whose Julia set is completely invariant and contains at least 2 points. Note that every smooth non-degenerate interval map that is topologically exact is in A , and that every interval map in A is a continuously diﬀerentiable multimodal map. We use the following important fact: For each map in A every Fatou component is mapped to a periodic Fatou component under forward iteration, and the number of periodic Fatou components is finite, see [24, Chapter IV, Theorem AB]. The following version of the Koebe principle follows from [41, Theorem C(2)(ii)]. As for non-degenerate smooth interval maps, a periodic point p of period n of a map f in A is hyperbolic repelling if |Df n (p)| > 1. Lemma 2.1 (Koebe principle). — Let f : I → I be an interval map in A all whose periodic points in J(f ) are hyperbolic repelling. Then there is δ0 > 0 such that for every K > 1 there is ε in (0, 1) such that the following property holds. Let J be an interval contained in I that intersects J(f ) and satisfies |J| ≤ δ0 . Moreover, let n ≥ 1 be an integer and W a diﬀeomorphic pull-back of J by f n . Then for every x and x′ in the unique pull-back of εJ by f n contained in W we have K −1 ≤ |Df n (x)|/|Df n (x′ )| ≤ K. The following general fact is used in the proof of the Main Theorem′ in §4. (7)

This definition is a variant of the usual definition of “child.” It is adapted to deal with the case where f has a critical connection.

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Fact 2.2. — If f is an interval map in A that is topologically exact on J(f ), then J(f ) contains a uniformly expanding set whose topological entropy is strictly positive. In particular, the Hausdorﬀ dimension of J(f ) is strictly positive. The following lemma is standard, see for example [36] for part 1. Lemma 2.3. — Let f : I → I be a multimodal map in A having all of its periodic points in J(f ) hyperbolic repelling. Then the following properties hold. 1. For every integer n ≥ 1, every pull-back W of B(x, δ1 ) by f n intersects J(f ), contains at most 1 critical point of f , and is disjoint from (Crit(f ) ∪ ∂I) \ J(f ). 2. For every κ > 0 there is δ2 > 0 such that for every x in J(f ), every integer n ≥ 1, and every pull-back W of B(x, δ2 ) by f n , we have |W | ≤ κ.

3. Exponential shrinking of components The purpose of this section is to prove the following proposition. It is the key step in the proof of the Main Theorem, which is given in the next section. Proposition 3.1. — Let f : I → I be a map in A that is topologically exact on J(f ). Suppose there is a point x0 of J(f ) and constants C > 0 and λ > 1 such that for every integer n ≥ 1 and every point x in f −n (x0 ) we have |Df n (x)| ≥ Cλn . Then every periodic point of f in J(f ) is hyperbolic repelling and for every λ0 in (1, λ) there is a constant δ2 > 0 such that the following property holds. Let J be an interval contained in I that intersects J(f ) and satisfies |J| ≤ δ2 . If J(f ) is not an interval, then assume that J is not a neighborhood of a periodic point in the boundary of a Fatou component of f . (8) Then for every integer n ≥ 1 and every pull-back W of J by f n , we have (3.1)

|W | ≤ λ−n 0 .

The proof of this proposition is at the end of this section. It is based on several lemmas. In this section, a critical point c of a map f in A is exposed, if for every integer j ≥ 1 the point f j (c) is not a critical point of f . Given c in Crit′ (f ), let s ≥ 0 be the largest integer such that f s (c) is in Crit(f ) and put o n Y ′ b ℓf j (c) and ℓd ℓbc := max := max ℓc : c ∈ Crit (f ) . j∈{0,...,s} f j (c)∈Crit(f )

(8)

There is an example showing that this hypothesis is necessary, see [36, Proposition A]. However, a qualitative result holds when this hypothesis is not satisfied, see [36, Theorem B].

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Lemma 3.2. — Let f : I → I be an interval map in A such that all of its periodic points in J(f ) are hyperbolic repelling. Then there are δ3 > 0 and C1 > 1 such that for every interval J that intersects J(f ) and satisfies |J| ≤ δ3 and C1 J ⊂ I, the following property holds: For every integer n ≥ 1 and every pull-back W of J by f n such that the pull-back of C1 J by f n containing W is a child of C1 J, we have n |W | ≤ 6ℓd max |J| max {|Df (a)| : a ∈ ∂W }

−1

.

Proof. — Let δ0 > 0 and ε in (0, 1) be given by Lemma 2.1 with K = 2 and let δ1 > 0 be given by Lemma 2.3. Since the critical points of f are non-ﬂat, there is δ∗ > 0 so that for each c in Crit′ (f ), each integer s ≥ 0 such that f s (c) is in Crit′ (f ), and each interval W contained in B(c, δ∗ ) we have |W | max |Df s+1 (a)| : a ∈ ∂W ≤ 3ℓbc |f s+1 (W )|. Let δ2 > 0 be given by Lemma 2.3(2) with κ = δ∗ . We prove the lemma with δ3 = ε min{δ2 , δ0 } and C1 = ε−1 . To do this, let J be an interval contained in I that intersects J(f ) and satisfies |J| ≤ δ2 and Jb := ε−1 J ⊂ I.

Moreover, let n ≥ 1 be an integer and let W be a pull-back of J by f n such that the c of Jb by f n containing W is a child of J. b Let c be the unique critical pull-back W c point of f contained in W and let s be the largest element of {0, . . . , n − 1} such c ′ of Jb by f n−s−1 that f s (c) is in Crit(f ). So either s = n − 1 or the pull-back W s+1 containing f (W ) is diﬀeomorphic. Then the Koebe principle (Lemma 2.1) implies that, if we denote by W ′ the pull-back of J by f n−s−1 containing f s+1 (W ), then −1 |W ′ | ≤ 2|J| max |Df n−s−1 (a′ )| : a′ ∈ ∂W ′ .

c ⊂ B(c, δ∗ ), so by our choice On the other hand, by our choice of δ2 we have W ⊂ W of δ∗ we have −1 |W | ≤ 3ℓbc |f s+1 (W )| max |Df s+1 (a)| : a ∈ ∂W −1 ′ s+1 ≤ 3ℓd (a)| : a ∈ ∂W . max |W | max |Df

The desired inequality is obtained by combining the last 2 displayed inequalities.

Lemma 3.3. — Let f : I → I be an interval map in A such that all of its periodic points in J(f ) are hyperbolic repelling. Suppose that none of the boundary points of I is a critical point of f and let C1 > 1 be the constant given by Lemma 3.2. Then, for every η > 1 there is a constant δ(η) > 0 such that for every interval Jb that b ≤ δ(η) and C1 Jb ⊂ I, the following properties hold for intersects J(f ) and satisfies |J| c of Jb by f n : every integer n ≥ 1 and every pull-back W b the number of pull-backs of J by f n contained 1. For every interval J contained in J, c is bounded from above by 2η n . in W n o−1 n b n c | ≤ 12ℓd c . 2. |W max η |J| max |Df (a)| : a ∈ ∂ W SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2020

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Proof. — Let δ0 > 0 and ε in (0, 1) be given by Lemma 2.1 with K = 2, let δ1 > 0 be given by Lemma 2.3(1), and let δ3 > 0 and C1 > 1 be given by Lemma 3.2. Enlarging C1 if necessary we assume C1 ≥ ε−1 . On the other hand, let L ≥ 1 be a suﬃciently large integer such that η L > 6ℓd max and let δ∗ > 0 be suﬃciently small so that for every exposed critical point c of f and every j in {0, . . . , L}, the point f j (c) is not in B(Crit(f ), δ∗ ). Finally, let δ2 be given by Lemma 2.3(2) with κ := C1−1 min {δ0 , δ1 , δ3 , δ∗ , dist(Crit(f ), ∂I)} . We prove the lemma with δ(η) = δ2 . To do this, let Jb be an interval that interc be b ≤ δ2 and C1 Jb ⊂ I, let n ≥ 1 be an integer, and let W sects J(f ) and satisfies |J| n b c b a pull-back of J by f . Put m0 := n and W0 := J and define inductively an integer k ≥ 0 and integers m0 > m1 > · · · > mk ≥ 0, ct of Jb by f n−mt containing f mt (W c ) is such that for each t in {1, . . . , k} the pull-back W contained in B(Crit(f ), κ). Note that by our choice of δ2 this last property implies ct ⊂ I. Recalling that m0 = n, let t ≥ 0 be an integer such that mt is that C1 W ct by f mt containing W c is already defined. If mt = 0, or if the pull-back of C1 W ′ diﬀeomorphic, then put k = t and stop. Otherwise, define mt+1 as the largest inct by f mt −m conc ′ of C1 W teger m in {0, . . . , mt − 1} such that the pull-back W t+1 c ) is not diﬀeomorphic. In view of Lemma 2.3(1), it follows that W c′ taining f m (W t+1 contains a unique critical point and that this critical point is in J(f ). Moreover, ct . Define mt+1 as the smallest integer m in {0, . . . , m′ } c ′ is a child of C1 W W t+1 t+1 ct by f mt −m containing f m (W c ) is a child of C1 W ct . such that the pull-back W∗ of C1 W ct+1 ⊂ W∗ ⊂ B(Crit(f ), κ). Clearly, W c is diﬀeomorphic; Note that if k = 0, then the pull-back of Cj Jb by f n containing W n c b in particular f : W → J is diﬀeomorphic. On the other hand, note that for every t c ′ is exposed. So, by definition of L in {1, . . . , k − 1} the unique critical point in W t+1 we have mt − mt+1 ≥ mt − m′t+1 ≥ L. c → Jb is a diﬀeoTo prove item 1 of the lemma, observe that if k = 0, then f n : W morphism and the desired assertion is trivially true. Suppose k ≥ 1 and let J be an b It follows from the definitions that for every t in {1, . . . , k} the interval contained in J. mt−1 −mt c ). Furthermore, an induction has at most one critical point in f mt (W map f t argument in t shows that there are at most 2 pull-backs of J by f n−mt contained in c ). Since the pull-back of Jb containing f mt (W 2k ≤ 2η (k−1)L ≤ 2η m1 −mk ≤ 2η n ,

the last assertion with t = k proves item 1 of the lemma in the case where mk = 0. ck by f mk If mk ≥ 1, then it follows from the definitions that the pull-back of C1 W n c containing W is diﬀeomorphic. So the number of pull-backs of J by f contained ASTÉRISQUE 416

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c is also bounded from above by 2η n . This completes the proof of item 1 of the in W lemma. To prove item 2, suppose first k = 0. Then the pull-back of C1 Jb by f n containc is diﬀeomorphic and the desired inequality follows from the Koebe princiing W n ple (Lemma 2.1) with 12ℓd max η replaced by 2. Suppose k ≥ 1 and observe that by Lemma 3.2 for each t in {1, . . . , k} we have n o−1 mt−1 −mt ct | ≤ 6ℓd c ct |W (a)| : a ∈ ∂ W . max |Wt−1 | max |Df By an induction argument we obtain,

Using we obtain

n o−1 k b n−mk ′ ck | ≤ (6ℓd ck |W (a )| : a′ ∈ ∂ W . max ) |J| max |Df k−1 (6ℓd < η (k−1)L ≤ η m1 −mk ≤ η n , max )

n o−1 n n−mk ck | ≤ 6ℓd ck |W (a) : a ∈ ∂ W . max η max |Df

This proves item 2 of the lemma in the case where mk = 0. If mk ≥ 1, then the ck by f mk containing W c is diﬀeomorphic and by the Koebe principle pull-back of C1 W (Lemma 2.1) we obtain n o−1 c | ≤ 2|W ck | max |Df mk (a)| : a ∈ ∂ W c |W n o−1 n b c ≤ 12ℓd . max |J| max |Df (a)| : a ∈ ∂ W

This completes the proof of item 2 and of the lemma.

The following lemma is more general than what we need for the proof of Proposition 3.1. It is used again in the proof of the Main Theorem in the next section. Lemma 3.4. — Let f : I → I be an interval map in A that is topologically exact on J(f ) and put χ0per (f ) := inf {χp (f ) : p periodic point of f in J(f )} . Then for every interval J contained in I that intersects J(f ) we have 1 (3.2) lim inf ln max |W | : W connected component of f −n (J) ≥ −χ0per (f ) n→+∞ n and for every point x0 of J(f ) we have 1 (3.3) lim sup ln min |Df n (x)| : x ∈ f −n (x0 ) ≤ χ0per (f ). n→+∞ n

Proof. — Let ℓ ≥ 1 be an integer and let p be a periodic point of f of period ℓ in J(f ). Suppose first p is hyperbolic repelling. Then there is δ > 0 and a uniformly contracting inverse branch φ of f ℓ that is defined on B(p, δ) and fixes p. It follows that φ(B(p, δ)) ⊂ B(p, δ) and that there is K > 1 such that for every integer k ≥ 1

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the distortion of φk on B(p, δ) is bounded by K. On the other hand, the hypothesis that f is topologically exact on J(f ) implies that there is an integer m ≥ 1 such that the intersection of f −m (J) and B(p, δ) contains an interval J ′ and such that there is a point x′0 in f −m (x0 ) contained in B(p, δ). Then we have (3.4)

lim inf

n→+∞

1 ln max |W | : W connected component of f −n (J) n 1 ≥ lim inf ln |φk (J ′ )| = −χp (f ) k→+∞ kℓ

and (3.5)

lim sup n→+∞

1 ln min |Df n (x)| : x ∈ f −n (x0 ) n

≤ − lim

k→+∞

1 ln |Dφk (x′0 )| = χp (f ). kℓ

Since p is an arbitrary hyperbolic repelling periodic point, this proves (3.2) and (3.3). It remains to consider the case where p is not hyperbolic repelling, so that Df 2ℓ (p) = 1. Without loss of generality we assume that for every δ > 0 the interval (p, p + δ) intersects J(f ). Let η > 1 be given and let δ > 0 be suﬃciently small so there is an inverse branch φ of f 2ℓ that is defined on B(p, δ), that fixes p, and that is strictly increasing on (p, p + δ). Reducing δ if necessary we assume we have |Df | < η on B(p, δ). As in the previous case there is an integer m ≥ 1 such that the intersection of f −m (J) and (p, p + δ) contains an interval J ′ and such that there is a point x′0 in f −m (x0 ) contained in (p, p + δ). Then we have (3.4) and (3.5) with χp (f ) replaced by ε. Since ε > 0 is arbitrary, these inequalities hold with χp (f ) = 0. The proof of the lemma is thus completed. Proof of Proposition 3.1. — By Lemma 3.4 all the periodic points of f in J(f ) are b0 in (λ0 , λ) there is a hyperbolic repelling. It is enough to show that for every λ constant C0 > 0 such that the proposition holds with the right hand side of (3.1) b−n . replaced by C0 λ 0 Let Ie be equal to I if J(f ) = I. Otherwise, for each periodic point y in the boundary of a Fatou component U of f , let y ′ be a point in U , let Uy be the open interval bounded by y and y ′ , and put [ Ie := I \ Uy , y

where the union runs through all the periodic points of in the boundary of a Fatou component of f . In all the cases Ie is a finite union of closed intervals. In part 1 below we show that for every y in J(f ) there is a constant Cy > 0 and an interval Jy contained in Ie that is a neighborhood of y in Ie and such that for every integer n ≥ 1 and every pull-back W of Jy by f n we have b−n . |W | ≤ Cy λ 0

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Since J(f ) is compact, this implies the proposition, except in the case where J(f ) is an interval having a boundary point in the interior of I that is a periodic point of f . This last case is treated in part 2. Let Ib be a compact interval containing I in its interior and let fb: Ib → Ib be an extension of f in A that is boundary anchored, such that all the critical points of fb S+∞ are contained in I, and such that n=0 fb−n (I) contains an interval whose closure contains J(fb). Note in particular that fb is essentially topologically exact on J(fb). Without loss of generality we assume that all the periodic points of fb in J(fb) are b0 )1/2 and let δ∗ > 0 be the constant δ(η) given hyperbolic repelling. Put η := (λ/λ by Lemma 3.3 with f replaced by fb. Moreover, let C1 > 1 be the constant given by Lemma 3.2. Reducing δ∗ if necessary we assume b δ∗ < C1−1 dist(I, ∂ I).

Note that this implies that for every interval J intersecting I and satisfying |J| ≤ δ∗ , b we have C1 J ⊂ I. 1. Suppose first y is not a boundary point of a Fatou component of f of length greater than or equal to δ∗ /2. Since f is topologically exact on J(f ), we can find an integer n0 ≥ 1 and points x and x′ in f −n0 (x0 ) such that x < y < x′ and |x − x′ | < δ∗ . Then the desired assertion follows with −1 Jy = (x, x′ ) and Cy = 12ℓd δ∗ , max C

by Lemma 3.3(2) with f replaced by fb and with Jb = (x, x′ ). Suppose y is a boundary point of a Fatou component of f and that y is not periodic. Then there is an integer N ≥ 1 such that every point in f −N (y) is either not in the boundary of a Fatou component or in the boundary of a Fatou component of length strictly smaller than δ∗ /2. Then the desired assertion follows from the previous case. It remains to consider the case where y is a periodic point in the boundary of a Fatou component of length greater than or equal to δ∗ /2. Let ℓ ≥ 1 be the period of y and let δ in (0, δ∗ /2) be suﬃciently small so that there is an inverse φ of fbℓ defined on B(y, δ), fixing y and such that φ(B(y, δ)) ⊂ B(y, δ). Since δ < δ∗ /2 and y is a boundary point of a Fatou component of f of length greater than or equal to δ∗ /2, it follows that φ is strictly increasing. Let n0 ≥ 1 be a suﬃciently large integer so that f −n0 (x0 ) intersects B(y, δ) and let y0 be a point of f −n0 (x0 ) in B(y, δ). For each integer j ≥ 1 put yj := φj (y0 ) and let Kj−1 be the closed interval bounded by yj−1 and yj . Note that the intervals (Kj )+∞ j=0 have pairwise disjoint interiors and that the closure of their union is equal to the e closed interval Jy bounded by y and y0 . Clearly Jy is a neighborhood of y in I. j On the other hand, for each integer j ≥ 1 the interval Kj is equal to φ (K0 ) and it is a pull-back of K0 by fbℓj . So, Lemma 3.3(2) with Jb = K0 , with f replaced SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2020

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by fb, and with n replaced by n + ℓj, shows that for every pull-back W of Kj by fbn we have

n o−1 n+jℓ |W | ≤ 12ℓd |K0 | max |Dfbn+jℓ (a)| : a ∈ ∂W max η n o n+jℓ ≤ 12ℓd δ∗ C −1 λ−(n+jℓ+n0 ) min |Dfbn0 (y0 )|−1 , |Dfbn0 +ℓ (y1 )|−1 . max η

On the other hand, by Lemma 3.3(1) with f replaced by fb and with Jb = Jy c of Jy by f n contains at most 2η n pull-backs and J = Kj , every pull-back W n of Kj by f . So, letting n o −1 −n0 C ′ := 12ℓd λ min |Dfbn0 (y0 )|−1 , |Dfbn0 +ℓ (y1 )|−1 max δ∗ C and using the definition of η we obtain

b−(n+jℓ) . c ∩ fb−n (Kj )| ≤ 2η n C ′ η n+jℓ λ−(n+jℓ) ≤ 2C ′ λ |W 0

Since Jy is the closure of c | ≤ 2C ′ |W

S

j≥0

+∞ X j=0

Kj , summing over j we get

b−n . b−(n+jℓ) = 2C ′ (1 − λ b−ℓ )−1 λ λ 0 0 0

b−ℓ )−1 . This proves the desired assertion with Cy = 2C ′ (1 − λ 0

2. Suppose that J(f ) is an interval having a boundary point y in the interior of I that is a periodic point of f . In view of part 1, it is enough to show that for each such point y there are δ > 0 and C > 0 such that for every integer n ≥ 1 b−n . By part 1 there and every pull-back W of B(y, δ) by f n , we have |W | ≤ C λ 0 are δ > 0 and C > 0 such that this property holds with B(y, δ) replaced by the interval J := B(y, δ) ∩ J(f ). Let O be the forward orbit of y. Note that O ⊂ ∂I, that the set O ′ := f −1 (O ) ∩ ∂J(f ) is forward invariant, and that f −1 (O ′ ) \ O ′ is contained in the interior of J(f ). Reducing δ if necessary assume that each pull-back of B(y, δ) by f or by f 2 that is disjoint from O ′ is contained in J(f ). It follows that for every integer n ≥ 1, each pull-back W of B(y, δ) by f n that is disjoint from O ′ is contained in J(f ) and therefore coincides with a pull-back of J by f n . By our b−n . It remains to consider those pullchoice of δ, in this case we have |W | ≤ C λ 0 ′ backs W of B(y, δ) that intersect O . Since by Lemma 3.4 the periodic point y satisfies χy (f ) ≥ ln λ, reducing δ if necessary we can assume that for every integer n ≥ 1 and every pull-back W of B(y, δ) by f n that intersects O ′ , we b−n . have |W | ≤ C λ 0

This completes the proof of the proposition.

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4. Quantifying asymptotic expansion The purpose of this section is to prove the following strengthened version of the Main Theorem. Given a compact space X and a continuous map T : X → X, we denote by M (X, T ) the space of Borel probability measures on X that are invariant by T . Main Theorem′ . — For an interval map f in A , the number χper (f ) := inf {χp (f ) : p hyperbolic repelling periodic point of f in J(f )} is equal to χinf (f ) := {χν (f ) : ν ∈ M (J(f ), f )} . If in addition f is topologically exact on J(f ), then there is δ ′ > 0 such that the following properties hold. Let J be an interval contained in I that intersects J(f ) and satisfies |J| ≤ δ ′ . In the case where χinf (f ) > 0 and where J(f ) is not an interval, assume in addition that J is not a neighborhood of a periodic point in the boundary of a Fatou component of f . Then: 1. For every χ < χinf (f ) there is a constant C > 0 independent of J, such that for every integer n ≥ 1 and every pull-back W of J by f n , we have |W | ≤ C exp(−nχ). 2. We have lim

n→+∞

1 ln max |W | : W connected component of f −n (J) = −χinf (f ). n

Finally, for each point x0 in J(f ) we have 1 lim sup ln min |Df n (x)| : x ∈ f −n (x0 ) ≤ χinf (f ), n n→+∞

and there is a subset E of J(f ) of zero Hausdorﬀ dimension such that for each point x0 in J(f ) \ E the lim sup above is a limit and the inequality an equality. Remark 4.1. — In the case where χinf (f ) > 0 and where J(f ) is not an interval, there is an example showing that the hypothesis in the Main Theorem′ that J is not a neighborhood of a periodic point in the boundary of a Fatou component, is necessary, see [36, Proposition A]. However, a qualitative result holds when this hypothesis is not satisfied, see [36, Theorem B]. The proof of the Main Theorem′ is given below, after the following lemmas from [33]. When f is a complex rational map the following lemma is a direct consequence of [33, Lemma 3.1]. Using Fact 2.2, the proof applies without change to the case where f is a map in A . Lemma 4.2. — Let f be an interval map in A that is topologically exact on J(f ) and such that χper (f ) > 0. Then there is a point x0 in J(f ) such that 1 lim inf ln min |Df n (x)| : x ∈ f −n (x0 ) ≥ χper (f ). n→+∞ n SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2020

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In the case where f is a complex rational map, the following is [33, Lemma 2.1 and Remark 2.2]. The proof applies without change to maps in A . Lemma 4.3. — Let f : I → I be a map in A . Then there are δ4 > 0 and a subset E of I of zero Hausdorﬀ dimension, such that for every interval J contained in I that intersects J(f ) and satisfies |J| ≤ δ4 and every point x0 in J \ E, we have lim inf

n→+∞

1 ln min |Df n (x)| : x ∈ f −n (x0 ) n 1 ≥ − lim sup ln max |W | : W connected component of f −n (J) . n→+∞ n

Proof of the Main Theorem′ . — To prove (4.1)

χinf (f ) = χper (f ),

suppose f is “infinitely renormalizable,” see [24] for the definition and for precisions. It follows easily from the a priori bounds in [41] that in this case we have χinf (f ) = χper (f ) = 0. So, to prove (4.1) it is enough to consider the case where f is at most finitely renormalizable. Then f can be decomposed into finitely many interval maps, each of which has a renormalization with a topologically exact restriction, see for example [24, §III, 4]. Thus, to prove the Main Theorem′ it is enough to consider the case where f is topologically exact. In part 1 below we prove item 1 of the theorem with χinf (f ) replaced by χper (f ) and in part 2 we prove χper (f ) = χinf (f ). We complete the proof of the theorem in part 3. 1. We prove item 1 of the theorem with χinf (f ) replaced by χper (f ). This statement being trivial in the case where χper (f ) = 0, we suppose χper (f ) > 0. Combining Lemma 4.2 and Proposition 3.1 we obtain that all the periodic points of f in J(f ) are hyperbolic repelling and that for every χ in (0, χper (f )) there is δ(χ) > 0 such that for every interval J that intersects J(f ), that is disjoint from each periodic Fatou component of f , and that satisfies |J| ≤ δ(χ), the following property holds: For every integer n ≥ 1 and every pull-back W of J by f n we have |W | ≤ exp(−nχ). Put δ ′ := δ(χper (f )/2) and let J be an interval that intersects J(f ), that is disjoint from the periodic Fatou components of f , and that satisfies |J| ≤ δ ′ . Given χ in (χper (f )/2, χper (f )), let N ≥ 1 be suﬃciently large so that exp(−N χ) ≤ δ(χ), let n ≥ N be an integer, and let W be a pull-back of J by f n . If we denote by W ′ the pull-back of J by f N containing f n−N (W ), then we have |W ′ | ≤ exp(−N χ) ≤ δ(χ). So the property above applied to W ′ instead of J implies |W | ≤ exp(−(n − N )χ).

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This proves item 1 of the theorem with C = exp(N χ) and with χinf (f ) replaced by χper (f ). 2. We prove χper (f ) = χinf (f ). To prove χper (f ) ≥ χinf (f ), let p be a hyperbolic repelling periodic point of f in J(f ) and let ν be the probability measure equidistributed on the orbit of p. Then ν is in M (J(f ), f ) and χν (f ) = χp (f ), so χp (f ) ≥ χinf (f ). This proves χper (f ) ≥ χinf (f ). To prove the reverse inequality we show that for every ν in M (J(f ), f ) we have χν (f ) ≥ χper (f ). By the ergodic decomposition theorem we can assume without loss of generality that ν is ergodic. By [30, Theorem B] or by Proposition A.1 in Appendix A, we have χν (f ) ≥ 0. We show that for every ε > 0 there is a point x in J(f ) such that for every suﬃciently large integer n ≥ 1 we have f n (B(x, exp(−(χν (f ) + 2ε)n))) ⊂ B(f n (x), exp(−εn)).

(4.2)

Using this estimate with a suﬃciently large n and combining it with part 1 we obtain χν (f ) + 2ε ≥ χper (f ). Since ν and ε are arbitrary, this proves χinf (f ) ≥ χper (f ), as wanted. To prove (4.2), note that by Birkhoﬀ’s ergodic theorem there is a point x0 in J(f ) and an integer n0 ≥ 1 such that for every n ≥ n0 we have (4.3) exp χν (f ) − 31 ε n ≤ |Df n (x0 )| ≤ exp χν (f ) + 31 ε n . On the other hand, since the critical points of f are non-ﬂat, there are constants C0 > 0 and α > 0 such that for every x in I we have |Df (x)| ≤ C0 dist(x, Crit(f ))α . Put ε′ :=

ε α.

Using the previous inequality with x = f n (x0 ), combined with Df n+1 (x0 ) = Df (f n (x0 )) · Df n (x0 ),

with (4.3) and with (4.3) with n replaced by n+1, we obtain that for every n ≥ n0 we have 1 dist(f n (x), Crit(f )) ≥ C0−1 exp(χν (f )) α exp − 23 ε′ (n + 1) .

This implies that there is an integer n1 ≥ n0 such that for every n ≥ n1 the 1 ′ n ′ distortion of f on B(f (x0 ), exp(−ε n)) is bounded by exp 3 ε . Let n2 ≥ n1 be suﬃciently large so that the distortion of f n1 on B(x0 , exp(−(χν (f ) + ε′ )n2 )) is bounded by exp 13 ε′ n1 . Then for every n ≥ n2 we have,

(4.4) f n1 (B(x0 , exp(−(χν (f ) + 2ε′ )n)))

⊂ B f n1 (x0 ), exp −(χν (f ) + 2ε′ )n + 13 ε′ n1 |Df n1 (x0 )| .

Fix n ≥ n2 . We prove by induction that for every j in {n1 , . . . , n} the inclusion above holds with n1 replaced by j. The desired assertion is obtained from this with j = n, combined with (4.3). Noting that the case j = n1 is given by (4.4) itself, let j in {n1 , . . . , n−1} be given and suppose (4.4) holds with n1 replaced by j. Then (4.4) with n1 replaced by j +1 is obtained by using that the right hand side

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of (4.4) with n1 replaced by j is contained in B(f j (x0 ), exp(−ε′ n)), combined with the fact that the distortion of f on this last set is bounded by exp 31 ε′ . This completes the proof of the induction step, and hence that χν (f ) ≥ χper (f ) and χinf (f ) = χper (f ). 3. So far we have shown item 1 of the theorem and the equality χinf (f ) = χper (f ). Let χ0per (f ) be as in the statement of Lemma 3.4. Clearly, χinf (f ) ≤ χ0per (f ) ≤ χper (f ) (cf., first part of part 2), so χ0per (f ) = χinf (f ). Thus, inequality (3.2) of Lemma 3.4 and item 1 of the theorem imply item 2 of the theorem. In turn, item 2 of the theorem together with (3.3) of Lemma 3.4 and with Lemma 4.3 imply the last assertion of the theorem. The proof of the theorem is thus complete. 5. Conjugacy to a piecewise affine map In this section we show that a conjugacy between 2 Lipschitz continuous multimodal maps that satisfy the Exponential Shrinking of Components condition (9) is bi-Hölder continuous (Proposition 5.2). Combined with Lemma 5.1 below, this proves implication 5 ⇒ 4 of Corollary A. A multimodal map f is expanding, if there is λ > 1 so that for every x and x′ contained in an interval on which f is monotonous, we have |f (x) − f (x′ )| ≥ λ|x − x′ |. In this case we say λ is an expansion constant of f . Lemma 5.1. — Every expanding multimodal map satisfies the Exponential Shrinking of Components condition. In this section, a turning point c of a multimodal map f is exposed if for every integer n ≥ 1 the point f n (c) is not a turning point of f . Proof. — Let f : I → I be an expanding multimodal map and let λ > 1 be an expansion constant of f . Let L ≥ 1 be a suﬃciently large integer so that λL > 2 and let δ† > 0 be suﬃciently small so that for every exposed turning point c of f and every j in {1, . . . , L} the set f j (B(c, δ† )) does not contain a turning point of f . Let δ∗ > 0 be suﬃciently small so that for every interval J contained in I that satisfies |J| ≤ δ∗ and every connected component W of f −1 (J) we have |W | ≤ δ† . We prove by induction on n ≥ 0 that for every interval J contained in I that satisfies |J| ≤ δ∗ /2, every j in {1, . . . , n}, and every pull-back W of J by f j we have 1 j |W | ≤ 2 L λ−1 δ∗ . (9)

The Exponential Shrinking of Components condition is defined in §1.2 for non-degenerate smooth interval maps. In this section we apply this definition to multimodal maps.

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This implies that f satisfies the Exponential Shrinking of Components condition. The case n = 0 being trivial, suppose that for some n ≥ 1 this assertion holds with n replaced by each element of {0, . . . , n − 1}. Let J be an interval contained in I that satisfies |J| ≤ δ∗ /2 and let W be a pull-back of J by f n . The induction hypothesis implies for every j in {1, . . . , n − 1} we have |f j (W )| ≤ δ∗ . Using the hypothesis |J| ≤ δ∗ /2 and the definition of δ∗ , we conclude that for every i in {0, . . . , n−1} we have |f i (W )| ≤ δ† . Using the definition of δ† , this implies that the number of those i in {0, . . . , n − 1} such that f i (W ) contains a turning point of f in its interior is at n n most L + 1. It thus follows that W can be partitioned into at most 2 L +1 intervals on n each of which f is injective. Using that λ is an expansion constant of f , we obtain n

n

|W | ≤ 2 L +1 λ−n |J| ≤ 2 L λ−n δ∗ . This completes the proof of the induction hypothesis and of the lemma. Proposition 5.2. — Let f : I → I be a Lipschitz continuous multimodal map and fe: Ie → Ie a multimodal map satisfying the Exponential Shrinking of Components condition. If h : I → Ie is a homeomorphism conjugating f to fe, then h is Hölder continuous. We deduce this proposition as an easy consequence of the following lemma.

Lemma 5.3. — Let f : I → I be a multimodal map satisfying the Exponential Shrinking of Components condition with constant λ > 1. Then for every A > (ln λ)−1 there is a constant δ5 > 0 such that for every interval J contained in I the following property holds: There is an integer m ≥ 0 that satisfies m ≤ max{−A ln |J|, 0} and an interval J0 contained in J, such that f m is injective on J0 and |f m (J0 )| ≥ δ5 . Proof. — Put χ := ln λ and let L be an integer satisfying L > (Aχ − 1)−1 A ln 2. Let δ† > 0 be suﬃciently small so that for every exposed turning point c of f and for every j in {1, . . . , L}, the set f j (B(c, δ† )) does not contain a turning point of f . Let δExp > 0 be the constant δ given by the Exponential Shrinking of Components condition, see §1.2. Reducing δExp if necessary we assume that for every interval J contained in I that satisfies |J| ≤ δExp , every integer n ≥ 1, and every pull-back W ∗ of J by f n we have |W | ≤ δ† . Let δExp > 0 be such that for every interval J contained in I that satisfies |J| ≥ δExp and for every connected component W of f −1 (J) we ∗ ∗ ∗ have |W | ≥ δExp . Reducing δExp if necessary we assume δExp ≤ δExp . Observing ln 2 that 1 + A L < χA, it follows that there is n0 ≥ 1 such that for every integer n ≥ n0 we have, ∗ δExp ln 2 (5.1) − A ln n ≤ χAn. + 1+A 2 L In part 1 below we show that every interval contains an interval that is mapped bijectively by an iterate of f onto a relatively large interval. In part 2 we use this fact to prove the lemma by induction.

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1. We prove that for every integer n ≥ 1 and every interval J contained in I that satisfies |J| ≥ exp(−(n + 1)χ), there is m in {0, . . . , n} and an interval J0 contained in J such that f m is injective on J0 and |f m (J0 )| ≥

∗ δExp m 2− L . 2

If |J| ≥ δExp , then the assertion follows with J0 = J and m = 0 from our ∗ assumption that δExp ≥ δExp . Assume |J| ≤ δExp and note that by the Exponential Shrinking of Components condition, for every integer m ≥ n + 1 we have |f m (J)| > δExp . So there is a largest integer m ≥ 0 such that |f m (J)| ≤ ∗ ∗ δExp and m satisfies m ≤ n. By definition of δExp we have |f m (J)| ≥ δExp . On the other hand, by our choice of δExp , for every j in {0, . . . , m − 1} we have |f j (J)| ≤ δ† . From the definition of δ† it follows that the number of those j in {0, . . . , m − 1} such that f j (J) contains a turning point in its interior is m L +1 bounded by m L + 1. This implies that J can be partitioned into at most 2 m ′ intervals on which f is injective. So, if we denote by J0 an interval J in this partition for which |f m (J ′ )| is maximal, then we have |f m (J0 )| ≥

(5.2) δ∗

∗ δExp m |f m (J)| 2− L . ≥ m +1 2 2L

n0

− L . We prove by induction that for every integer n ≥ 1 the 2. Put δ5 := Exp 2 2 lemma holds for every interval J that satisfies |J| ≥ exp(−(n + 1)χ). Part 1 implies that this holds for every integer n ≥ 0 satisfying n ≤ n0 . Let n ≥ n0 be an integer for which the lemma holds for every interval J that satisfies |J| ≥ exp(−nχ). To prove the inductive step, let J be a given interval contained in I that satisfies

exp(−(n + 1)χ) ≤ |J| ≤ exp(−nχ). Let m be the integer in {0, . . . , n} and J0 the interval contained in J given by part 1. So f m is injective on J0 and |f m (J0 )| ≥

∗ ∗ δExp δExp m n 2− L ≥ 2− L . 2 2

Together with (5.1) this implies |f m (J0 )| ≥ exp(−nχ), so we can apply the induction hypothesis with J replaced by f m (J0 ). Therefore there is an interval J0′ contained in f m (J0 ) and an integer m′ ≥ 0 satisfying m′ ≤ max{−A ln |f m (J0 )|, 0}, ′ ′ such that f m is injective on J0′ and |f m (J0′ )| ≥ δ5 . If m′ = 0, then |f m (J0 )| ≥ |J0′ | ≥ δ5 . Together with m ≤ n ≤ −χ−1 ln |J| < −A ln |J|, this completes the proof of the induction step in the case where m′ = 0. Suppose m′ ≥ 1 and let Je0 be the connected component of f −m (J0′ ) contained in J0 , ASTÉRISQUE 416

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′ so that f m is injective on Je0 and f m (Je0 ) = J0′ . Then f m+m is injective on Je0 ′ ′ and |f m+m (Je0 )| = |f m (J0′ )| ≥ δ5 . On the other hand, we have by (5.1) and (5.2) ∗ δExp ln 2 ′ m m + m ≤ m − A ln |f (J0 )| ≤ −A ln m ≤ χAn ≤ −A ln |J|. + 1+A 2 L

This completes the proof of the induction step with m replaced by m + m′ and J0 replaced by Je0 .

The proof of the lemma is thus complete.

Proof of Proposition 5.2. — Denote by M a Lipschitz constant of f , let A and δ5 be as in Lemma 5.3 with f replaced by fe and let δ5∗ > 0 be such that for every interval J ∗ contained in Ie that satisfies |J ∗ | ≥ δ5 , we have |h−1 (J ∗ )| ≥ δ5∗ . To prove that h is Hölder continuous, let J be an interval contained in I and let m ≥ 0 be the integer and J0 the interval given by Lemma 5.3 with J replaced by h(J), so that m ≤ max{−A ln |h(J)|, 0}, J0 ⊂ h(J), |fem (J0 )| ≥ δ5 , and so that fem is injective on J0 . It follows that f m is injective on h−1 (J0 ), so by the

definition of δ5∗ we have

|J| ≥ |h−1 (J0 )| ≥ M −m |h−1 (fem (J0 ))| ≥ min{|h(J)|A ln M , 1} · δ5∗ .

This proves that h is Hölder continuous of exponent (A ln M )−1 . 6. Nonuniform hyperbolicity conditions

The purpose of this section is to prove Corollaries A, D and E. Proof of Corollary A. — To prove that conditions 1–7 are equivalent, remark first that the equivalence between conditions 1, 2, 5 and 6 is given by the Main Theorem′ , using Fact 2.2 for the implication 5 ⇒ 6. When f is a complex rational map, the implication 5 ⇒ 3 is [31, Theorem C]. The proof applies without change to the case where f is a non-degenerate smooth interval map that is topologically exact. (10) When f is unicritical, the implication 3 ⇒ 2 is [28, Lemma 8.2]. The proof applies without change to the general case. We complete the proof that conditions 1–6 are equivalent by showing the implications 5 ⇒ 4 and 4 ⇒ 2. For the implication 5 ⇒ 4, recall that by the general theory of Parry [29] and of Milnor and Thurston [26], the map f is conjugated to a piecewise aﬃne expanding map. That the conjugacy is biHölder follows from the combination of Lemma 5.1 and Proposition 5.2. When f is unicritical, the implication 4 ⇒ 2 is [28, Lemma 8.4]. The proof applies without change to the general case. This completes the proof that conditions 1–6 are equivalent. (10)

For a proof written for maps in A , see [37, Corollary 2.19]. If in addition f satisfies ColletEckmann condition and J(f ) = I, see also [16, 45] if f is unicritical, [6] if all the critical points of f are of the same order and [12, Theorem 6] if f is real analytic.

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To complete the proof that conditions 1–7 are equivalent, we prove that condition 7 is equivalent to condition 4. First notice that the conjugacy h : I → [0, 1] to the piecewise aﬃne model is Hölder continuous by Lemma 5.1 and Proposition 5.2. Thus condition 4 is equivalent to the condition that h−1 is Hölder continuous. The conjugacy h is defined in terms of its unique maximal entropy measure ρf , as follows: If we denote by a the left end point of I, then for every x in I we have h(x) = ρf ([a, x]). Thus, it readily follows that condition 4 is equivalent condition 7. To prove the final statement, note that the Backward Collet-Eckmann condition implies condition 6 trivially. On the other hand, the Collet-Eckmann condition implies condition 2 by [7, Corollary 1.1]. Remark 6.1. — Conditions 1, 2, 5 and 6 of Corollary A have natural formulations for maps in A . The Main Theorem′ implies that, for maps that are essentially topologically exact on their Julia sets, these conditions are equivalent, using Fact 2.2 for the implication 5 ⇒ 6. Using conformal measures, a condition analogous to condition 3 of Corollary A can also be stated for a general interval map in A . Our results imply that in this more general setting condition 3 is equivalent to conditions 1, 2, 5 and 6. In fact, the implication 5 ⇒ 3 is again given by either [31, Theorem C] or [37, Corollary 2.19]. The proof of the implication 3 ⇒ 2 for unicritical maps in [28, Lemma 8.2] does not apply directly to this more general setting, as it uses that the reference measure is the Lebesgue measure. Using Frostman’s lemma, the argument can be adapted to deal with the case where the reference measure is a conformal measure, as in [31, Theorem D] for complex rational maps. Remark 6.2. — Both, the Collet-Eckmann and the Backward Collet-Eckmann condition have natural formulations for maps in A . In this more general setting each of these conditions implies conditions 1–3, 5, and 6 of Corollary A, see Remark 6.1. In fact, the Backward Collet-Eckmann condition implies condition 6 trivially and the Collet-Eckmann condition implies condition 2 by [7, Corollary 1.1]. We note also that for a map in A the Collet-Eckmann condition implies the Backward Collet-Eckmann condition at each critical point of maximal order: For complex rational maps this is given by [13, Theorem 1]; the proof applies without change to maps in A . (11) Proof of Corollary D. — We show that for a non-degenerate smooth map f : I → I having only hyperbolic repelling periodic points, an iterate of f has an exponentially mixing acip if and only if: (*) There is an interval J contained in I and an integer s ≥ 1, such that f s (J) ⊂ J and such that f s : J → J is a topologically exact map that satisfies the TCE condition. Since (*) is clearly invariant under topological conjugacy preserving critical points, this implies the corollary. (11)

In fact, the proof for maps A is slightly simpler, as the arguments involving shrinking neighborhoods can be replaced by the one-sided Koebe principle.

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If (*) is satisfied, then f s | is non-injective and therefore it is a non-degenerate J smooth interval map. Then Corollary C implies that f s | , and hence f s , has an J exponentially mixing acip. Suppose there is an integer s ≥ 1 such that f s has an exponentially mixing acip ν, and denote by J the support of ν. Then J is an interval, f s (J) ⊂ J, and f s | is J topologically exact, see [41, Theorem E(2)]. It follows that fJs is non-injective and therefore that f s | is a non-degenerate smooth interval map. Thus Corollary C implies J that f s | satisfies the TCE condition. This proves that f satisfies (*), and completes J the proof of the corollary. Remark 6.3. — The proof of Corollary D applies without change to maps in A . Proof of Corollary E. — Denote by I the domain of f . Recall from §1.5 that P in nonincreasing, that it has at least one zero, and that its first zero t0 is in (0, 1]. The implication 2 ⇒ 1 is trivial, and the implication 2 ⇒ 3 is a direct consequence of the fact that P is nonincreasing. Since P has at least one zero, the implication 3 ⇒ 2 also follows from the fact that P is nonincreasing. To prove the implication 2 ⇒ 4, suppose 2 holds. Since the first zero of P is in (0, 1], we have P (2) = 0. So for each χ > 0 there is an ergodic measure ν in M (I, f ) satisfying hν (f ) − 2χν (f ) ≥ −χ. By [30, Theorem B] or Proposition A.1, we have χν (f ) ≥ 0. Combined with Ruelle’s inequality hν (f ) ≤ max{0, χν (f )} = χν (f ), see [38], we obtain 2χν (f ) ≤ hν (f ) + χ ≤ χν (f ) + χ and χν (f ) ≤ χ. Since χ is arbitrary, this shows that χinf (f ) = 0 and completes the proof of the implication 2 ⇒ 4. To prove the implication 4 ⇒ 3, suppose χinf (f ) = 0, and let t > t0 and χ > 0 be given. Then there is a measure ν in M (I, f ) such that χν (f ) < χ, so P (t) ≥ hν (f ) − tχν (f ) ≥ −tχ. Since χ > 0 is arbitrary we conclude that P (t) ≥ 0 and hence that P is nonnegative. We complete the proof of the corollary by showing the implication 1 ⇒ 4. Suppose χinf (f ) > 0, so that t+ := sup{t > 0 : P (t) > −tχinf (f )} satisfies t+ > t0 . By [32, Theorem A] the function P is real analytic on (0, t+ ), and hence at t = t0 . This proves that f does not have a high-temperature phase transition, and completes the proof of the implication 1 ⇒ 4 and of the corollary. Remark 6.4. — Each of the conditions 1–4 of Corollary E have natural formulations in the case where f is an interval map in A . The proof of Corollary E applies without change in this more general setting.

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Appendix A Lyapunov exponents are nonnegative In this appendix we prove the following general result characterizing those invariant measures whose Lyapunov exponent is strictly negative (possibly infinite). For smooth interval maps with a finite number of non-ﬂat critical points, this was shown by Przytycki in [30, Theorem B]. We give a proof of this important fact that avoids the Koebe principle and applies to continuously diﬀerentiable maps. It is considerably shorter than the proof in [30] and extends without change to complex rational maps. For a continuously diﬀerentiable interval map f , a periodic orbit of f of period n is strictly attracting, if for each point p in this orbit |Df n (p)| < 1. For a Borel measure ν on a topological space X, we use supp(ν) to denote the support of ν, which is by definition the set of all points in X such that the measure of each of its neighborhoods is strictly positive. Proposition A.1. — Let f be a continuously diﬀerentiable interval map and let ν be an ergodic invariant probability measure. Then either χν (f ) ≥ 0 or ν is supported on a strictly attracting periodic orbit of f . Proof. — Suppose χν (f ) < 0. By the dominated convergence theorem there exists L > 0 such that the function ϕ := max{ln |Df |, −L} satisfies A :=

R

ϕ dν < 0. Fix χ in (0, −A/3) and for each integer n ≥ 1 put Sn (ϕ) := ϕ + ϕ ◦ f + · · · + ϕ ◦ f n−1 .

1. We show that for every point x in the domain I of f satisfying lim 1 Sn (ϕ)(x) n→+∞ n

= A,

there exists τ > 0 such that for every suﬃciently large integer n we have |Df n | ≤ exp(−χn) on B(x, τ ). Fix such x in I and let δ > 0 be such that we have |Df | ≤ exp(−L) on B(Crit(f ), δ). As f is continuously diﬀerentiable there is ε in (0, δ/3) such that the distortion of f on an interval of length at most ε and disjoint from B(Crit(f ), δ/3) is at most exp(χ). By our choice of χ there is τ > 0 so that for every n ≥ 0 we have τ exp(Sn (ϕ)(x) + 3nχ) < ε/2. Finally, for each n ≥ 0 put rn := τ exp(Sn (ϕ)(x) + nχ) and Bn := B(f n (x), rn ). Note that we have |Bn | = 2rn ≤ ε exp(−2nχ). We show that for every n ≥ 0 we have |Df | ≤ exp(ϕ(f n (x)) + χ) on Bn . This implies that f (Bn ) ⊂ Bn+1 and by induction that on B(x, τ ) we have |Df n | ≤ exp(Sn (ϕ)(x) + χn) ≤ τ −1 (ε/2) exp(−2nχ).

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It then follows that for large n we have |Df n | ≤ exp(−χn) on B(x, τ ), as wanted. Case 1. — f n (x) 6∈ B(Crit(f ), 2δ/3). Since the length of Bn is less than ε < δ/3, it follows that the interval Bn is disjoint from B(Crit(f ), δ/3) and that the distortion of f on Bn is bounded by exp(χ). So on Bn we have |Df | ≤ |Df (f n (x))| exp(χ) ≤ exp(ϕ(f n (x)) + χ). Case 2. — f n (x) ∈ B(Crit(f ), 2δ/3). Then Bn ⊂ B(Crit(f ), δ) and by our choice of δ we have |Df | ≤ exp(−L) on Bn . 2. By Birkhoﬀ’s ergodic theorem the set of points x satisfying the property described in part 1 has full measure with respect to ν. We can thus find such a point x in supp(ν), such that in addition its orbit is dense in supp(ν). Let τ > 0 be given by the property described in part 1 for this choice of x. Then there is an integer n ≥ 1 such that |Df n | ≤ exp(−nχ) ≤ 14 on B(x, τ ) and such that f n (x) is in B(x, τ /4). Then f n (B(x, τ )) ⊂ B(f n (x), τ /2) and f n is uniformly contracting on B(x, τ ). This implies that x is asymptotic to a strictly attracting periodic point of f . Since x is in supp(ν) and ν is ergodic, it follows that ν is supported on a strictly attracting periodic orbit of f . References [1] J. F. Alves, J. M. Freitas, S. Luzzatto & S. Vaienti – “From rates of mixing to recurrence times via large deviations”, Adv. Math. 228 (2011), p. 1203–1236. [2] M. Aspenberg – “The Collet-Eckmann condition for rational functions on the Riemann sphere”, Math. Z. 273 (2013), p. 935–980. [3] A. Avila & C. G. Moreira – “Statistical properties of unimodal maps: the quadratic family”, Ann. of Math. 161 (2005), p. 831–881. [4] V. Baladi – Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, vol. 16, World Scientific Publishing Co., River Edge, NJ, 2000. [5] M. Benedicks & L. Carleson – “On iterations of 1 − ax2 on (−1, 1)”, Ann. of Math. 122 (1985), p. 1–25. [6] H. Bruin, S. Luzzatto & S. Van Strien – “Decay of correlations in one-dimensional dynamics”, Ann. Sci. École Norm. Sup. 36 (2003), p. 621–646. [7] H. Bruin & S. van Strien – “Expansion of derivatives in one-dimensional dynamics”, Israel J. Math. 137 (2003), p. 223–263. [8] D. Coronel & J. Rivera-Letelier – “Low-temperature phase transitions in the quadratic family”, Adv. Math. 248 (2013), p. 453–494. [9]

, “High-order phase transitions in the quadratic family”, J. Eur. Math. Soc. (JEMS) 17 (2015), p. 2725–2761.

[10] B. Gao & W. Shen – “Summability implies Collet-Eckmann almost surely”, Ergodic Theory Dynam. Systems 34 (2014), p. 1184–1209.

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[11] S. Gouëzel – “Berry-Esseen theorem and local limit theorem for non uniformly expanding maps”, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), p. 997–1024. [12] J. Graczyk & S. Smirnov – “Non-uniform hyperbolicity in complex dynamics”, Invent. math. 175 (2009), p. 335–415. [13] J. Graczyk & S. Smirnov – “Collet, Eckmann and Hölder”, Invent. math. 133 (1998), p. 69–96. [14] J. Graczyk & G. Świa¸tek – “Harmonic measure and expansion on the boundary of the connectedness locus”, Invent. math. 142 (2000), p. 605–629. [15] M. V. Jakobson – “Absolutely continuous invariant measures for one-parameter families of one-dimensional maps”, Comm. Math. Phys. 81 (1981), p. 39–88. [16] G. Keller & T. Nowicki – “Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps”, Comm. Math. Phys. 149 (1992), p. 31–69. [17] H. Li – “Topological invariance of the Collet-Eckmann condition for one-dimensional maps”, Nonlinearity 30 (2017), p. 2010–2022. [18] H. Li & W. Shen – “Topological invariance of a strong summability condition in one-dimensional dynamics”, Int. Math. Res. Not. 2013 (2013), p. 1783–1799. [19] S. Luzzatto – “Stochastic-like behaviour in nonuniformly expanding maps”, in Handbook of dynamical systems. Vol. 1B, Elsevier B. V., 2006, p. 265–326. [20] S. Luzzatto & L. Wang – “Topological invariance of generic non-uniformly expanding multimodal maps”, Math. Res. Lett. 13 (2006), p. 343–357. [21] M. Lyubich – “Almost every real quadratic map is either regular or stochastic”, Ann. of Math. 156 (2002), p. 1–78. [22] I. Melbourne & M. Nicol – “Almost sure invariance principle for nonuniformly hyperbolic systems”, Comm. Math. Phys. 260 (2005), p. 131–146. [23]

, “A vector-valued almost sure invariance principle for hyperbolic dynamical systems”, Ann. Probab. 37 (2009), p. 478–505.

[24] W. de Melo & S. van Strien – One-dimensional dynamics, Ergebn. Math. und ihrer Grenzg., vol. 25, Springer, 1993. [25] N. Mihalache – “Two counterexamples in rational and interval dynamics”, preprint arXiv:0810.1474. [26] J. Milnor & W. Thurston – “On iterated maps of the interval”, in Dynamical systems (College Park, MD, 1986–87), Lecture Notes in Math., vol. 1342, Springer, 1988, p. 465– 563. [27] T. Nowicki & F. Przytycki – “Topological invariance of the Collet-Eckmann property for S-unimodal maps”, Fund. Math. 155 (1998), p. 33–43. [28] T. Nowicki & D. Sands – “Non-uniform hyperbolicity and universal bounds for Sunimodal maps”, Invent. math. 132 (1998), p. 633–680. [29] W. Parry – “Symbolic dynamics and transformations of the unit interval”, Trans. Amer. Math. Soc. 122 (1966), p. 368–378. [30] F. Przytycki – “Lyapunov characteristic exponents are nonnegative”, Proc. Amer. Math. Soc. 119 (1993), p. 309–317. [31] F. Przytycki & J. Rivera-Letelier – “Statistical properties of topological ColletEckmann maps”, Ann. Sci. École Norm. Sup. 40 (2007), p. 135–178.

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, Geometric pressure for multimodal maps of the interval, Mem. Amer. Math. Soc., vol. 259, 2019.

[33] F. Przytycki, J. Rivera-Letelier & S. Smirnov – “Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps”, Invent. math. 151 (2003), p. 29–63. [34] M. Rees – “Positive measure sets of ergodic rational maps”, Ann. Sci. École Norm. Sup. 19 (1986), p. 383–407. [35] J. Rivera-Letelier – “The maximal entropy measure detects non-uniform hyperbolicity”, Math. Res. Lett. 17 (2010), p. 851–866. [36]

, “On the asymptotic expansion of maps with disconnected Julia set”, preprint arXiv:1206.2376.

[37] J. Rivera-Letelier & W. Shen – “Statistical properties of one-dimensional maps under weak hyperbolicity assumptions”, Ann. Sci. Éc. Norm. Supér. 47 (2014), p. 1027– 1083. [38] D. Ruelle – “An inequality for the entropy of diﬀerentiable maps”, Bol. Soc. Brasil. Mat. 9 (1978), p. 83–87. [39] W. Shen – “On stochastic stability of non-uniformly expanding interval maps”, Proc. Lond. Math. Soc. 107 (2013), p. 1091–1134. [40] S. Smirnov – “Symbolic dynamics and Collet-Eckmann conditions”, Int. Math. Res. Not. 2000 (2000), p. 333–351. [41] S. van Strien & E. Vargas – “Real bounds, ergodicity and negative Schwarzian for multimodal maps”, J. Amer. Math. Soc. 17 (2004), p. 749–782. [42] M. Tsujii – “Absolutely continuous invariant measures for expanding piecewise linear maps”, Invent. math. 143 (2001), p. 349–373. [43] M. Tyran-Kamińska – “An invariance principle for maps with polynomial decay of correlations”, Comm. Math. Phys. 260 (2005), p. 1–15. [44] Q. Wang & L.-S. Young – “Nonuniformly expanding 1D maps”, Comm. Math. Phys. 264 (2006), p. 255–282. [45] L.-S. Young – “Decay of correlations for certain quadratic maps”, Comm. Math. Phys. 146 (1992), p. 123–138. [46]

, “Recurrence times and rates of mixing”, Israel J. Math. 110 (1999), p. 153–188.

J. Rivera-Letelier, Department of Mathematics, University of Rochester. Hylan Building, Rochester, NY 14627, U.S.A. • E-mail : [email protected]

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Astérisque 416, 2020, p. 65–132 doi:10.24033/ast.1111

ON ROTH TYPE CONDITIONS, DUALITY AND CENTRAL BIRKHOFF SUMS FOR I.E.M. by Stefano Marmi, Corinna Ulcigrai & Jean-Christophe Yoccoz

Abstract. — We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m.) and translation surfaces: the absolute Roth type condition is a weakening of the notion of Roth type i.e.m., while the dual Roth type condition is a condition on the backward rotation number of a translation surface. We show that results on the cohomological equation previously proved in [38] for restricted Roth type i.e.m. (on the solvability under finitely many obstructions and the regularity of the solutions) can be extended to restricted absolute Roth type i.e.m. Under the dual Roth type condition, we associate to a class of functions with subpolynomial deviations of ergodic averages (corresponding to relative homology classes) distributional limit shapes, which are constructed in a similar way to the limit shapes of Birkhoff sums associated in [36] to functions which correspond to positive Lyapunov exponents. 2010 Mathematics Subject Classification. — 37E05; 11K60, 37A20, 37E35. Key words and phrases. — Diophantine conditions, rotation number, Roth type, interval exchange maps, Birkhoff sums, deviation of ergodic averages. In August 2010 J.C.Y. wrote a first text (12 pages) containing the results obtained on dual Birkhoff sums with the title “On Birkhoff sums for i.e.m.” (a further version of the same text was written in March 2011). This constitutes the heart of Section 4 of this paper. Notable progress was made during a visit of S.M. and J.C.Y. to C.U. in Bristol in October 2014, when the homological interpretation emerged and J.C.Y. wrote a new version of this draft introducing the notion of KZ-hyperbolic translation surfaces (this homological interpretation is included in Section 6). Motivated by the discussions in Bristol, the notion of absolute Roth type i.e.m. emerged and was developed during further meetings in of S.M. with J.C.Y. in Paris in December 2014, March 2015 and September 2015. This notion, together with the completeness of backward rotation numbers, is the object of the text “Absolute Roth Type and Backward Rotation Number” (14 pages) written by J.C.Y. in April 2015. This text formed the basis for Section 3 and parts of it are here included as two Appendixes (Appendix A and Appendix B). The final version of this manuscript was prepared by S.M. and C.U. who are fully and solely responsible for any mistake or imprecision. Jean-Christophe discussed publicly the results obtained in our collaboration in his talk “Problèmes de petits diviseurs pour les échanges d’intervalles” on May 20, 2015 given at the “Journée Surfaces plates” held at the Institut Galilée of the University of Paris 13 (as Carlos Matheus informed us) and also during his talk “Diophantine conditions for interval exchange map” on September 28, 2015 given in Oxford as part of the Worshop Geometry and Dynamics on Moduli Spaces at the 2015 Clay Research conference.

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Résumé (Sur les conditions de type Roth, la dualité et les sommes de Birkhoff centrées pour les échange d’intervalles) Nous introduisons deux conditions diophantiennes pour les nombres de rotation des transformations d’échange d’intervalles (i.e.m.) et des surfaces de translation: la condition absolue de type Roth est un affaiblissement de la notion i.e.m de type Roth, tandis que la condition duale de type Roth est une condition sur le nombre de rotation en arrière d’une surface de translation. Nous montrons que les résultats sur l’équation cohomologique prouvés précédemment dans [38] pour les i.e.m. de type Roth restreint (sur la solvabilité en supposant un nombre fini d’obstructions et la régularité des solutions) peuvent être étendues aux i.e.m. de type Roth absolu restreint. Sous la condition duale de type Roth, nous associons des formes limites (limit shapes) distributionnelles à une classe de fonctions avec des déviations sous-polynomiales des moyennes ergodiques (correspondantes aux classes d’homologie relatives), qui sont construites de manière similaire aux formes limites des sommes de Birkhoff associées dans [36] aux fonctions qui correspondent aux exposants de Lyapunov positifs.

1. Introduction Diophantine conditions play a central role in the study of the dynamics of rotations of the circle, diffeomorphisms of the circle and more in general area-preserving flows on tori. These conditions, which convey information on how well a rotation number α can be approximated by rationals (and hence on small divisors problems), are often expressed in terms of growth rates for the entries of the continued fraction expansion of α. In the study of dynamics on surfaces, one often requires similar Diophantine conditions on interval exchange maps and linear flows. Passing from genus one to higher genus, a natural generalization of linear flows on tori is indeed provided by linear flows on translation surfaces (see § 2.2 for definitions); interval exchange maps, which will be shortened throughout this paper by i.e.m., are piecewise isometries which, analogously to rotations in genus one, arise are Poincaré maps of linear flows (see § 2.1 for the definition). An algorithm which plays in this context an analogous role to the continued fraction expansion is the Rauzy-Veech induction, first introduced [40, 45] and used since then as an essential tool for proving many results on the ergodic and spectral properties of i.e.m., flows on surfaces and rational billiards, see for example [46, 53, 35, 25, 2, 1, 43, 44, 37, 12, 11, 41, 29]. Diophantine conditions on i.e.m. can be prescribed imposing conditions on the behaviour of the Rauzy-Veech induction matrices and the related (extended) KontsevichZorich cocycle. In this spirit, in their work [35] on the cohomological equation for i.e.m., S. M. , P. Moussa and J.-C. Y., define a Diophantine condition on i.e.m. which generalize the notion of Roth type rotation and under which they show that the cohomological equation can be solved under finitely many obstructions (see Theorem 3.14 for a generalized statement). After Forni’s celebrated paper [22] on the cohomological

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equation associated to linear ﬂows on surfaces of higher genus, this was the first result giving an explicit diophantine condition suﬃcient to guarantee the existence of a solution to the cohomological equation. The i.e.m. which satify this condition are called Roth type i.e.m. (or i.e.m. of Roth type) and have full measure (as shown in [35], see also [34] for the sketch of a simpler proof based on the results in [3]). A reformulation and a strenghthening of the Roth type condition (namely, restricted Roth type) were then defined in [37] also for generalized i.e.m. and provide the Diophantine condition under which a linearization result is proved in [37]. Two new Diophantine conditions related to the Roth type condition for i.e.m. are introduced in this paper, for the applications that we explain in § 1.1 and § 1.2 below. More precisely (in § 3) we introduce the notion of absolute Roth type i.e.m., thus defining a class of i.e.m. which include and generalize Roth type i.e.m. but for which the results mentioned before on linearization and the cohomological equation still hold. We then define the notion of dual Roth type for a translation surface (or more precisely, for its suspension data or backward rotation number, see § 4). This is a Diophantine condition which is dual to the Roth type condition for i.e.m. in a sense which will be made precise further on (see § 4 and § 6). Let us now explain the motivation for introducing these conditions and the results which we proved assuming them, starting with the notion of dual Roth type. 1.1. Dual Roth type and distributional limit shapes. — Results on deviations of ergodic averages and ergodic integrals are a central part of the study of i.e.m. and translation ﬂows, see for example the works by Zorich [53, 52], Forni [23], Avila-Viana [4] and Bufetov [11] among others. Deviations of ergodic averages, i.e., the oscillaPn−1 tions of the Birkhoﬀ sum Sn f (x) := k=0 f (T k x) of a function f : [0, 1] → R of zero average over the orbit of (typical) point x ∈ [0, 1] under a i.e.m. T : [0, 1] → [0, 1] are of polynomial nature. In [53] Zorich shows for example that for a typical i.e.m. T and any mean-zero function f constant on the intervals exchanged by T , we have Sn f (x) = O(xν ) for some power exponent ν < 1; more precisely for a full measure set of T there exists ν = ν(f ) such that for all x ∈ [0, 1], (1.1)

lim sup n→+∞

log Sn f (x) = ν. log n

Remark also that if f is a coboundary for T with bounded transfer function, i.e., f = g ◦ T − T where the transfer function g : [0, 1] → R is bounded, then Birkhoﬀ sums Sn f (x) are uniformely bounded (and in particular ν = 0). The power exponent can be understood in terms of Lyapunov exponents of (a suitable acceleration) of the Kontsevich-Zorich cocycle associated to the Rauzy-Veech induction [54, 52]. In particular, ν is a ratio of Lyapunov exponents and depends on the position of the piecewise constant function f (identified with a vector of Rd ) with respect to the Oseledets filtration of the Kontsevich-Zorich cocycle. Using this interpretation, it follows from the work of Forni [23] (see also [4]) that, for a typical choice of function f , ν is positive; furthermore one also has ν < 1 as an immediate consequence of the work

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of Veech [48] (see also [23] for a more general result)). A powerful result of similar nature for ergodic integrals of smooth area-preserving ﬂows was proved by Forni in [23]: the power spectrum of ergodic integrals is related to Lyapunov exponents of the Kontsevich-Zorich cocycle and Forni’s invariant distributions. A finer analysis of the behaviour of Birkhoﬀ sums or integrals, beyond the size of oscillations, appears in the works [11, 36]. In [36], motivated by the study of wandering intervals in aﬃne i.e.m., S. M., P. Moussa and J.C.Y. introduced an object called limit shape and used it to describe the shape of ergodic sums (see § 3.4 and § 3.7.3 in [36]). Roughly speaking these are obtained by looking at suitably rescaled Birkhoﬀ sums, where time is renormalized according to the leading Laypunov exponent of the Kontsevich-Zorich cocycle, whereas the range of the sum is renormalized using one of the other positive exponents, according to the choice of f . After this double rescaling one obtains a sequence of shapes exponentially converging (in the Hausdorﬀ metric) to the graph of a Hölder function. In [11], Bufetov studies limit theorems for ergodic integrals of translation ﬂows and describe weak limit distributions in terms of objects that he calls Hölder cocycles (or, in the context of Markov compacta, finitelyadditive measures) and turn out to be dual to Forni’s invariant distributions (see [11] for details). We remark that limit shapes and Hölder cocycles, despite having been introduced independently, are intrinsically related: limit shapes are essentially graphs of Hölder cocycles along ﬂow leaves. Let us remark that similar results can also be proved for horocycle ﬂows on negatively curved surfaces, see [10] were the existence of Hölder cocycles is proved in this context. Both limit shapes and Hölder cocycles are associated to functions which display truly polynomial deviations, i.e., for which the exponent ν in (1.1) is strictly positive. More precisely, from the work of Forni [23] and Avila-Viana [4] it follows that for a typical i.e.m. T with d exchanged subintervals, the extended Kontsevich-Zorich cocycle has g positive Lyapunov exponents, g negative and s−1 zero ones, where d = 2g+s−1 and g and s can be computed from the combinatorics of T (g is the genus and s is the number of marked points of any translation surface which suspends T , see § 2.2). For typical i.e.m. T , functions which are coboundaries with bounded transfer functions and hence have bounded Birkhoﬀ sums, can be associated to the stable space of the Kontsevich-Zorich cocycle, which correspond to negative Lyapunov exponents. We construct in this paper objects similar to the limit shapes introduced in [36] for functions which display subpolynomial deviations of ergodic averages, i.e., functions for which ν = ν(f ) in (1.1) is equal to zero, but are not coboundaries. An important example of this type of function arise when considering rotations of the circle (which correspond to i.e.m. with d = 2) and a mean zero function χ = χ[0,β] − β, where χ[0,β] is the characteristic function of the interval [0, β]. The function χ can be seen as a piecewise constant function on a i.e.m. with d = 3. It is well known that in this case Sn χ display logarithmic deviations (which can be described for example in terms of the Ostrowski expansion of β with respect to α, see for example [7]). Results on the subdiﬀusive behaviour of these Birkhoﬀ sums were proved for example in [27, 14, 15] (see also [8] for a related result in the context of substitutions). Celebrated results

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on limit distributions for the Birkhoﬀ sums Sn χ under additional randomness were proved for example by Kesten in [30, 31] (where the rotation number is randomized) and by Beck in [5, 6] (where time is randomized, see also [17, 39, 9, 19] for recent extensions of Beck’s temporal limit distribution results and [17, 18] for results in the context of horocycle ﬂows). More in general, we show that for a typical i.e.m. T (for any number of exchanged intervals d > 3) the characteristic function χ[0,β] can be corrected by adding a function constant on the intervals exchanged by T in order to display subpolynomial deviations (see Propostion 5.3) and hence these seem to be natural functions for future investigations towards a generalizations of Kesten’s and Beck’s results. Functions with subpolynomial deviations can more generally be associated to relative homology classes in H1 (M, Σ, R) where M is a (typical) translation surface with conical singularities Σ (see § 6). We prove in § 4 that for a full measure set of translation surfaces, to each function with subpolynomial deviations (or more in general, in § 6, to each relative homology class with subpolynomial deviations), we can associate limit distribution on Hölder functions, which is constructed in a similar way to the limit shapes in [36] (see § 5.2 and § 5.4 for details). More precisely, limit shapes in [36] are Hölder functions defined as pointwise limit of a sequence of piecewise aﬃne functions obtained by plotting suitably rescaled graphs of Birkhoﬀ sums. Convergence exploits positivity of the Lyapunov exponent, which implies that oscillations on diﬀerent scales are of exponentially smaller orders. We define similarly a sequence of piecewise aﬃne functions in our setup, but they fail to converge pointwise since all oscillations are of the same magnitude. We prove on the other hand that there is convergence in the sense of distributions, when integrating these graphs against Hölder continuous functions (see Theorem 5.6). The Diophantine condition that we need to ensure distributional convergence of this sequence of piecewise aﬃne functions turns out to be a Roth type-like condition, but not for the usual Kontsevich-Zorich cocycle, but for a dual cocycle. This is the (weak) dual Roth type condition that we define in § 4.3. In the proof of convergence, we also need to introduce the notion of dual special Birkhoﬀ sums (see § 4.2). This is based on a duality between horizontal and vertical ﬂow and future and past of the Teichmueller geodesic ﬂow. The same duality is also exploited crucially in the paper by Bufetov [11]. This duality has also a homological intepretation, which we explain in Section 6. The construction of limit distributions can also be done at the level of homology bases, as explain in Section 6. Distributional limit objects can hence be associated to relative homology classes Υ ∈ H1 (M, Σ, R) which have non trivial image ∂Υ by the boundary operator ∂ : H1 (M, Σ, R) → H0 (M \Σ, R) (see § . In view of the connection between the limit shapes in [36] and Bufetov’s Hölder cocycles [11] and the duality of the latter and Forni’s invariant distributions (explained in [11], see ADD), our distributional limit objects are presumably connected to invariant distributions in relative homology.

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1.2. Absolute Roth type. — To motivate the notion of absolute Roth type i.e.m., consider the following example. Let [0, β) be an interval contained in the domain of an i.e.m. T0 and let χ be the (possibly corrected, as explained above) characteristic function of [0, β). To study the Birkhoﬀ sums Sn χ of χ under T0 , it is convenient to add the discontinuity β of χ as marked point (or fake discontinuity), so that the function χ can be considered as function which is constant over the subintervals exchanged by an i.e.m. T of d + 1 of intervals. It is natural for many questions to fix χ and vary the i.e.m. T0 . One would hence like to exploit a Diophantine condition on T which only depends on the underlying T0 and not on the position of the fake discontinuity β. Let us recall that the number of exchanged intervals d0 coincides with the dimension of the relative homology H(S0 , Σ0 , R), where M0 is any translation surface which suspends the i.e.m. T0 (see § 2.2 for definitions) and Σ0 is the set of conical singularities of M0 . The Roth type condition for i.e.m. is defined in terms of the Rauzy-Veech cocycle, which acts on the relative homology H(M0 , Σ0 , R). If T0 in the above example is such that d0 = 2g, where g is the genus of M0 , i.e., d0 equals the dimension of the absolute homology H(M0 , R), the growth of Sn χ, where χ is a corrected characteristic function, is given by the growth of a relative homology class in H(M, Σ, R), where M suspends T and has singularities Σ. One would hence like to introduce Diophantine conditions on T which depends only on the absolute homology of M . These considerations led us to define the notion of absolute Roth type i.e.m. (given in § 3). This is a Diophantine condition on an i.e.m. T defined in terms of the RauzyVeech cocycle, and hence on the action on H(M, Σ, R), but which depends on how the cocycle acts on the absolute homology H(M, R), which can be identified with a subspace of Rd defined in terms of the combinatorial data of T (see § 3 for details). We stress that the absolute Roth type condition is weaker than Roth type, i.e., Roth type i.e.m. satisfy the absolute Roth type condition (see Lemma 3.6 in § 3.3). While we were thinking of this notion, Chaika and Eskin [13] proved a result on Oseledets genericity of the Kontsevich-Zorich cocycle which, as an application, implies that given any translation surface S, for a.e. direction θ the i.e.m. obtained as Poincaré maps of the linear ﬂow in direction θ to a segment in good position (see § 2.2 for definitions) satisfy the absolute Roth type condition. This motivated us to prove that several of the results on the cohomological equation for i.e.m. of (restricted) Roth type, in particular the main results in [38], are still valid under the weaker (restricted) absolute Roth type condition. These results are presented in § 3.5 (see also § 3.6 and Appendix B). 1.3. Outline of the paper. — We first recall background material, in particular definitions and notations for i.e.m. and Rauzy Veech induction (see Section 2). The reader who is already familiar with this can skip to Section 2.8 where we simply summarize the notation. In Section 3 we first recall the Roth type condition introduced by [35] and we then define absolute Roth type. We then states the results, first proved for Roth type, that still hold under this weaker condition. We prove in this section two crucial estimates

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that shows that Roth type allows to control the lengths of inducing subintervals for the positive acceleration of Rauzy-Veech induction. The proofs of the results which extend from Roth type to absolute Roth type can then be adapted with minor modifications, which, for completeness, we discuss in an appendix (see Appendix B). We then recall the result by Chaika and Eskin in [13] and get as a corollary that the results on the cohomological equation holds for every surface in a.e. direction. In Section 4 we introduce the second variation of Roth type condition studied in this paper, namely dual Roth type. We first introduce the notion of dual Birkhoﬀ sums, which are used to give the definition. We also prove that some classical results, in particular estimates on the growth of special Birkhoﬀ sums of BV fucntions, also holds for special dual Birkhoﬀ sums under this dual Roth type condition. In Section 5 we study a distributional analogue of limit shapes for functions with subpolynomial deviations of ergodic averages. We first show how a natural class of functions belonging to the central space of the can be obtained by correcting indicatrix functions. We then define the aﬃne graphs which plot the behaviour of central Birkhoﬀ sums. Under the Roth type DC that we introduced in the previous section, we when show that these graphs converge in the sense of distributions (see Theorem 5.6). Finally, in Section 6, we give a homological interpretation of distributional limit shapes. We define suitable dual bases of relative homology and show that the graphs which we studied in the previous section and their convergence have a homological interpretation. The Appendix A contain the involved proof that backward rotation numbers are infinitely complete. The Appendix B, as already mentioned, contains a summary of the structure of the proof of the results on the cohomological equation with an indication of the changes one has to do to adapet them to absolute Roth type i.e.m. 1.4. Acknowledgements. — We would like to thank Jon Chaika, Alex Eskin and Carlos Matheus for several useful discussions. We also thank the anonimous referee for useful comments. C.U. is partially supported by the ERC Starting Grant ChaParDyn and also acknowledges the support of the Leverhulme Trust through a Leverhulme Prize and the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award. S.M. acknowledges support from UnicreditBank through the Dynamics and Information Research Institute at the Scuola Normale Superiore. The authors would like thank the University of Bristol, Centro di Ricerca Matematica Ennio de Giorgi, Mittag-Leﬄer Institute and IHES for hospitality during the visits that made this collaboration possible. The research leading to these results has received funding from the European Research Council under the European Union Seventh Framework Program (FP/2007-2013) / ERC Grant Agreement n. 335989. 2. Backgound material and notations In this section we recall basic definitions and notation for interval exchange maps and Rauzy-Veech induction. We refer to the lecture notes [49, 51, 50, 54] for further

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information. The reader who is already familiar with this material might want to skip it and to refer to § 2.8 which provides a brief summary of the notation used. 2.1. Interval exchange maps. — Let I ⊂ R be a bounded open interval. An interval exchange map (i.e.m.) i.e.m. T on I is defined by the following data. Let A be an alphabet with d > 2 symbols. Consider two partitions (modulo 0) of I into d open intervals indexed by A (the top and bottom partitions): G G I= Iαt = Iαb α∈A

α∈A

F such that |Iαt | = |Iαb | for every α ∈ A. The map T is defined on Iαt so that its restriction to each Iαt is a translation onto the corresponding Iαb . The partitions are determined by combinatorial data on one side, lengths on the other side, as follows. The combinatorial data is a pair π = (πt , πb ) of bijections from A onto {1, . . . , , d} which indicates in which order the intervals are met in the top and in the bottom partition respectively. We always assume that the combinatorial data are irreducible: for 1 6 k < d, we have (2.1)

πt−1 ({1, . . . , , k}) 6= πb−1 ({1, . . . , , k}).

The length data prescribe the lengths (λα )α∈A of the partition elements, i.e., |Iαt | = |Iαb | = Pλα for α ∈ A. The i.e.m. T = Tπ,λ determined by these data acts on I := (0, α∈A λα ); the subintervals of the top partition are X X λβ λβ , Iαt = Iαt (T ) := πt β πt (β), πb (α) < πb (β), 0 otherwise. ASTÉRISQUE 416

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The points ut1 < · · · < utd−1 separating the Iαt are called the singularities of T . The points ub1 < · · · < ubd−1 separating the Iαb are called the singularities of T −1 . We also define u0 , ud so that I = (u0 , ud ). We will also write uti (T ) and ubi (T ) when we want F to stress the dependence on T . The action of T on Iαt ⊂ I can be extended to I by right-continuity. A connection is a triple (uti , ubj , m), where m is a nonnegative integer, such that T m (ubj ) = uti . 2.2. Translation surfaces and suspension data. — Let M be a compact connected orientable topological surface of genus g ≥ 1 and let Σ = {A1 , . . . , , As } be a non-empty finite subset of M . A structure of translation surface on (M, Σ) is a maximal atlas ζ for M − Σ of charts by open sets of C ≃ R2 which satisfies two properties: any coordinate change between two charts is locally a translation of R2 and for every 1 6 i 6 s, there exists an integer κi > 1 and a ramified covering π : (Vi , Ai ) → (Wi , 0) of degree κi such that every injective restriction of π is a chart of ζ. Here Vi and Wi ⊂ R2 are neighborhoods respectively of Ai and 0. A translation surface structure naturally provides a complex structure and an holomorphic 1-form ω which does not vanish on M − Σ and has at each point Ai a zero of order κi − 1. It also provides a ﬂat metric on M − Σ: the metric exhibits a (true) singularity at each Ai such that κi > 1 and the total angle around each Ai ∈ Σ is 2πκi . The genus g of the surface and the order of the zeros of ω are related by the Ps Riemann-Roch formula 2g − 2 = i=1 (κi − 1). The geodesic ﬂow of the ﬂat metric on M − Σ gives rise to a 1-parameter family of constant unitary directional ﬂows on M − Σ, called linear ﬂows, containing in particular a vertical ﬂow ∂/∂y and a horizontal ﬂow ∂/∂x. An orbit of the vertical (horizontal) ﬂow which ends (resp. starts) at a point of Σ is called an ingoing (resp. outgoing) vertical (horizontal) separatrix. A vertical connection (resp. horizontal connection) is an orbit of the vertical (resp. horizontal) ﬂow which both starts and ends at a point of Σ. The group GL(2, R) acts naturally on traslation surfaces: given the structure of translation surface ζ on (M, Σ) and g ∈ GL(2, R), one defines a new structure g · ζ by postcomposing the charts of ζ by g. There are two distinguished one-parameter subgroups of GL(2, R) whose actions will turn out of be important (for example in § 3.6): the Teichmüller geodesic ﬂow (gt )t∈R and the group of rotations (rθ )θ∈S1 , which are given respectively by ! ! cos θ sin θ et 0 , θ ∈ S1. , t ∈ R; rθ = gt = − sin θ cos θ 0 e−t The rotation rθ has the eﬀect of rotating the ﬂat surface (M, Σ, ζ) by the angle θ. Given a translation surface (M, Σ, ζ), an open bounded horizontal segment I is in good position if 1. I meets every vertical connexion;

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2. the endpoints of I are distinct and each of them either belongs to Σ or is connected to a point of Σ by a vertical segment not meeting I. If there is no vertical connexion, or no horizontal connexion, then such segments always exist. One may even ask that the left endpoint of I is in Σ ([Yo4, Proposition 5.7, p.16]) In particular, one can always find g ∈ GL(2, R), preserving the vertical direction, and a segment in good position which is horizontal for g.ζ. The Poincaré first return map of the vertical ﬂow on a horizontal segment in good position is a i.e.m. with minimal possible number of exchanged intervals. Conversely, starting from an interval exchange map T , one can construct, following Veech [46], a translation surface MT for which T appears as a return map on a standard interval and the vertical ﬂow on M becomes a suspension ﬂow on TI . Furthermore, if M is a translation surface with no horizontal and no vertical saddle connections and I is a horizontal interval in good position, the return map TI of the vertical ﬂow of M on I is an i.e.m. of minimal number of exchanged intervals and the translation surface (M, Σ, κ, ζ) can be recovered from TI and appropriate suspension data via Veech’s zippered rectangles construction ([46], see e.g., [50], Section 4). We will limit ourselves here to explain a simple version of this construction. Let π = (πt , πb ) be a combinatorial data. A vector τ ∈ RA is a suspension vector (for π) if it satisfies the following inequalities X X (2.3) τα > 0 , τα < 0 for all 1 < k 6 d. πt (α) ℑVi for 0 < i < d. Assume that the two lines do not intersect except from their endpoints. This is in particular the case for τ = τ can . Then we can construct a translation surface M (λ, π, τ ) considering the closed polygon bounded by the two lines and identifying

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for each α ∈ A the ζα side of the top line with the ζα side of the bottom line through the appropriate translation. This produces a translation surface, whose singularity set Σ is by definition the image of the vertices of the polygon. The above non-intersection assumption, unfortunately, is not valid for all values of the data, but one can still associate to data (λ, π, τ ) as above a translation surface M (λ, π, τ ). Consider the vector q = −Ωπ τ and observe that, for all α ∈ A , since X X qα = τβ − τβ , πt (β) ubd−1 ). One then writes π b = Rt (π) (resp. π b = Rb (π)). A Rauzy class on the alphabet A is a nonempty set of irreducible combinatorial data which is invariant under Rt , Rb and minimal with respect to this property. A Rauzy diagram is a graph whose vertices are the elements of a Rauzy class and whose arrows connect a vertex π to its images Rt (π) and Rb (π). Each vertex is therefore

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the origin of two arrows. As Rt , Rb are invertible, each vertex is also the endpoint of two arrows. An arrow connecting π to Rt (π) (respectively Rb (π)) is said to be of top type (resp. bottom type). The winner of an arrow of top (resp. bottom) type starting at π = (πt , πb ) with πt (αt ) = πb (αb ) = d is the letter αt (resp. αb ) while the loser is αb (resp. αt ). To an arrow γ of a Rauzy diagram D starting at π of top (resp. bottom) type, is associated the matrix Bγ ∈ SL(ZA ) defined by Bγ = I + Eαb αt

(resp. Bγ = I + Eαt αb ),

where Eαβ is the elementary matrix whose only nonzero coeﬃcient is 1 in position αβ. A path γ in a Rauzy diagram is complete if each letter in A is the winner of at least one arrow in γ; it is k-complete if γ is the concatenation of k complete paths. An infinite path is ∞-complete if it is the concatenation of infinitely many complete paths. For a path γ in D consisting of successive arrows γ1 , . . . , , γn , we set Bγ = Bγn · · · Bγ1 . It belongs to SL(ZA ) and has nonnegative coeﬃcients. Writing π for the origin of γ and π ′ for the endpoint of γ, one has Bγ Ωπ tBγ = Ωπ′ .

(2.6)

Notice that from this relation it follows that Bγ Im Ωπ = Im Ωπ′ ,

t

Bγ−1 ker Ωπ = ker Ωπ′ .

Setting, for v, w ∈ ImΩπ , := t v Ωπ w defines a symplectic structure on Im Ωπ , and similarly on Im Ωπ′ . Thus, (2.6) shows that the cocycle Bγ is indeed symplectic relative to the (degenerate) symplectic structure induced by the matrices Ωπ and Ωπ′ . 2.4. Iterations of the Rauzy-Veech map and the Zorich algorithm. — Let T = T (0) be an i.e.m. with no connection. We denote by A the alphabet for the combinatorial data π (0) of T (0) and by D the Rauzy diagram on A having π (0) as a vertex. The i.e.m. T (1) , with combinatorial data π (1) , deduced from T (0) by the elementary step of the Rauzy–Veech algorithm has also no connection. It is therefore possible to iterate this elementary step indefinitely and get a sequence T (n) of i.e.m. with combinatorial data π (n) , acting on a decreasing sequence I (n) of intervals. Remark that for m, n ∈ Z with n 6 m, I (m) is a subinterval of I (n) with the same left endpoint, and T (m) is the first return map into I (m) under iteration of T (n) . We also define a sequence γn , n ∈ N, of arrows in D associated to the successive steps of the algorithm, so that γn is the arrow from π (n−1) to π (n) corresponding to the n-th step. For n < m, we also write γ(n, m) for the path from π (n) to π (m) composed by γ(n, m) = γn+1 ⋆ · · · ⋆ γm−1 ⋆ γm ,

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where ⋆ denotes the juxtapposition of paths. In particular γ(n − 1, n) coincides with the n-th arrow γn . We write γ = γ(T ) for the infinite path γ1 ⋆ γ2 ⋆ · · · starting from π (0) formed by the γn , n > 1. It is ∞-complete (cf. [35], p. 832). We call γ(T ) the rotation number of T . Conversely, an ∞-complete path is equal to γ(T ) for some i.e.m. with no connection. Given T with no connection, let γ(T ) be its rotation number. For any integers m > n, let B(n, m) := Bγ(n,m) = Bγm Bγm−1 · · · Bγn+1 be the matrix associated to the path γ(n, m) = γn+1 ⋆ · · · γm−1 ⋆ γm from π (n) to π (m) . In particular, for n > 1, B(n − 1, n) is the elementary matrix associated to the n-th arrow γn of γ(T ) connecting π (n−1) to π (n) . For any p 6 n 6 m the following cocycle relation holds: (2.7)

B(p, m) = B(n, m)B(p, n).

We also denote by (2.8)

Bα (n, m) :=

X

B(n, m)αβ ,

m > n,

α, β ∈ A .

β∈A

The entries of the matrix B(n, m), m > n, have the following dynamical interpretation. For any α, β ∈ A , the entry B(n, m)α,β gives the number of times in the (m) (n) orbit x, T (n) (x), . . . , of any point x ∈ Iα under T (n) visits Iβ up to the first return time of x to I (m) . Hence, Bα (n, m) gives the first return time to I (m) of any point (m) x ∈ Iα under T (n) . Following Zorich [52], it is often convenient to group together in a single Zorich step successive elementary steps of the Rauzy–Veech algorithm whose corresponding arrows have the same type (or equivalently the same winner); we therefore introduce a sequence 0 = n0 < n1 < · · · such that for each k all arrows in γ(nk , nk+1 ) have the same type and this type is alternatively top and bottom. For n > 0, the integer k such that nk 6 n < nk+1 is called the Zorich time and denoted by Z(n). 2.5. Dynamics of the continued fraction algorithms. — Let R be a Rauzy class on an alphabet A . The elementary step of the Rauzy–Veech algorithm, ˆ , (π, λ) 7→ (ˆ π , λ) considered up to rescaling, defines a map from R × P((R+ )A ) to itself, denoted by QRV . There exists a unique absolutely continuous measure mRV invariant under these dynamics ([46]); it is conservative and ergodic but has infinite total mass, which does not allow all ergodic–theoretic machinery to apply. Replacing a Rauzy–Veech elementary step by a Zorich step gives a new map QZ on R × P((R+ )A ). This map has now a finite absolutely continuous invariant measure mZ , which is ergodic ([52]).

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It is also useful to consider the natural extensions of the maps QRV and QZ , defined through the suspension data which serve to construct translation surfaces from i.e.m. For π ∈ R , let Θπ be the convex open cone of suspension vectors in RA , see (2.3). Define also X X Θtπ = {τ ∈ Θπ , τα < 0} , Θbπ = {τ ∈ Θπ , τα > 0}. α

α

Let γ : π → π ˆ be an arrow in the Rauzy diagram D associated to R . Then Bγ−1 sends Θπ isomorphically onto Θtπˆ (resp. Θbπˆ ) when γ is of top type (resp. bottom type). + A ˆ RV is the defined on F The natural extension Q π∈R {π} × P((R ) ) × P(Θπ ) by t

(π, λ, τ ) 7→ (ˆ π ,t Bγ−1 λ,t Bγ−1 τ )

ˆ RV where γ is the arrow starting at π, associated to the map QRV at (π, λ). The map Q has again a unique absolutely continuous invariant measure m ˆ RV ; it is ergodic, conˆ Z for QZ ; it has a servative but infinite. One defines similarly a natural extension Q unique absolutely continuous invariant measure m ˆ Z , which is finite and ergodic. The sequences (π (n) )n∈Z , (λ(n) )n∈Z , (τ (n) )n∈Z defined by the Rauzy-Veech algoritm satisfy, for m 6 n X X (m) (m) B(m, n)∗α β τβ , where B ∗ =t B(m, n)−1 . λ(n) B(m, n)∗α β λβ , τα(n) = α = β

β

The associated sequences δ (n) = Ωπ(n) λ(n) and q (n) = Ωπ(n) τ (n) , for n > 0, satisfy X X (m) Bα β . qα(n) = B(m, n)α β δβ , δα(n) = β

β

2.6. Special Birkhoff sums and the cocycle. — For a real Fextended Kontsevich-Zorich Q r r number r ≥ 0 we denote by C ( Iα ) the product α∈A C (Iα ). If r = 0 we someF times drop the exponent in C r ( Iα ). Let (T (n) )n∈Z be a sequence of i.e.m. acting on intervals I (n) related by the RauzyF (m) Veech algorithm as in § 2.8. Let m 6 n. For a function ϕ ∈ C( Iα ), we define the t,(n) special Birkhoﬀ sum S(m, n)ϕ ∈ C(⊔Iα ) by (2.9)

S(m, n)ϕ(x) =

X

ϕ((T (m) )j (x)),

∀α ∈ A , ∀x ∈ Iαt,(n) ,

06j n and we have Z Z ′ ψ. S(n, n )ψ = I (n′ )

I (n)

2.7. The boundary operator. — Let π be irreducible combinatorial data over the alphabet A . Define a 2d-element set S by S

= {U0 = V0 , U1 , V1 , . . . , , Ud−1 , Vd−1 , Ud = Vd }.

These symbols correspond to the vertices of the polygon produced by the suspension of an i.e.m. T with combinatorial data π (cf. § 2.8). Going anticlockwise around the vertices (taking the gluing into account) produces a permutation (1) σ of S : σ(Ui ) = Vj σ(Vj ′ ) = Ui′

with πb−1 (j + 1) = πt−1 (i + 1) for 0 6 i < d for 0 < j ′ 6 d. with πt−1 (i′ ) = πb−1 (j ′ )

The cycles of σ in S are canonically associated to the marked points on the translation surface MT obtained by suspension of T . We will denote by Σ the set of cycles of σ, by s the cardinality of Σ (cf. § 2.8). F Let ϕ ∈ C 0 ( Iα ). We write ϕ(uti −0) (resp. ϕ(uti +0)) for its value at uti considered as a point in [uti−1 , uti ] (resp. [uti , uti+1 ]). We will use the convention that ϕ(u0 − 0) = ϕ(ud + 0) = 0. Let T be an i.e.m. with combinatorial data π, and let Σ be the set of cycles of the associated permutation σ of S . The boundary operator ∂ is the linear F operator from C 0 ( Iαt ) to RΣ defined by X (ϕ(uti − 0) − ϕ(uti + 0)), (2.10) (∂ϕ)C := 06i6d, Ui ∈C

0

F for any ϕ ∈ C ( Iα ), C ∈ Σ.

Remark 2.2. — The name boundary F operator is due to the following homological interpretation. The space Γ(T ) ⊂ C 0 ( Iα ) is naturally isomorphic to the first relative homology group H1 (MT , Σ, R) of the translation surface MT : the characteristic

(1)

We remark that the presentation of the permutation σ is diﬀerent from [37].

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function of Iαt (T ) corresponds to the homology class [ζα ] of the side ζα (oriented rightwards) of the polygon giving rise to MT (see § 2.2). Through this identification, the restriction of the boundary operator to Γ(T ) is the usual boundary operator ∂ : H1 (MT , Σ, R) −→ H0 (Σ, R) ≃ RΣ . We will denote by Γ∂ (T ) the kernel of the restriction of ∂ to Γ(T ). We note in the following proposition several useful properties of the cycles and of the boundary operator, for which we refer to [37] (and in particular to Proposition 3.2, p. 1597). Proposition 2.3. — Let T be an i.e.m. with combinatorial data π: 1. Let ψ ∈ C 0 ([u0 , ud ]). One has ∂ψ = ∂(ψ ◦ T ). F 2. If ϕ ∈ C 0 ( Iα ) satisfies ϕ(uti + 0) = ϕ(uti − 0) for 0 ≤ i ≤ d then ∂ϕ = 0. F 3. The boundary operator ∂ : C 0 ( Iα ) → RΣ is onto.

4. The restriction of ∂ to Γ(T ) has as kernel Γ∂ (T ) the image of Ωπ . P Σ 5. The image of the restriction of ∂ to Γ(T ) is RΣ 0 := {x ∈ R , C xC = 0}.

6. Let n ≥ 0, let T (n) be the F i.e.m. obtained from T by n steps of the Rauzy–Veech algorithm. For ϕ ∈ C 0 ( Iα ) one has ∂(S(0, n)ϕ) = ∂ϕ ,

where the left–hand side boundary operator is defined using the combinatorial data π (n) of T (n) . 2.8. Summary of notation. — For convenience of the reader, in particular the reader who is already familiar with Rauzy-Veech induction, we summarize in this section the notation used in the rest of the paper. Interval exchange maps — — — — — — —

— — — —

is the alphabet used to index the intervals. d := #A . π = (πt , πb ) are the combinatorial data of an i.e.m. T , acting on an interval I. Iα := Iαt = Iαt (T ), Iαb = Iαb (T ), for α ∈ A are the subintervals in the top and bottom subdivisions of I. λ = (λα )α∈A are the length data, i.e., λα = |Iαt | = |Iαb | for all α ∈ A . δ = (δα )α∈A is the translation vector of T , i.e., T : Iαt → Iαb is given by x 7→ x + δα . Ωπ = (Ωα,β )α,β∈A is the antisymmetric matrix associated by the combinatorial data π, given by (2.2) (the length vector λ and the translation vector δ are related by δ = Ωλ). u0 = u0 (T ), u1 = u1 (T ) endpoints of I = I(T ) = (u0 , ud ). ut1 < · · · < utd−1 singularities of T . We also write uti (T ) = uti (T ), 1 6 i 6 d − 1. ub1 < · · · < ubd−1 singularities of T −1 . We also write ubi (T ) = ubi (T ), 1 6 i 6 d−1. T has no connection if there is no m nonnegative integer, such that T m (ubj ) = uti for some 1 6 i, j < d. A

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Translation surfaces and suspension data — τ = (τα )α∈A is a suspension vector for the combinatorial data π. — M (π, λ, τ ) translation surface obtained from the zippered rectangle construction with data (π, λ, τ ). — MT = M (π, λ, τ can ) canonical suspension associated to the canonical suspension vector τ can in (2.4). — ζα := λα + iτα , for α ∈ A are the periods of M (π, λ, τ ). — ω is the canonical holomorphic 1-form on M . — g = g(π) is the genus of M ; it is also the half-rank of Ω. — Σ is the set of marked points on M . — s := #Σ depends only on π. One has d = 2g + s − 1. — S := {U0 = V0 , U1 , V1 , . . . , , Ud−1 , Vd−1 , Ud = Vd } vertices of the suspension (singualarities of M (π, λ, τ )). — T is the return map to I for the upwards vertical ﬂow of M . — q = (qα )α∈A given by q = −Ωτ is the vector of heights of the rectangles in the zippered rectangle construction of M (π, λ, τ ); qα is the return time of Iαt to I under the vertical ﬂow on M (π, λ, τ ). The Rauzy-Veech renormalization algorithm — R is the Rauzy class of the combinatorial data π of T . — D is the associated Rauzy diagram. The set of vertices of D is R . — αt is the element of A such that πt (αt ) = d; αb is the element of A such that πb (αb ) = d. — Each arrow γ in D has a type top or bottom and is associated to two elements of A , called winner and loser; if γ if of type type top (resp. bottom), the winner is αt and the loser αb (resp. the winner is αb and the loser αt ). — Eαβ is the elementary matrix whose only nonzero coeﬃcient is 1 in position αβ. — Bγ , where γ is an arrow starting at π of top (resp. bottom) type, is the matrix Bγ = I + Eαb αt (resp. Bγ = I + Eαt αb )). — Bγ = Bγn · · · Bγ1 if γ is a path γ in D consisting of successive arrows γ1 , . . . , , γn . — T (n) := (λ(n) , π (n) ) for n ∈ N is the sequence of (non-renormalized) i.e.m. obtained by applying the Rauzy-Veech algorithm starting from T (0) := (λ(0) , π (0) ) with no connection. — (π (n) , λ(n) , τ (n) ) for n ∈ Z is the sequence produced by the Rauzy-Veech induction from π (0) , λ(0) , τ (0) := (π, λ, τ ) s.t. M (π, λ, τ ) has neither horizontal nor vertical connections. — the interval I (n) is the domain of the i.e.m. T (n) = π (n) , λ(n) . (n) — Iα = Iαt (T (n) ), α ∈ A , are the subintervals exchanged by T (n) . — γn = γ(n − 1, n) for n ∈ N (or n ∈ Z) is the arrow of D from π (n−1) to π (n) . — γ(n, m), for n < m, is the path in R from π (n) to π (m) given by γn+1 ⋆ · · · ⋆ γm−1 ⋆ γm .

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— γ(T ) for T = (π, λ) with no connection, is the rotation number of T ; γ(T ) is the juxtapposition γ1 ∗ γ2 ∗ of the arrows γ1 , γ2 , . . . , — B(m, n), where m < n is the matrix product B(m, n) := Bγ (m, n) = Bγn Bγn−1 · · · Bγm+1 . P (n) into I (n) — Bα (m, n) := β∈A B(m, n)α β is the return time of any x ∈ Iα under T (m) . — QRV : R × P((R+ )A ) → R × P((R+ )A ) Rauzy-Veech map (rescaled step of Rauzy-Veech algorithm). — QZ : R × P((R+ )A ) → R × P((R+ )A ) Zorich map, acceleration of the RauzyVeech map; — Z(n) Zorich time, i.e., the integer k such that nk 6 n < nk+1 where nk are k such that QkZ (T ) = QnRV (T ). — mRV (resp. mZ ) unique absolutely continuous measure invariant under QRV (resp. QZ ). ˆ RV (resp. Q ˆ Z ), defined on R × P((R+ )A ) × P(Θπ ): natural extension of QRV — Q (resp. QZ ). ˆ RV (resp. Q ˆ Z ); — m ˆ Q (resp. m ˆ Z ) invariant measures for Q Functional spaces, special Birkhoﬀ sums and boundary operator F F — C( FIα ) space of functions on Iα which α ∈ A. F belong to C(Iα ) for every r r — C ( Iα ) for r ≥ 1 space of functions on Iα which belong to C (Iα ) for every α ∈ A. F — Γ(T ) ⊂ C( Iα ) d-dimensional space of functions which are constant on each (n) Iα . — Given T with no connection, Γ(n) := Γ(T (n) ), where T (n) = R n (T ), for n ∈ Z. F (n) F (m) F (n) — S(m, n) : C( Iα ) → C( Iα ) special Birkhoﬀ sums operator; for ϕ ∈ C( Iα ), P (n) for α ∈ A and x ∈ Iα , S(m, n)ϕ(x) = 06j 0, ||B(0, n)χ|| = O (||B(0, n)||−σ )}.

As Γs (T ) is finite-dimensional, there exists an exponent σ > 0 which works for every χ ∈ Γs (T ). We fix such an exponent in the rest of the paper. The subspace Γs (T ) is contained in Im Ω(π), and is an isotropic subspace of this symplectic space. We have Γs (n) := Γs (T (n) ) = B(0, n)Γs (T ),

for n > 0.

Let us first define a suitable acceleration of the Rauzy-Veech algorithm. Consider a rotation number γ = γ0 ⋆ γ1 ⋆ · · · , i.e., an infinite ∞-complete path in D obtained concatenating successive arrows γ0 , γ1 , . . . Define n ˆ 0 = 0. Define inductively n ˆ k for k > 1 as the smallest integer n > n ˆ k−1 such that the path γnˆ k−1 +1 ⋆ · · · ⋆ γnˆ is complete (see 2.3). Following [38], Roth type i.e.m. are those i.e.m. whose rotation number satisfy three conditions (a) (matrix growth), (b) (spectral gap), (c) (coherence). Furthermore, i.e.m. which in addition satisfy also a fourth condition (d) (hyperbolicity), are called of restricted Roth type. We now recall the definition of these four conditions. The sequence (ˆ nk ) that appears in Condition (a) is the sequence of accelerated times defined just above. Definition 3.1. — An i.e.m. T with no connection is of restricted Roth type if the following four conditions are satisfied:

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(a) (matrix growth) ∀τ > 0, ∃ Cτ > 0, ∀k > 0, kB(ˆ nk−1 , n ˆ k )k 6 Cτ kB(ˆ n0 , n ˆ k−1 )kτ . (b) (spectral gap) There exists θ > 0 such that ||B(0, n)|Γ0 (T ) || = O (||B(0, n)||1−θ ) . (c) (coherence)

For 0 6 m 6 n, let

Bs (m, n) : Γs (T (m) ) → Γs (T (n) ), B♭ (m, n) : Γ(T (m) )/Γs (T (m) ) → Γ(T (n))/Γs (T (n) ) be the operators induced by B(m, n). We ask that, for all τ > 0, ||Bs (m, n)|| = O (||B(0, n)||τ ),

||(B♭ (m, n))−1 || = O (||B(0, n)||τ ) .

(d) (hyperbolicity) dim Γs (T ) = g. The i.e.m. with no connection that satisfy conditions (a), (b) and (c) are called Roth type i.e.m. Finally, we will say that i.e.m. with no connection that satisfy conditions (a) and (b) only are i.e.m. of weak Roth type. We will sometimes in this paper refer to these conditions as direct Roth type conditions to distinguish it from the dual Roth type condition which we will introduce later. In [35], condition (a) had a slightly diﬀerent (but equivalent) formulation, as explained in the remark below. Remark 3.2. — Condition (a) can be rephrased in the following way. Define n e0 = 0. Define inductively n ek for k > 0 as the smallest integer n > n ek−1 such that the matrix B(e nk−1 , n) has positive coeﬃcients. Then condition (a) is equivalent to (3.2)

∀τ > 0, ∃ Cτ > 0, ∀k > 0, kB(e nk−1 , n ek )k 6 Cτ kB(e n0 , n ek−1 )kτ .

Indeed, let γ be a finite path in D. If γ is not complete, at least one of the coeﬃcients of Bγ is equal to 0. Conversely, if γ is (2d − 3)-complete (or, when d = 2, 2-complete), all coeﬃcients of Bγ are positive, see [35] (more precisely, see the lemma in § 1.2.2, page 833). These two facts imply the equivalence of the two formulations, namely of condition (a) and condition (3.2). It is the condition above which was used in [38]. Let us remark that an i.e.m. T with no connection satisfying condition (b) is uniquely ergodic. Let us also recall that for any combinatorial data, the set of i.e.m. of relative restricted Roth type has full measure. First, it is obvious that almost all i.e.m. have no connection. That condition (c) is almost surely satisfied is a consequence of Oseledets theorem applied to the Kontsevich-Zorich cocycle. Condition (d) follows from the hyperbolicity of the (restricted) Kontsevich-Zorich cocycle, proved by Forni ([23]). A proof that condition (a) has full measure is provided in [35], but much better diophantine estimates were later obtained in [3]. A simpler proof where full measure is deduced from the estimates in [3] is sketched in [34]. Finally, the fact that condition (b) has full measure is a consequence from the fact that the larger Lyapunov exponent of the Kontsevich-Zorich cocycle is simple (see [47]).

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3.2. The cones C (π). — We give in this section some preliminary definitions which we need in order to define absolute Roth type i.e.m. We denote by C the positive open cone of RA . Remark first that for m < n, the matrix B(m, n) has positive coeﬃcients iﬀ B(m, n)(C ) ⊂ C ∪ {0}. For π ∈ R , let Ωπ be the antisymmetric matrix associated to π (see § 2.1) and let H(π) ⊂ RA the image of Ωπ . The dimension of H(π) is 2g and the codimension of H(π) is s − 1, where g is the genus of the translation surfaces associated to R and s is the number of marked points (see § 2.2). We are interested in the case s > 1 where H(π) $ RA . We denote by C (π) the intersection C ∩H(π). Let us recall that, from the properties of the extended KZ-cocycle, for any oriented path γ : π → π ′ in D, one has (3.3)

Bγ (H(π)) = H(π ′ ),

Bγ (C (π)) ⊂ C (π ′ ).

Let us explain the connection between H(π) and absolute homology. Let M be a translation surface with combinatorial data π, namely a surface of the form A M = M (π, λ, τ ) where λ ∈ (R+ ) and τ ∈ Θπ . Denote by Σ the set of marked points of M . Recall from § 2.2 that the homology classes [ζα ], α ∈ A of the vectors ζα defined in (2.5) provide a base of relative homology H1 (M, Σ, Z) (see Remark 2.1). When we identify RA to H1 (M, Σ, R) as explained in Remark 2.1, the subspace H(π) is identified with the absolute homology group H1 (M, R) ⊂ H1 (M, Σ, R), see for example [51]. This homological interpretation will be discussed further in Section 6. The following result is well known, we recall its proof for convenience. Proposition 3.3. — For any π ∈ R , the open cone C(π) ⊂ H(π) is not empty. Proof. — Recall that an element π = (πt , πb ) is standard if there exist letters αt , αb ∈ A such that πt (αt ) = πb (αb ) = d and πt (αb ) = πb (αt ) = 1. Every Rauzy class contains at least one standard vertex. By (3.3), it is suﬃcient to prove that C(π) is not empty when π is standard. Consider the (absolute) homology class ζ ∈ H1 (M, Z) associated to the loop obtained by concatenation of paths in the polygon joining U0 to Ud and U1 to Ud−1 (this is a loop because π is standard). One has X X ζ=2 xα ζα . ζα − ζαt − ζαb =: α∈A

As ζ is an absolute homology class, one has x ∈ H(π). Obviously, x belongs also to C . For later reference, we also state the

Proposition 3.4. — There exists a constant C = C(R ) > 1 such that, for any path γ : π → π ′ in D, one has, for B := Bγ kB|H(π) k 6 kBk 6 CkB|H(π) k.

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Proof. — Recall that we choose as norm the operator norm for the ℓ∞ -norm on RA , see (3.1). The first inequality is trivial. The second isP a consequence of Proposition 3.3. Indeed, fix a vector v in C (π); one has kBk = maxα β Bα β . Hence, writing w = Bv, X Bα β vβ = max wα 6 kB|H(π) k max vβ , min vβ kBk 6 max α

α

β

β

β

max v

which gives the required estimate with C = maxπ ( minββ vββ ). 3.3. Absolute Roth type i.e.m. — To define the (restricted) absolute Roth type Diophantine condition for i.e.m. we will modify the definition of restricted Roth type recalled in the previous section. We will not change the last three conditions (b), (c) and (d), but replace Condition (a) by a weaker Condition (a)’. Let γ be the rotation number of an i.e.m. with no connection and let π (n) , n > 0 be the vertices of the paths and B(m, n), n > m, the associated Rauzy-Veech matrices (see § 2.3). To introduce condition (a)’, we set n0 = 0 and define nk for k > 0 as the smallest integer n > nk−1 such that the matrix B(nk−1 , n) satisfies B(nk−1 , n)(C (π (nk−1 ) )) ⊂ C (π (n) ) ∪ {0}. Then we ask that (a)′

∀τ > 0, ∃ Cτ > 0, ∀k > 0, kB(nk−1 , nk )k 6 Cτ kB(n0 , nk−1 )kτ .

Definition 3.5. — An i.e.m. with no connection is of absolute Roth type if its rotation number satisfies conditions (a)′ , (b), (c), (d). The following lemma shows that the absolute Roth type condition is weaker than the Roth type condition. Lemma 3.6. — A rotation number which satisfies condition (a) also satisfies condition (a)′ . Thus, (restricted) Roth type i.e.m. are (restricted) absolute Roth type i.e.m. Proof. — Observe first that, if B(m, n), where 0 6 m < n, has positive coeﬃcients, we have (3.4)

B(m, n)(C (π (m) )) ⊂ C (π (n) ) ∪ {0}.

Let (nk )k>0 be the subsequence of times used to defined absolute Roth type condition (a)′ and let (e nk )k>0 be the subsequence of times defined in Remark 3.2 so that the matrix B(e nk−1 , n ek ) has positive coeﬃcients. Fix some k > 0. Let j := j(k) be the smallest l such that n el > nk . We claim that we have the inequalities n ej−1 < nk 6 n ej ,

nk+1 6 n ej+1 .

The first set of inequalities follow from the definition of j = j(k); to prove the last inequality, notice that, since B(e nj , n ej+1 ) > 0, also B(nk , n ej+1 ) > 0 and thus by (3.4) we have that B(nk , n ej+1 )(C (π (nk ) )) ⊂ C (π (enj+1 ) ) ∪ {0}. This, by definition of the sequence (nk )k>0 , implies that nk+1 6 n ej+1 .

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Recall that by Remark 3.2, condition (a) is equivalent to asking that for all τ > 0, there exists Cτ > 0 such that kB(e nk−1 , n ek )k 6 Cτ kB(e n0 , n ek−1 )kτ for all positive integers k. Thus, using the inequalities among indexes and the cocycle identity, we get kB(nk , nk+1 )k 6 kB(e nj−1 ,e nj+1 )k 6 Cτ2 kB(0, n ej−1 )kτ kB(0, n ej )kτ

6 Cτ2 kB(0, n ej−1 )kτ kB(0, nk )kτ kB(nk , n ej )kτ

2

6 Cτ2 kB(0, nk )k2τ kB(e nj−1 , n ej )kτ 6 C ′ kB(0, nk )k2τ +τ .

for some C ′ = C ′ (τ ). Since τ > 0 was arbitrary, this shows that (a)′ holds in this case. Remark 3.7. — On the other hand, the converse of Lemma 3.6 is in general not true (for s > 1), namely there exists i.e.m. which are of absolute Roth type but not of Roth type. For instance when d = 3, one can construct i.e.m. which are obtained from a Roth type rotation number by marking a point and there is a set of measure zero but full Hausforﬀ dimension of marked points for which the 3 i.e.m. fails to be Roth type. Finally, let us remark that since (restricted) Roth type i.e.m. have full measure, from Lemma 3.6 it also follows that (restricted) absolute Roth type i.e.m. also have full measure. 3.4. Two crucial estimates. — We state and prove in this section two crucial estimates on the lengths of induced subintervals (see Corollary 3.11 and Proposition 3.13) which are needed in particular to prove results on the cohomological equation, but are also of independent interest (see for example [32, 33]). These estimates were proved for i.e.m. of Roth type in [35]. They require a diﬀerent proof for i.e.m. of absolute Roth type, which is given in this section. Once these estimates are proved, one can easily generalize all the main results on the cohomological equation proved in [35, 38] for (restricted) Roth type i.e.m. to (restricted) absolute Roth type i.e.m. (see § 3.5 and Appendix B). We present the proof of these estimates this section, since they show how the absolute Roth type condition is used. Let T be an i.e.m. with combinatorial data in R and no connection. Let γ be the rotation number of T . Let (T (n) )n>0 be the sequence of i.e.m. obtained from T =: T (0) by the Rauzy-Veech algorithm. Recall that, for n > 1, T (n) acts on I (n) ⊂ I (n−1) (n) exchanging the subintervals Iα := Iαt (T (n) ), α ∈ A . Recall from [35] (see also Lemma 3.14 of [38]) that condition (a) in the Roth type definition implies the following estimates on the lengths of induced subintervals. Proposition 3.8. — Assume that γ satisfies condition (a). Then, for any τ > 0, there exists Cτ > 0 such that, for any α ∈ A , any n > 0, one has Cτ−1 |I (0) | kB(0, n)k−1−τ 6 |Iα(n) | 6 Cτ |I (0) | kB(0, n)k−1+τ .

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Remark 3.9. — One can also see (cf. [37] Appendix C, Proposition C1, p. 1639) that condition (a) of Roth type is equivalent to the following: for all τ > 0, there exists C > 0 such that for all n ∈ N, we have 1−τ (n) (n) . max |Iα | 6 C min |Iα α∈A

α∈A

We will find analogues estimates to those provided by Prop. 3.8 when γ satisfies only the weaker condition (a’). We deal first with the upper bound, which is the easiest one (and the same than in Proposition 3.8). The upper bound will follow from the following proposition (see Corollary 3.11 below). Proposition 3.10. — Assume that γ satisfies condition (a’). Then, for any τ > 0, there exists Cτ > 0 such that, for any α ∈ A , any n > 0, one has X Bα β (0, n) > Cτ−1 kB(0, n)k1−τ . β∈A

Let us state and prove the corollary giving the upper bound before giving the proof of Propostion 3.10. Corollary 3.11. — For any τ > 0, there exists Cτ > 0 such that, for any α ∈ A , any n > 0, one has |Iα(n) | 6 Cτ |I (0) | kB(0, n)k−1+τ . Proof. — The upper bound follows immediately from Proposition 3.10 combined with the simple observation that X Bα β (0, n) |Iα(n) | = |I (0) |. α,β∈A

For the proof of Proposition 3.10, we need the following lemma. Lemma 3.12. — There exists a constant C = C(R ) with the following property. Let γ : π → π ′ be a finite path in D such that Bγ (C (π)) ⊂ C (π ′ ) ∪ {0}. For any vector w ∈ Bγ (C (π)), one has max wα 6 CkBγ k min wα . α

α

Proof. — Fix π ∈ R . As the subspace H(π) is rational, one can find primitive integral vectors v i ∈ H(π) which span the extremal rays of C (π). Let C(π) be the supremum over i of the kv i k. Let γ : π → π ′ be as in the lemma, and let wi := Bγ v i . Each wi is an integral vector with positive coordinates. We have therefore max wαi 6 kBγ k kv i k 6 C(π)kBγ k 6 C(π)kBγ k min wαi . α

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P For v ∈ C (π), w = Bγ v, we write v = i ti v i with ti > 0. For α, β ∈ A , we obtain X X ti wβi = C(π)kBγ k wβ . ti wαi 6 C(π)kBγ k wα = i

i

Therefore the conclusion of the lemma holds with C(R ) := maxπ∈R C(π). Proof of Proposition 3.10. — We first prove the inequality of the proposition when n = nk . Fix u ∈ C (π (n0 ) ); let v = B(n0 , nk−1 )u,

w = B(n0 , nk )u = B(nk−1 , nk )v.

For α, α′ ∈ A , using Lemma 3.12, condition (a)’ and that kB(0, n)k is non decreasing in n, one has that X X Bα β (0, nk ) 6 (min uβ )−1 Bα β (0, nk )uβ = (min uβ )−1 wα β

β

β∈A

β∈A

6 CkB(nk−1 , nk )k(min uβ )−1 wα′ β

6 CCτ′ kB(0, nk−1 )kτ (min uβ )−1 β

X

Bα′ β (0, nk )uβ

β∈A

6

CCτ′ kB(0, nk )kτ

maxβ uβ X Bα′ β (0, nk ). minβ uβ β∈A

As α,β Bα β (0, nk ) is of the order of kB(0, nk )k, we obtain the inequality of the proposition for n = nk . When nk < n < nk+1 , we use that each Bα,β (0, n) is a non-decreasing function of n. For α ∈ A , we have X X Bα β (0, n) > Bα β (0, nk ) > Cτ−1 kB(0, nk )k1−τ . P

β∈A

β∈A

On the other hand, we have eτ kB(0, nk )k1+τ . kB(0, n)k 6 kB(0, nk )kkB(nk , nk+1 )k 6 C

As τ > 0 is arbitrary, we obtain the estimate of the proposition.

(n)

The estimate of |Iα | from below, as stated in Proposition 3.8, is not a consequence of condition (a′ ) (even for d = 3) . The true statement is slightly more sophisticated: Proposition 3.13. — Assume that γ satisfies condition (a′ ). Then, for any τ > 0, there exists Cτ > 0 such that, for any α ∈ A , any ℓ > 0, one has |Iα(nℓ ) | > Cτ−1 |I (0) | kB(0, nk )k−τ −1 where k is the largest integer greater or equal to ℓ such that

P

β

Bβ α (nℓ , nk ) < s.

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Proof. — Fix α ∈ A . Let nk be as in the proposition. We have (n ) Iα ℓ

P

β

Bβ α (nℓ , nk+1 ) > s.

It means that the interval decomposes into at least s subintervals of level nk+1 . (n ) Therefore there are at least s + 1 points (including the endpoints of Iα ℓ = Iαt (T (nℓ ) )) bounding these subintervals and two of them must correspond to the same marked point. The (horizontal) segment inside the transversal which has these two points as endpoints hence give an absolute homology class ζ, which we express in the (n ) base ζα k+1 , α ∈ A , as X (n ) xβ ζβ k+1 . ζ := β

The coeﬃcients xβ are integral, non-negative and the vector x = (xβ ) belongs to H(π (nk+1 ) ). Let us also represent ζ at level nk+2 as X (n ) ζ := Xβ ζβ k+2 . β

The vectors x and X are related by

X = B(nk+1 , nk+2 )x. As x ∈ C (π (nk+1 ) ) and the coeﬃcients Xβ are integers, we get from the definition of nk+2 that Xβ > 1 for all β ∈ A. We therefore have Z X (n ) XZ (nℓ ) Re ω = λβ k+2 > |I (nk+2 ) |. |Iα | > Re ω > (n ) ζ

β

ζβ

k+2

β

One has also

|I (0) | =

X

(nk+2 )

Bβ µ (0, nk+2 )|Iβ

β,µ

and we conclude from condition (a’),

| 6 |I (nk+2 ) |

X

Bβ µ (0, nk+2 ),

β,µ

|Iα(nℓ ) | > C −1 |I (0) | kB(0, nk+2 )k−1 > Cτ−1 |I (0) | kB(0, nk )k−τ −1 . 3.5. Results on the cohomological equation for absolute Roth type i.e.m. — We show in this paper that the main results on the cohomological equations for i.e.m. which were proved in [38] under the (restricted) Roth type diophantine condition, actually hold under the weaker (restricted) absolute Roth type condition. One of the motivations for this weakening is shown in § 3.6. As recalled in the introduction, motivated by Forni’s celebrated paper [22] on the cohomological equation for area preserving ﬂows on surfaces of higher genus, in [35] Roth type i.e.m. were introduced in order to provide an explicit diophantine condition suﬃcient to guarantee the existence of a solution to the cohomological equation up to finitely many obstructions. More precisely, in [35] it is proven that, given a suﬃciently regular datum (namely, for functions which are of absolutely continuous on each of the intervals Iαt with derivative of bounded variation and of zero mean on I), after subtracting (to the datum) a correction function which is constant on each Iαt , the

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cohomological equation has a bounded solution. In [37] this result was used for proving a linearization theorem for generalized interval exchange maps with a rotation number of restricted Roth type. More recently, for restricted Roth type i.e.m., in [38] a more refined regularity result was proved, namely the Hölder regularity of the solutions of the cohomological equation with C r , r > 1 data. In this section we limit ourselves to the statement of the generalization of the main theorem of [38] (Theorem 3.10, p.127). In Appendix B we will give the proof of this theorem, as well as the statement and the proof of the generalizations of other two results of [38]: the growth estimate for special Birkhoﬀ sums with C 1 data (Theorem 3.11, p. 128) and the regularity result for the solutions of the cohomological equation in higher diﬀerentiability (Theorem 3.22, p. 138). Let T be an i.e.m. and let r > 0 be a real number. The boundary operator ∂ which appears in the following result is the operator defined in in § 2.7. We denote F F by C0r ( Iα ) the kernel of the boundary operator ∂ in C r ( Iα ). We recall from § 2.7 that the coboundary of a continuous function belongs to the kernel of the boundary operator (it follows from properties (1) and (2) of Proposition 2.3). The following result generalizes Theorem 3.10 of [38] to absolute restricted Roth type i.e.m. Theorem 3.14 (Hölder solutions to the cohomological equation) Let T be an i.e.m. of absolute restricted Roth type. Let Γu (T ) be a subspace of Γ∂ (T ) which is supplementing Γs (T ) in Γ∂ (T ). Let r > 1 be a real number. There exist δ¯ > 0 F ¯ and bounded linear operators L0 : ϕ 7→ ψ from C0r ( Iα ) to the spaceFC δ ([u0 , ud ]) of r ¯ Hölder continuous functions of exponent δ and L1 : ϕ 7→ χ from C0 ( Iα ) to Γu (T ) F such that any ϕ ∈ C0r ( Iα ) satisfies Z ud ϕ=χ+ψ◦T −ψ , ψ(x)dx = 0. u0

The operators L0 and L1 are uniquely defined by the conclusions of the theorem. Remark 3.15. — The exponent δ¯ depends only on r and the constants θ, σ appearing in § 3.1. Thus, since in the diﬀerence of definition between absolute Roth type and Roth type does not concern the two costants θ (spectral gap) e σ (in the definition of stable space), the exponent δ¯ in this Theorem 3.14 is the same than the exponent in Theorem 3.10 of [38]. One can show that δ¯ tends to 0 as r > 1 tends to 1 and conjecture that one can take any δ¯ > 0 for any fixed r > 1. This is indeed the case in the Sobolev scale, as proved by Forni in [25]. Very recently, this optimal loss of derivatives was also proved in the Hölder class in the pseudo-Anosov case (namely in the case of periodic Rauzy-Veech induction), see [20]. The proofs of these theorems for i.e.m. of restricted absolute Roth type follow essentially the same proofs given in the respective papers for restricted Roth type. The main diﬀerence is in the proof of two crucial estimates on lengths of subintervals of i.e.m. in a Rauzy-Veech orbit, presented above in § 3.4. Once these estimates are

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proved, the proofs of the theorems for restricted Roth type i.e.m. can be followed and adapted with minor modifications to the case of restricted absolute Roth type i.e.m. For completeness, in the Appendix B we follow the original proof and indicate, for convenience of the interested reader, where and which modifications are needed. 3.6. Results on the cohomological equation for translation surfeces in a.e. direction. — The notion of (absolute) Roth type i.e.m. naturally leads to define (absolute) Roth type translation surfaces: the following definition of Roth type translation surfaces was given in [37] and we now extend it to absolute Roth type. Definition 3.16. — Let (M, Σ, ζ) be a translation surface. It is of absolute Roth type if there exists some open bounded horizontal segment I in good position such that the return map TI of the vertical ﬂow on I is an i.e.m. of absolute Roth type. Absolute restricted Roth type, restricted Roth type and Roth type translation surfaces are defined analogously. In Appendix C of [37] it is proved that for a (restricted) Roth type translation surface, TI will actually be of (restricted) Roth type for any horizontal segment I in good position. Analogously, one can show that for an absolute (restricted) Roth type translation surface, TI will be of absolute (restricted) Roth type for any horizontal segment I in good position. In [13] Chaika and Eskin prove that, for all translation surfaces (M, Σ, ζ) and for almost every angle θ, the translation surface (M, Σ, rθ · ζ) obtained by rotating by rθ the translation structure ζ is generic for the Teichmüller geodesic ﬂow and Oseledets generic for the Kontsevich-Zorich cocycle. As an application of this result, they prove the following lemma, which we now state using the definition of absolute Roth type translation surface just given above. (2) Lemma 3.17 ( [13]). — For all translation surfaces (M, Σ, ζ), for almost every angle θ, the translation surface (M, Σ, rθ · ζ) is of absolute Roth type. Remark 3.18. — Furthermore, it also follows from [13] that (3) if the orbit closure M := SL(2, R) · (M, Σ, ζ) under the linear action of SL(2, R) is such that λg > 0, where ±λi , for 1 6 i 6 g here denote the Lyapunov exponents of the KontsevichZorich cocycle restricted to the locus M , then, for almost every angle θ, the translation surface (M, Σ, rθ · ζ) is of restricted absolute Roth type. (2)

In Section 1.2.2 of [13], the authors define a diophantine conditon on the vertical ﬂow of the translation surface, called in the paper Roth type and consisting of three conditions (a), (b) and (c) (see Definition 1.13, [13]). Their definition is given in a geometric language, but one can check that it is equivalent to our notion of absolute Roth type translation surface (Definition 3.16 above), and not to the definition of Roth types surface given in [35]. Lemma 1.16 in [13] shows that Condition (a) holds for a.e. θ, while the proof that (b) and (c) also holds for a.e. θ is given at the end of Section 1.2.2. (3) More precisely, it follows from Lemma 1.16 and Thm 1.4 of [13], see the comment at the very end of § 1.2.2 in [13].

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Combining this remark with the Theorem 3.14 proved in this paper, we get the following corollary (4). Corollary 3.19. — Assume that (M, Σ, ζ) is such that the orbit closure SL(2, R) · (M, Σ, ζ) has λg > 0. Then, for almost every angle θ, any i.e.m. T obtained as first return map of the linear ﬂow in direction θ on (M, Σ, ζ) on a segment in good position satisfies the conclusion ofFTheorem 3.14. Thus, for any r > 1 there exists δ¯ > 0 such that for any ϕ ∈ C0r ( Iαt (T )), there exists a Hölder ¯ continuous function ψ ∈ C δ ([u0 (T ), ud (T )]) with zero mean and a piecewise constant correction χ ∈ Γu (T ) such that ϕ = χ + ψ ◦ T − ψ. Remark 3.20. — We remark that results on the cohomological equation for almost all directions on a fixed translation surface were first proved by Forni in [22, 25]. The result in this corollary, compared with the latter results, improves on the loss of derivatives. However, a stronger assumption is assumed, namely hyperbolicity of the KZ cocycle (5) While this paper was under review, two new results on the cohomological equation have appeared: in [26] the Diophantine condition from [35] is weakened to allow the presence of zero Lyapunov exponents, but the loss of derivatives is worst than in our (or [35]) results; in [20], the optimal loss of derivatives without hyperbolicity assumptions is proved, but only in the pseudo-Anosov (periodic Rauzy-Veech induction) special case (a measure zero class). 4. Dual Roth Type We define in this section another Diophantine condition for translation surfaces which we call dual Roth type. In order to define it, we need first to introduce dual special Birkhoﬀ sums (see § 4.2). This is a notion which we believe is of independent interest, which is based on a form of duality for Rauzy-Veech induction (which heuristically correspond to considering backward time in the Teichmueller geodesic ﬂow and ﬂipping the role of horizontal and vertical ﬂow on translation surfaces). It should be noticed that this type of duality was exploited also in the work of Bufetov [11], who discovered a duality between invariant distributions for the horizontal (vertical) ﬂow and finitely additive functionals for the vertical (horizontal) ﬂow. An analogous duality was later found by Bufetov and Forni for horocycle ﬂows on hyperbolic surfaces in (4)

In [13], a result on the solvability of the cohomological equation (under finitely many obstructions) for all translation surfaces in a.e. direction is stated (see the paragraph after Theorem 1.14 in [13]) and deduced from Lemma 3.17 and [35], but since the Definition 1.13 in [13] is not equivalent to Roth type (but to absolute Roth type, see previous footnote), the results from [35] cannot be directly applied, but require the strenghthening proved in this paper. In private communications, though, Chaika and Eskin told us that their proof can be modified to actually show that all translation surfaces are of Roth type for a.e. θ. (5) We already recalled that the Masur-Veech measure is hyperbolic, as proved in [23]. Results on the presence (or absence) of zero Lyapunov exponents and hyperbolicity for other SL(2, R) invariant measures appear in several works in the literature, see for example [24, 21].

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their work [10]. Our motivation for introducing a dual special Birkhoﬀ sums operator and dual Roth type conditions in the combinatorial set up of Rauzy-Veech induction came from the results proved in Section 5. 4.1. Backward Rauzy-Veech induction and backward rotation numbers. — Let M := M (π, λ, τ ) be a translation surface constructed from combinatorial data π ∈ R , length data λ ∈ C and suspension data τ ∈ Θπ . We assume in this section that M has no horizontal saddle connections. Then, starting from (π (0) , λ(0) , τ (0) ) := (π, λ, τ ), we can iterate indefinitely the Rauzy-Veech backward algorithm and (using the notation (n) (n) (n) recalled in § 2) construct the sequence as ζα = λα + iτα for n 6 0 follows: P (n) — when α τα < 0, we perform the inverse of an elementary top step of the Rauzy-Veech algorithm π (n−1) = Rt−1 π (n) ζα(n−1) = ζα(n) + ζα(n) , w w ℓ

ζα(n−1) = ζα(n) ,

∀α 6= αw ,

where αw , αℓ are the letters such that (n)

πt (αw ) = d,

(n)

(n)

πb (αℓ ) = πb (αw ) + 1;

P (n) — when α τα > 0, we perform the inverse of an elementary bottom step of the Rauzy-Veech algorithm π (n−1) = Rb−1 π (n) ζα(n−1) = ζα(n) + ζα(n) , w w ℓ

ζα(n−1) = ζα(n) ,

∀α 6= αw ,

where αw , αℓ are the letters such that (n)

πb (αw ) = d,

(n)

(n)

πt (αℓ ) = πt (αw ) + 1.

P The case α τα (n) = 0 never occurs because there are no horizontal saddle connections. We obtain in this way a backward rotation number, i.e., an infinite path γ in D with terminal point π (0) of the form γ = · · · ⋆ γn−1 ⋆ γn ⋆ · · · γ−1 ⋆ γ0 , where γn , n > 0, is the arrow from π (n−1) to π (n) . (n)

(n)

(n)

Remark 4.1. — Remark that the equations for ζα = λα + iτα split into independent sets of equations for λ(n) and τ (n) . Since the type of backward move (top or bottom) depends only on τ (n) , it follows that the backward rotation number γ only depends on (π, τ ).

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Remark 4.2. — The backward rotation number is defined as soon as the preferred rightwards separatrix in M (issued from the leftmost vertex O of the polygon) is not a saddle connection. Other horizontal connections are apparently not detected by the algorithm (i.e the algorithm does not stop in this case). In particular, if τ is such that its components τα , α ∈ A , are irrationally independent over Q, then the backward Rauzy algorithm does not stop. Thus, to any π and a.e. τ ∈ Θπ , one can associate to the pair (π, τ ) a backward rotation number. Recall that a finite path in D is complete if every letter in A is the winner of at least one of its arrows and that an infinite path is ∞-complete if it is the concatenation of infinitely many finite complete paths. Proposition 4.3 (Completeness of backward rotation numbers) When M = M (π, λ, τ ) has no horizontal connection, the backward rotation number γ associated to (π, τ ) is ∞-complete. We give the proof of this proposition in the Appendix A, since the proof is quite long and rather involved combinatorially. It would be interesting to know whether the converse to the above proposition is true (see Remark 4.2 above). 4.2. Dual special Birkhoff sums. — Let M (π, λ, τ ) be a translation surface as above that has no horizontal saddle connections. Thus, we have well defined data (π (n) , λ(n) , τ (n) ) for any n ∈ Z. For any n ∈ Z we denote by T (n) the i.e.m. with data (π (n) , λ(n) ). Let I (n) = I(T (n) ) be the sequence of intervals on which T (n) acts for n ∈ Z. Let us recall that special Birhoﬀ sums (see (2.9)) are operators S(m, n), where F (m) F (n) m 6 n, which map from functions defined on Iα to functions defined on Iα and that, when acting on the space of piecewise constant functions, S(m, n) : Γ(m) → Γ(n) is represented by the KZ cocycle matrix B(m, n) (see § 2.6). We will now define a dual operator, which we call dual special Birkhoﬀ sums. It is useful to remark that this operator plays for the horizontal linear ﬂow on a translation surface the role of the special Birkhoﬀ sums play for the vertical ﬂow. Let q (n) be the vectors given by q (n) = −Ωπ(n) τ (n) , so that the translation surface M (π (n) , λ(n) , τ (n) ) can be represented as a zippered rectangle over T (n) with heights q (n) (see § 2.2). For n ∈ Z, α ∈ A , let G (n) L(n) := 0, q (π, τ ) , L(n) := L(n) α α α . α∈A

The disjoint union L(n) will be the domain of the functions on which the n-th special (n) Birkhoﬀ sum operator will act. Geometrically, one should think of the intervals Lα , α ∈ A , as vertical intervals, one for each zippered rectangle representation with heights q (n) of the translation surface M (π (n) , λ(n) , τ (n) ). Remark that, for any n 6 0, (n) qα depends only on π and τ (see Remark 4.1).

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(n)

For n′ 6 n, α, ∈ A , we can write the interval Lα as a disjoint union (modulo 0) (n′ ) of a number of translated copies of the Lβ as follows (see also Figure 1). P ′ (n) Let us recall that Bα (n′ , n) := β∈A B(n , n)αβ gives the return time in I (n)

′

of Iα under T (n ) (see § refsec:algorithms). For 0 6 j < Bα (n′ , n), let β(α, j) be the letter in A such that ′

(n)

′

(T (n ) )j (Iα(n ) ) ⊂ Iβ(α,j) . Then, for any α ∈ A , we can write qα(n) =

X

(n′ )

qβ(α,j) .

06j 0 given by Condition (a) of the Roth type condition or Condition (a) of the absolute Roth type condition (see § 3.4). Proposition 4.8. — For all ε > 0 there exists C0 = C0 (ε) > 0 and C1 = C1 (ε) > 0 such that, for all β ∈ A , (m)

C0 ||B(m, 0)||−1−ε 6 |Lβ | 6 C1 ||B(m, 0)||−1+ε . We remark that the estimate from below is not needed for our immediate purposes, but useful in a more general context. We conclude this section by proving this proposition. The proof using the following auxiliary lemma. Lemma 4.9. — For every ε > 0, there exists C ′ = C ′ (ε) > 0 such that, for all m 6 0, all α, β ∈ A max (1, Bα, β (m, 0)) > C ′ ||B(m, 0)||1−ε . Proof. — Let (ˆ nk )k>0 be the subsequence of induction times defined at the beginning of § 4.3. Without loss of generality, we can assume that m 6 n ˆ −4d+5 , since if n ˆ −4d+5 < m 6 0, the conclusion is true as soon as we take C ′ 6 ||B(ˆ n−4d+5 , 0)||−1 . Therefore, if k the unique integer such that n ˆ k−1 < m 6 n ˆ k , we can assume that k 6 −4d + 5. By [35] (see lemma in § 1.2.2 of [35]), all coeﬃcients of the matrix X := B(ˆ nk , n ˆ k+2d−3 ) are strictly positive, hence at least equal to 1; the same is true for the coeﬃcients of the matrix Y := B(ˆ n−2d+3 , 0). Moreover, Setting V := B(ˆ nk+2d−3 , n ˆ −2d+3 ) (which is well defined since k < −4d + 6), we can write (recalling the cocycle relation (2.7)) U := B(ˆ nk , 0) = B(ˆ n−2d+3 , 0)B(ˆ nk+2d−3 , n ˆ −2d+3 )B(ˆ nk , n ˆ k+2d−3 ) = Y V X. Thus, we have that, for all α, β, α′ , β ′ ∈ A (using also that B(n, 0)α,β is a non decreasing function of n and the definition of k), Bα, β (m, 0) > Uα, β > Yα, α′ Vα′ , β ′ Xβ ′ , β > Vα′ , β ′ . Choosing α , β such that Vα′ , β ′ = maxα,β Vα,β , since maxα,β Vα,β is a norm on RA and all norms on Rd are equivalent, we get that there exists c > 0 such that ′

′

Bα, β (m, 0) > c||V ||. On the other hand, from condition (a) in the Definition 4.5 of dual Roth type and the choice of k, we have that for any ε > 0 there exists C˜0 = C˜0 (ε) and C˜1 = C˜1 (ε) such that ||B(m, 0)|| 6 ||B(ˆ nk−1 , 0)|| 6 ||Y V || ||B(ˆ nk−1 , n ˆ k+2d−3 )|| 6 C˜0 ||Y V ||1+ε 6 C˜1 ||V ||1+ε ,

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giving the estimate of the lemma. Proof of Proposition 4.8. — If follows from the geometric decomposition (4.1) that, for every α ∈ A , for every n′ < n′′ X (n′ ) (n′′ ) (4.3) |Lα |= Bα, β (n′ , n′′ )|Lβ |, β∈A

which gives that for every β ∈ A , for every n′ < n′′ (4.4) ′′ ′′ (n′ ) ) ) | 6 max(Bα, β (n′ , n′′ ))−1 max|L(n |. |Lβ | 6 max (Bα, β (n′ , n′′ ))−1 |L(n α α α∈A

α∈A

α∈A

The estimate from above follows from Lemma 4.9 and the inequality (4.4) above for n′ := m and n′′ := 0. To prove the estimate from below, we define k60 as in the proof of the previous lemma, i.e., so that n ˆ k−1 < m 6 n ˆ k . Writing (recalling the cocycle Equation (2.7)) V =: B(ˆ nk−2d+2 , 0) = B(m, 0)B(ˆ nk−2d+2 , m) = B(m, 0)X, for X := B(ˆ nk−2d+2 , m), we know again (from m > n ˆ k−1 , k − 1 − (k − 2d + 2) = 2d − 3 and the lemma in § 1.2.2 of [35]) that all coeﬃcients of X are at least equal to 1. We have then that for all β ∈ A (by (4.3) for n′ := n ˆ k−2d+2 , n′′ := m) (m)

nk−2d+2 ) |Lβ | > max |L(ˆ |. α α∈A

From (4.3) for n := n ˆ k−2d+2 , n := 0 and the definition of || · ||, we get that there exists c > 0 such that ′

′′

nk−2d+2 ) max |L(ˆ | > c||B(ˆ nk−2d+2 , 0)||−1 . α α∈A

Finally, using repeatingly condition (a) of the dual Roth type Definition 4.5 (and then m 6 n ˆ k and monotonicity of n 7→ ||B(n, 0)||), we have that for every ε there exists C(ε) such that ||B(ˆ nk−2d+2 , 0)|| 6 ||B(m, 0)|| ||B(ˆ nk−2d+2 , n ˆ k )|| 6 C(ε)||B(ˆ nk , 0)||1+ε 6 C(ε)||B(m, 0)||1+ε . The three previous inequalities imply the required estimate. 4.5. Estimates on dual special Birkhoff sums of Hölder functions. — The main result in this section is a bound on the growth of dual special Birkhoﬀ sums for Hölder continuous functions under the dual Roth type condition. The main result proved in this section is the following proposition. The analogous estimate for growth of (direct) special Birkhoﬀ sums for Hölder continuous was given in [35] (see the proposition at the end of § 2.3 in [35]).

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Proposition 4.10. — Assume that (π, τ ) is of dual Roth type. Then, for every function ψ on L(0) which has mean value 0 and is Hölder continuous for some exponent η ∈ (0, 1), the dual Birkhoﬀ sums S ♯ (0, n)ψ satisfy 1

||S ♯ (0, n)ψ||C 0 6 C(η)||ψ||C η ||B(n, 0)||1− 20 θη , where θ is the exponent that appears in the definition of dual Roth type (see (b) in Definition 4.5). The rest of the section is devoted to the proof of this proposition. We obtain an estimate on the C 0 -norm of S ♯ (0, n)ψ by the same method used for (direct) Birkhoﬀ sums. Let us remark that following the same methods in [35], the Proposition 4.10 can also be proved for functions of bounded variation. Proof of Proposition 4.10. — Let (π, τ ) be of dual Roth type and let ψ be a function on L(0) as in the assumptions of the proposition. We first write ψ = ψ0 + ν0 , where ν0 belongs to the space Γ♯0 (0) (defined in § 4.2) and ψ0 has mean value 0 on each (0) interval Lα . For 0 > m > n, we write S ♯ (m, m − 1)ψm = ψm−1 + νm−1 , (m−1)

where νm−1 ∈ Γ♯0 (m − 1) and ψm−1 has mean value 0 on each interval Lα have then X (4.5) S ♯ (0, n)ψ = ψn + S ♯ (m, n)νm .

. We

n6m60

To estimate the terms in this sum, we first observe that, for n 6 m 6 0 and each (m) β ∈ A and x, y ∈ Lβ , using that ψ is Hölder-continuous, we have that X (m) Bα, β (m, 0). |ψm (x) − ψm (y)| = |S ♯ (0, m)ψ(x) − S ♯ (0, m)ψ(y)| 6 ||ψ||C η |Lβ |η α∈A

(m)

As ψm has mean 0 in each Lβ , this gives (m)

||ψm ||C 0 6 ||ψ||C η |Lβ |η

X

Bα, β (m, 0).

α∈A

The estimate from above in Lemma 4.8 gives, for all ε > 0 ||ψm ||C 0 6 C(ε)||ψ||C η ||B(m, 0)||1−η(1−ε) , and thus also, as S ♯ (m, m − 1)ψm = ψm−1 + νm−1 , ||νm−1 ||C 0 6 C(ε)||ψ||C η ||B(m, 0)||1−η(1−ε) . From the Formula (4.5) above, we get, taking ε = ♯

||S (0, n)ψ||C 0 6 C||ψ||

Cη

n X

m=0

In this sum, we distinguish two cases:

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θ

Case 1. — When ||B(m, 0)|| 6 ||B(n, 0)|| 3 , we write B0 (n, m) = B0 (m, 0)−1 B0 (n, 0) and obtain, from condition (b) and the symplecticity of B(m, 0) θ

1

||t B0 (n, m)|| ||B(m, 0)||1− 2 η 6 ||B0 (m, 0)−1 || ||B0 (n, 0)|| ||B(n, 0)|| 3 θ

6 ||B(m, 0)−1 || ||B0 (n, 0)|| ||B(n, 0)|| 3 θ

6 C||B(m, 0)|| ||B(n, 0)||1−θ ||B(n, 0)|| 3 θ

6 C||B(n, 0)||1− 3 . θ

Case 2. — When ||B(m, 0)|| > ||B(n, 0)|| 3 , we use the trivial bound of ||t B0 (n, m)|| by ||B(n, m)|| and the following inequality, which is proved below: for all ε > 0 ||B(n, m)|| ||B(m, 0)|| 6 C(ε)||B(n, 0)||1+ε . We then obtain, taking ε =

1 12 θη 1

1

|| t B0 (n, m)|| ||B(m, 0)||1− 2 η 6 ||B(n, m)|| ||B(m, 0)||1− 2 η 1

1

6 C||B(n, 0)||1+ 12 θη ||B(n, 0)||− 6 θη 1

6 C||B(n, 0)||1− 12 θη . To prove that ||B(n, m)|| ||B(m, 0)|| 6 C(ε)||B(n, 0)||1+ε , we may assume that n6n ˆ −2d+4 . Then, there exist integers k, l with n 6 n ˆk 6 m 6 n ˆ l such that l = k + 2d − 3 so that all coeﬃcients of B(ˆ nk , n ˆ l ) are positive (by the lemma in § 2.3 of [35]) and hence at least equal to 1. Furthermore, by condition (a), there exists ε C1 = C1 (ε) such that ||B(ˆ nk , n ˆ l )|| 6 C1 ||B(n, 0)|| 2 . This gives ||B(n, m)|| ||B(m, 0)|| 6 C(ε)2 ||B(n, n ˆ k )|| ||B(ˆ nl , 0)|| ||B(n, 0)||ε 6 C(ε)2 ||B(n, 0)||1+ε , as required. We conclude that 1

||S ♯ (0, n)ψ||C 0 6 C||ψ||C η n ||B(n, 0)||1− 12 θη . Since we also know that ||B(n, 0)|| grows exponentially fast with relation to Zorich time, this is suﬃcient to conclude that n||B(n, 0)||−ε converges to 0 for any ε > 0. This concludes the proof of the proposition. 5. Distributional limit shapes Let R be a Rauzy class on an alphabet A of cardinality d, D be the associated diagram. Let g be the genus of the associated translation surfaces and s the cardinality of the marked set on such surfaces. The rank of the matrices Ωπ , π ∈ R is equal to 2g and we have d = 2g + s − 1 (see § 2.2). We assume throughout this section that s > 1. Consider a point T = (π, λ, τ ) in R ×(R+ )A ×Θπ which is typical for the dynamics ˆ RV (defined in 2.4) and typical (in the sense of Oseledets of the natural extension Q genericity) for the Kontsevich-Zorich cocycle. Let T := Tπ,λ be the associated i.e.m.

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acting on the interval I. For n ∈ Z, let T (n) = (π (n) , λ(n) , τ (n) ) be the image of T ˆ n , and let T (n) be the associated i.e.m. acting on the interval I (n) . For any under Q RV ′ ′ integers n, n′ ∈ Z with n > n′ , T (n ) is the first return map of T (n) in I (n ) . (n) (n) (n) For each n ∈ Z, we denote respectively by Γs , Γu , Γc the stable, unstable and (n) (n) (n) central subspaces of the KZ-cocycle at (π , λ , τ ) (see § 3). By hyperbolicity of the Kontsevich-Zorich cocycle (proved in [23]), we have (n) dim(Γ(n) s ) = dim(Γu ) = g,

dim(Γ(n) c ) = s − 1,

(n) Γ(n) s ⊕ Γu = Im (Ωπ (n) ) . (0)

We will say that χ := (χ(n) )n∈Z is a sequence of central functions if χ(0) = χ ∈ Γc and χ(n) = S(0, n)χ, (n)

χ

−1

= S(n, 0)

for n > 0, χ,

for n < 0.

A particularly interesting example of central functions can be constructed from characteristic functions as shown in § 5.1. 5.1. Corrected characteristic functions. — Assume only for this subsection that s = 1. Fix a point ξ (0) ∈ I (0) . Definition 5.1. — A sequence ξ := (ξ (n) )n∈Z is compatible with ξ (0) ∈ I (0) if: — for each n ∈ Z, ξ (n) is a point of I (n) ; ′ ′ — for n > n′ , ξ (n) is the first entry point in I (n) when iterating (T (n ) )−1 from ξ (n ) . We will say in this case that ξ is a compatible sequence. Remark 5.2. — It is clear that ξ (n0 ) determines ξ (n) for all n > n0 . On the other hand, given ξ (n) for n > n0 , one can for instance choose ξ (n) = ξ (n0 ) for n 6 n0 . In particular, by Remark 5.2, sequences compatible with ξ (0) do exist. For any such sequence, for any n ∈ Z, let χ e(n) be the function defined on I (n) and equal to the characteristic function of the interval (0, ξ (n) ). Let Γ(n) be the d-dimensional vector space (canonically isomorphic to RA ) of func(n) tions defined on I (n) and constant on each Iα . Finally, let Γ(n) (ξ) be the d-dimensional aﬃne space given by Γ(n) (ξ) := χ e(n) + Γ(n) .

The following proposition shows that any sequence (e χ(n) )n∈Z of characteristic functions as above can be corrected by adding a function constant on the subintervals exchanged by T (0) , to produce a sequence (χ(n) )n∈Z of central functions: Proposition 5.3 (Projection to central sequences). — For any sequence ξ := (ξ (n) )n∈Z compatible with ξ (0) ∈ I (0) , for any intergers n, n′ with n > n′ , the special Birkhoﬀ ′ sum operator S(n′ , n) (cf. § 2.6) is an isomorphism from Γ(n ) onto Γ(n) and from ′ Γ(n ) (ξ) onto Γ(n) (ξ).

ASTÉRISQUE 416

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Furtheremore, there exists a unique sequence (χ(n) )n∈Z such that χ(n) belongs to Γ(n) (ξ) for any n ∈ Z, ′ S(n′ , n)(χ(n ) ) = χ(n) for all n, n′ ∈ Z with n > n′ and 1 log ||χ(n) ||∞ = 0, (5.1) lim n→±∞ Z(n) where Z(n) is the Zorich time (see § 2.3). Proof. — The first part of the assertion is already known (see [35]). For the second ′ part, let n > n′ and consider S(n′ , n)e χ(n ) . It is locally constant on I (n) , except at ξ (n) and at the singularities of T (n) . Moreover, the diﬀerence between the left and right ′ limits at ξ (n) is equal to 1. Therefore S(n′ , n)e χ(n ) belongs to Γ(n) (ξ) and the first part of the proposition follows. (n) (n) Let Γ(n) = Γs ⊕ Γu be the Oseledets decomposition into stable and unstable subspaces. Write then for n ∈ Z (n)

(n)

S(n − 1, n)e χ(n−1) = χ e(n) + ∆e χ(n) χ(n) s + ∆e u (n)

(n)

e(n) is bounded by 1 and n χu ∈ Γu . Observe that, since χ with ∆e χs ∈ Γs and ∆e here indexes Rauzy-Veech elementary steps, we have that ||S(n − 1, n)e χ(n−1) ||∞ 6 2. (n) (n) (n) (n) We look for χ = χ e + ∆χs + ∆χu satisfying the requirements of the proposition. In particular, we must have (n−1) ∆χ(n) = ∆e χ(n) , s s + S(n − 1, n)∆χs

(n−1) χ(n) . ∆χ(n) u = ∆e u + S(n − 1, n)∆χu This leads to the formulas X ′ ) ∆χ(n) = S(n′ , n)∆e χ(n , s s n′ 6n

∆χ(n) u =−

X

′

) (S(n, n′ ))−1 ∆e χ(n u .

n′ >n

We will now show that the convergence of these series follows from the hyperbolicity of the Kontsevich-Zorich cocycle and this will conclude the proof. Let us first remark that, as T = (π, λ, τ ) is typical and, as already observed, ||S(n − 1, n)e χ(n−1) ||∞ 6 2, there is, for every ε > 0, a constant C := C(ε, T ) such that, for all n ∈ Z ||∆e χ(n) χ(n) s ||∞ + ||∆e u ||∞ 6 C exp(ε|Z(n)|). As the cocycle is hyperbolic, there exists a constant C ′ := C ′ (ε, T ) such that, for all (m) (n) n > m, all vs in Γs , all vu in Γu , denoting by θg the smallest positive exponent of ˆ Z , we have the KZ-cocycle w.r.t the Zorich dynamics Q ||S(m, n)vs ||∞ 6 C ′ exp[(Z(m) − Z(n))θg + (|Z(m)| + |Z(n)|)ε]||vs ||∞ , ||(S(m, n))−1 vu ||∞ 6 C ′ exp[(Z(m) − Z(n))θg + (|Z(m)| + |Z(n)|)ε]||vu ||∞ . These equations show that the series above converge.

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As ε is arbitrary, the sequence χ(n) is well-defined and satisfies the required properties. The uniqueness of the sequence with the required properties follows from the hyperbolicity of the KZ-cocycle: the diﬀerence between two solutions would grow subexponentially (in Zorich time) and thus must be equal to 0. Remark 5.4. — This proposition uses the assumption, assumed throughout this secˆ RV and Oseledets generic for tion, that T = (π, λ, τ ) is typical for the dynamics of Q the Kontsevich-Zorich cocycle. In § 6 we will prove a similar result in the homological context under weaker assumptions on (π, λ, τ ) than Oseledets genericity (see in particular the notion of KZ hyperbolic introduced in § 6.3). Furthermore, one can prove the existence of a sequence Γ(n) (ξ) as in the conclusion of Proposition 5.3 but satisfying (5.1) only for n → +∞ (resp. n → −∞) assuming only negative (resp. positive) KZ hyperbolicity (see, in § 6.3, Propositions 6.8 and 6.9 respectively). In this case (n) (n) though uniqueness only holds modulo Γs (resp. Γu ). (n)

5.2. The functions Ωα (π, τ, χ). — We will now define an object analogous to the limit shapes defined in [36], but suited to study Birkhoﬀ sums of central functions (0) χ ∈ Γc . The largest Lyapunov exponent of the Kontsevich-Zorich cocycle is simple and the associated 1-dimensional eigenspace F1 (π, τ ) does not depend on the λ variable. Moreover, F1 (π, τ ) is generated by a vector q(π, τ ) contained in the positive cone (R+ )A . We normalize q(π, τ ) by asking that its ℓ2 -norm is equal to 1. For n 6 0, we write (n)

(S(n, 0))−1 q(π, τ ) =: q (n) (π, τ ) = Θ1 q(π (n) , τ (n) ) (n)

with Θ1

∈ R+ satisfying 1 (n) log Θ1 = θ1 . n→−∞ Z(n) lim

(θ1 is the largest Lyapunov exponent of the KZ-cocycle.) (0) Let χ ∈ Γc . Write χ(n) := (S(n, 0))−1 (χ) for n 6 0. For any α ∈ A and any (n) (n) n 6 0 , we define a function Ωα = Ωα (π, τ, χ) on the interval [0, qα (π, τ )] by the following requirements: (0)

1. recall that Bα (n, 0) (defined in (2.8)) is the return time in I (0) of Iα under T (n) ; (0) for 0 6 j < Bα (n, 0), let β(α, j) be the letter in A such that (T (n) )j (Iα ) ⊂ (n) Iβ(α,j) ; for 0 6 ℓ 6 Bα (n, 0), set X (n) Sq(ℓ) = qβ(α,j) (π, τ ). 06j 0, kχ(n) k 6 CkB(n, 0)k−σ kχ(0) k, ∀n < 0}. Notice that both Γs and Γu are subspaces of H1 (M, R) (more precisely of the image of H1 (M, R) in H1 (M, Σ, R)). Moreover, when H1 (M, R) is equipped with the symplectic intersection form, both Γs and Γu are isotropic. Notice also that, as H1 (M, Σ, R) is finite-dimensional, one can choose an exponent σ > 0 which works for all vectors in Γs and all vectors in Γu . (0) When we identify H1 (M, Σ, R) with RA through the basis (ζα ), The subspace Γs u only depend on (π, λ) and the subspace Γ on (π, τ ).

ASTÉRISQUE 416

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113

Definition 6.2. — The translation surface M = M (π, λ, τ ) is positively KZ-hyperbolic if M has no vertical connection and the following properties hold 1. for any τ > 0, one has X

kB(0, n)k−τ < +∞;

n>0

2. the dimension of Γs is equal to g; 3. (coherence) For any τ > 0, there exists Cτ > 0 such that, for any Υ ∈ Γs , any 0 6 m 6 n, one has kχ(n) k 6 Cτ kB(0, n)kτ kχ(m) k. Definition 6.3. — The translation surface M = M (π, λ, τ ) is negatively KZ-hyperbolic if M has no horizontal connection and the following properties hold 1. for any τ > 0, one has X

kB(0, n)k−τ < +∞;

n 0, there exists Cτ > 0 such that, for any Υ ∈ Γu , any 0 > m > n, one has kχ(n) k 6 Cτ kB(0, n)kτ kχ(m) k. Definition 6.4. — The translation surface M = M (π, λ, τ ) is KZ-hyperbolic if it is both positively and negatively KZ-hyperbolic and one has H1 (M, R) = Γs ⊕ Γu . Remark 6.5. — By Forni’s results in [23], for any combinatorial data π, almost all data (λ, τ ) produce KZ-hyperbolic translation surfaces. Remark 6.6. — Assume that M is positively KZ-hyperbolic. For Υ ∈ H1 (M, Σ, R) and n > 0, define X kΥ kn,⋆s := min{ kχk∞ | Υ − χα ζα(n) ∈ Γs }.

It follows from the symplecticity of the restricted KZ-cocycle that one has, for Υ ∈ H1 (M, R), n > m > 0, and any τ > 0 kΥ k0,⋆s 6 CkB(0, n)k−σ kΥ kn,⋆s ; kΥ km,⋆s 6 Cτ kB(0, n)kτ kΥ kn,⋆s .

The proof is an adaptation of the results discussed in Section 3.2 of [38].

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Remark 6.7. — Similarly, assume that M is negatively KZ-hyperbolic. For Υ ∈ H1 (M, Σ, R) and n 6 0, define X kΥ kn,⋆u := min{ kχk∞ | Υ − χα ζα(n) ∈ Γu }. Then one has, for Υ ∈ H1 (M, R), n 6 m 6 0, and any τ > 0 kΥ k0,⋆u 6 CkB(n, 0)k−σ kΥ kn,⋆u ; kΥ km,⋆u 6 Cτ kB(n, 0)kτ kΥ kn,⋆u . Proposition 6.8. — Assume that M is positively KZ-hyperbolic. Then, for every P (n) (n) υ ∈ RΣ χα ζα ∈ H1 (M, Σ, R) such that ∂Υ = υ and 0 , there exists a class Υ = log(kχ(n) k/kχ(0) k) = 0. n→+∞ log kB(0, n)k lim

Moreover, two such classes diﬀer by an element in Γs . Therefore the proposition des fines a linear section from RΣ 0 to H1 (M, Σ, R)/Γ . The proof of this proposition is given below. Remark that the content of the proposition is non-empty only when s > 2. We also have the following symmetric proposition. Proposition 6.9. — Assume that M is negatively KZ-hyperbolic. Then, for every P (n) (n) υ ∈ RΣ χα ζα ∈ H1 (M, Σ, R) such that ∂Υ = υ and 0 , there exists a class Υ = log(kχ(n) k/kχ(0) k) = 0. n→−∞ log kB(n, 0)k lim

Moreover, two such classes diﬀer by an element in Γu . Therefore the proposition u defines a linear section from RΣ 0 to H1 (M, Σ, R)/Γ . We omit the proof of Proposition 6.9 since it is analogous to the proof of Proposition 6.8 given below. Finally, combining the two propositions, we have the following corollary. Corollary 6.10. — Assume that M is KZ-hyperbolic. Then, for every υ ∈ RΣ 0 , there P (n) (n) exists a unique class Υ = χα ζα ∈ H1 (M, Σ, R) such that ∂Υ = υ and log(kχ(n) k/kχ(0) k) log(kχ(n) k/kχ(0) k) = lim = 0. n→+∞ n→−∞ log kB(0, n)k log kB(n, 0)k lim

The section υ 7→ Υ from RΣ 0 to H1 (M, Σ, R) is linear. Proof of Proposition 6.8. — Uniqueness modulo Γs is clear from the first inequality in Remark 6.6. To prove existence, we choose, for each n > 0, a class Ξ(n) = P (n) (n) bα ζα ∈ H1 (M, Σ, R) such that ∂Ξ(n) = υ and αχ kb χ(n) k 6 C = C(υ).

ASTÉRISQUE 416

ON ROTH TYPE CONDITIONS, DUALITY AND CENTRAL BIRKHOFF SUMS

115

We have therefore (as the coeﬃcients of B(n − 1, n) are equal to 0 or 1) kΞ(n) − Ξ(n − 1)kn,⋆s 6 C. As Ξ(n) − Ξ(n − 1) belongs to H1 (M, R), we deduce from the first inequality in Remark 6.6 that kΞ(n) − Ξ(n − 1)k0,⋆s 6 CkB(0, n)k−σ . P (0) We can therefore choose a class Ξ(n − 1, n) = xα (n)ζα ∈ H1 (M, R) such that Ξ(n) − Ξ(n − 1) − Ξ(n − 1, n) ∈ Γs and kx(n)k∞ 6 CkB(0, n)k−σ . Therefore the series

P

n>0

Ξ(n − 1, n) is convergent. We will see that X Ξ(n − 1, n) Υ := Ξ(0) + n>0

satisfies the conclusions of the proposition. Clearly we have ∂Υ = υ. Next we estimate kΥ km,⋆s . One has X Υ = Ξ(m) + Ξ(n − 1, n) mod. Γs . n>m

From the choice of Ξ(m), one has kΞ(m)km,⋆s 6 C. To estimate kΞ(n − 1, n)km,⋆s , there are two cases — If kB(0, n)kσ/2 6 kB(0, m)k, we use the second inequality in Remark 6.6 to get, for any τ > 0 kΞ(n − 1, n)km,⋆s 6 Cτ kB(0, m)kτ . — If kB(0, n)kσ/2 > kB(0, m)k, we have kΞ(n − 1, n)km,⋆s 6 kB(0, m)kkΞ(n − 1, n)k0,⋆s 6 CkB(0, m)kkB(0, n)k−σ 6 CkB(0, n)k−σ/2 . From the first condition in Definition 6.2, we have X kB(0, n)k−σ/2 6 C,

and

X

kB(0, m)kτ 6 Cτ kB(0, m)k2τ .

kB(0,n)kσ/2 6kB(0,m)k

After renaming τ , we get kΥ km,⋆s 6 Cτ kB(0, m)kτ . Thus, for each m > 0, we can write X Υ = yα (m)ζα(m) + Υ (m),

with ky(m)k∞ 6 Cτ kB(0, m)kτ and Υ (m) ∈ Γs . We can also take Υ (0) = 0. As the coeﬃcients of B(m − 1, m) are equal to 0 or 1, we have, for m > 0 X Υ (m) − Υ (m − 1) = yeα (m)ζα(m) , ke y (m)k∞ 6 Cτ kB(0, m)kτ . SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2020

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P (m) From the coherence property in Definition 6.2, we finally obtain Υ (m) = ybα (m)ζα with X kb y (m)k∞ 6 Cτ′ kB(0, m)kτ kB(0, ℓ)kτ 6 Cτ′′ kB(0, m)k3τ . ℓ6m

This concludes the proof of the proposition.

6.4. Piecewise-affine paths in H1 (M \ Σ, R). — We assume that M has no horizontal connection. Let n < 0; denote by αw the winner of the arrow γ(n, n + 1), by αℓ its (n+1) (n) loser. Recall that we have θα = θα for α 6= αℓ and θα(n+1) = θα(n) + θα(n) . w ℓ ℓ To the arrow γ = γ(n, n + 1), we associate a substitution transformation ςγ on the alphabet A defined as follows: — If γ is of top type, ςγ is defined by αℓ 7→ αℓ αw ,

α 7→ α,

∀α 6= αℓ .

— If γ is of bottom type, ςγ is defined by αℓ 7→ αw αℓ ,

α 7→ α,

∀α 6= αℓ .

For α ∈ A , we then define inductively a sequence of words (Wα (n))n60 by Wα (0) = α,

Wα (n) = ςγ(n,n+1) (Wα (n + 1)),

∀n < 0.

One can see that for any n 6 0 and α, β ∈ A , the number of occurrences of β in the word Wα (n) is equal to (B(n, 0))α β . The relation to the Rauzy-Veech algorithm is given by the following proposition. Proposition 6.11. — Write Wα (n) = β0 · · · βN −1 . Then we have, for 0 6 j < N (n)

(T (n) )j (Iα(0) ) ⊂ Iβj . The proof is clear from the definition of the Rauzy-Veech algorithm. Remark 6.12. — Equivalently, for any n < 0 and α, β ∈ A , we also have the following (0) geometric interpretation of the word Wα (n) = β0 · · · βN −1 . The rectangle Rα can be seen as a union of rectangles (of the same width, but shorter height) each fully (b) contained in a rectangle Rβ for some β ∈ A (refer to the bottom part of Figure 1). (0)

These rectangles, in the order from the bottom to the top of Rα , are contained (n) in Rβℓ for ℓ = 0, 1, . . . , , N − 1. In other words, if one considers a leaf of the vertical (0)

(0)

ﬂow from a point x ∈ Iα of length qα , it crosses I (n) exactly N times, and the (n) (n) (n) intersections belong, in order, to Iβ0 , Iβ1 , . . . , , IβN −1 . In other words (cf. Remark 6.12), if one considers a leaf of the horizontal ﬂow (0) (0) starting from a point x ∈ Lα and crossing all the rectangle Rα horizontally (see for example the top part of Figure 1), then its intersections with L(n) will be contained, (n) in order, in the intervals Lβℓ , for ℓ = 0, 1, . . . , , N − 1.

ASTÉRISQUE 416

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ON ROTH TYPE CONDITIONS, DUALITY AND CENTRAL BIRKHOFF SUMS

To the word Wα (n) = β0 · · · βN −1 , we associate a broken line Wα (n) in H1 (M \ Σ, R) as follows: P (n) — for 0 6 j 6 N , let Wα,j (n) := 06i 0 min qα(n) 6 Cτ−1 kB(n, 0)k−1−τ . α

On the other hand, for two consecutive vertices Wα,j (n), Wα,j+1 (n) of Wα (n), one has (n)

|i(Υ ′ − Υ, Wα,j+1 (n)) − i(Υ ′ − Υ, Wα,j (n))| = i(Υ ′ − Υ, θβj ) 6 CkB(n, 0)k−σ . Using the space decomposition of [38] (see Section 3.8 in [38]), one gets the conclusions of the theorem from these estimates. (n)

Since by Proposition 6.13, for any α ∈ A the sequence Ωα (Υ ) converge to a distribution on Holder continous functions, Proposition 6.14 shows that the limit distribution is uniquely determined, up to the addition of Holder functions, by the projection of Υ ∈ H1 (M, Σ, R) onto the stable space Γs (along the unstable space). 6.6. Piecewise-affine paths in H1 (M, Σ, R). — In this section, we exploit the duality between horizontal and vertical ﬂows and between H1 (M \Σ, R) and H1 (M, Σ, R) in order to describe a construction and results which are dual to the one in the previous two sections, namely § 6.4 and § 6.7. In particular we describe how to construct distributional limit shapes for relative homology classes with subpolynomial deviations for the horizontal ﬂow. This dual result requires the KZ negative hyperbolicity assumption, together with the direct Roth type condition (and is based on the study of special Birkohﬀ sums instead than dual ones). Let us first construct piecewise-aﬃne paths in H1 (M, Σ, R), in a dual way to the definition of piecewise-aﬃne paths in H1 (M \Σ, R) in § 6.4. We assume that M has no vertical connection. Let n > 0. To the arrow γ = γ(n, n + 1), we associate a substitution transformation ςγ on the alphabet A defined as follows. Denote by αw the winner of arrow γ, by αℓ its loser. — If γ is of top type, ςγ is defined by αw 7→ αw αℓ ,

α 7→ α,

∀α 6= αℓ .

— If γ is of bottom type, ςγ is defined by αw 7→ αw αℓ ,

ASTÉRISQUE 416

, αℓ 7→ αw

α 7→ α,

∀α ∈ / {αℓ , αw }.

ON ROTH TYPE CONDITIONS, DUALITY AND CENTRAL BIRKHOFF SUMS

119

For α ∈ A , we then define inductively a sequence of words (Wα∗ (n))n>0 by Wα∗ (0) = α,

Wα∗ (n + 1) = ςγ(n,n+1) (Wα∗ (n)),

∀n > 0.

One can see that for any n > 0 and α, β ∈ A , the number of occurrences of β in the word Wα∗ (n) is equal to (B(0, n))α β . The relation to the Rauzy-Veech algorithm is the following. For any n > 0 and (0) (0) α ∈ A the rectangle Rα of base Iα can be seen as union of rectangles (of same (0) height but shorter width) fully contained in the rectangles Rβ , β ∈ A (refer to the (0)

right hand side of Figure 2). In particular, the interval Iα can be seen as a union of (n) translates of the intervals Iβ . If we write Wα∗ (n) = β0 · · · βN −1 , (0)

(n)

then we have that Iα is union, from left to right, of translates of the intervals Iβ0 , (n)

(n)

Iβ1 , . . . , , IβN −1 , or, more precisely [ (0,n) (n) jα,ℓ (Iβℓ ), Iα(0) =

where

(0,n)

jα,ℓ (x) = x +

X

(n)

λβi .

06i 0,

Hkb (π, τ ) < 0,

∀1 6 k < d.

As usual, we write ζα = λα + iτα . Since M has no horizontal saddle connections, by iterating the backward Rauzy Veech algorithm defined in § 4.1, we obtain a backward rotation number, i.e., an infinite path γ = · · · γ−1 ⋆ γ0 in D with terminal point π (0) . We want to show that it is ∞-complete. A.1. The quantity H(n). — Given T with no connection, let us denote as usual by T (n) the i.e.m. with data (π (n) , λ(n) ) obtained via Rauzy-Veech induction. We define for n 6 0 the quantities Hkt (n) := Hkt (π (n) , λ(n) ),

for 1 6 k < d;

Hkb (π (n) , λ(n) ),

for 1 6 k < d;

Hbt (n)

:=

H(n) :=

X

Hkt (n) − Hkb (n).

16k 0 when n goes to −∞. Lemma A.1. — For any α ∈ A , any n 6 0, one has |τα(n) | 6 H(n). Proof. — Indeed, if for instance πt (n)(α) =: k < d, one has t |τα(n) | 6 |Hkt (n) − Hk−1 (n)|, t with Hkt (n), Hk−1 (n) ∈ [0, H(n)].

A.2. Switching times Lemma A.2. — The path γ contains infinitely many arrows of each type. Proof. — If the conclusion of the lemma does not hold, we may assume, after truncating γ, that all arrows γ(n, n + 1) have the same type. Assume for instance that all (0)

arrows are of top type. Then they have all the same winner αw . Let πb (αw ) = k. We

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b must have k = 1 because Hk−1 (n) does not depend on n and therefore must vanish (n)

(as H(n) is convergent). The τα

τα(n−d+1) w

with α 6= αw do not depend on n. One also has X = τα(n) + τα , ∀n 6 0. w α6=αw

However, we have a contradiction.

P

α6=αw

τα =

t Hd−1

> 0, hence ταw cannot be negative for all n 6 0,

Denote by T (resp. B) the set of n 6 0 such that γ(n, n + 1) is of top type (resp. bottom type). Lemma A.3. — One has lim

n→−∞,n∈T

t Hd−1 (n) =

lim

n→−∞,n∈B

b Hd−1 (n) = 0.

Proof. — This follows from the formulas relating H(n) and H(n − 1). Lemma A.4. — If γ(n, n + 1) is of top type, one has t t Hd−1 (n − 1) = Hd−1 (n),

b Hd−1 (n − 1) = Hd (n).

If γ(n, n + 1) is of bottom type, one has b b Hd−1 (n − 1) = Hd−1 (n),

t Hd−1 (n − 1) = Hd (n).

Proof. — Clear from the formulas of an elementary Rauzy-Veech step. Definition A.5. — A negative integer n is a switching time if γ(n, n + 1) and γ(n + 1, n + 2) are of diﬀerent types. Let Sw be the set of switching times. It is infinite by Lemma A.2. Lemma A.6. — One has lim

n→−∞,n∈Sw

t Hd−1 (n) =

lim

n→−∞,n∈Sw

b Hd−1 (n) = 0.

Proof. — This follows from Lemmas A.3 and A.4. Lemma A.7. — One has also lim

n→−∞,n∈Sw

Hd (n) = 0.

Proof. — Let n ∈ Sw . Assume for instance that γn+1 = γ(n, n + 1) is of top type and γn+2 = γ(n + 1, n + 2) is of bottom type. Let αw be the winner of γn+2 and define k := πb (n + 1)(αw ) ∈ [1, d). We have b Hd (n) = H(n + 1) − H(n) + Hk+1 (n + 1).

When n is large, H(n) and H(n + 1) are close, Hd (n) is positive (as γ(n) is of bottom b type) and Hk+1 (n + 1) is negative (even when k = d − 1 as γn+2 is of top type). Therefore Hd (n) is small.

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A.3. The subset A ′ ⊂ A . — Let A ′ be the set of letters in A which are the winners of at most finitely many arrows of γ. After cutting a terminal subpath oﬀ γ, we may assume that none of the letters in A ′ is the winner of an arrow of γ. Therefore the (n)

quantities τα , for α ∈ A ′ , do not depend on n. Observe that, as M has no horizontal saddle-connection, these quantities are diﬀerent from 0. Lemma A.8. — For any α ∈ A \ A ′ , we have lim

n→−∞,n∈Sw

τα(n) = 0.

Proof. — Let α ∈ A \ A ′ . Between consecutive switching times, all arrows have the same winner. Let 0 > n1 > · · · > nk > · · · be the switching times such that α is the winner of γ(nk , nk+1 ), and let n′k < nk be the switching time immediately inferior (n) to nk . The value of τα stays the same for nk+1 6 n 6 n′k . On the other hand, it follows from Lemmas A.6 and A.7 that lim τα(nk ) = 0.

k→+∞

(n)

Remark A.9. — At this stage of the proof we do not claim that τα n′k < n < nk .

is small for

We have to prove that A ′ is empty. We assume by contradiction that A ′ is not empty. Lemma A.10. — One has X

τα = 0.

α∈A ′

Proof. — Indeed, as the switching time n goes to −∞, the quantity Hd (n) is alter(n) nately positive and negative and the τα with α ∈ A \ A ′ converge to zero. A.4. Decompositions of A \ A ′ . — Let n be a switching time. Define X ′ t τβ = 0}, A0 (n) = {α ∈ A \ A | (n)

πt

t

A>0 (n)

= {α ∈ A \ A ′ | (n)

πt b

A0 (n)

= {α ∈ A \ A ′ | (n)

πb b

A0 (nk+1 ). t — Assume that the letter αw belongs to A>0 (nk ). Then we must have (from Lemmas A.6 and A.7) that the loser αℓ of γnk+1 +2 , which is also the winner (n ) (n ) of γnk+1 +1 , satisfies αℓ ∈ A \ A ′ , πt k (αw ) < πt k (αℓ ) 6 d − 2 and X τβ = 0 β∈A ′

(n ) (n ) (nk ) (αw )0 (n), n ∈ Sw , is eventually empty. Indeed, a letter t which would belong to A>0 (n) for all n ∈ Sw is the winner of γn for infinitely many γn , n ∈ Sw . But we have seen that it cannot be the winner of infinitely many arrows of top type, and that each time that it is the winner of an arrow of bottom t type, the cardinality of A>0 (n) is decreasing. b We prove similarly that A m such that β ∈ A ′ for m 6 πt (β) 6 m′ , and X X τβ = 0, τβ > 0, ∀m 6 m1 < m′ . m6πt (n)(β)6m′

(n)

m6πt

(β)6m1

Then we have an horizontal saddle connection!

Appendix B Adaptation of the proofs of results on cohomological equation for absolute Roth type i.e.m. Our goal in this appendix is to check that the conclusion of Theorem 3.10 of [38], namely Theorem 3.14, as well as the conclusions of Theorem 3.11 and Theorem 3.22 of [38], are still valid under the assumption that the i.e.m. is of restricted absolute Roth type (instead that under the stronger requirement of being of restricted Roth type). The generalization of Theorem 3.10 of [38] was already stated as Theorem 3.14 in § 3.5. We begin the appendix by stating the generalizations of Theorems 3.11 and 3.22 of [38] (stated as Theorem B.1 and B.4 below), then review their proofs indicating which (small) modifications are needed. B.1. Estimates for special Birkhoff sums with C 1 data. — The following result extends Theorem 3.11 of [38], which was proved for i.e.m. of restricted Roth type, to i.e.m. of restricted absolute Roth type. Theorem B.1 (Generalization of Theorem 3.11 of [38]). — Let T be an i.e.m. which satisfies conditions (a’), (c), (d) of Sections 3.1 and 3.3. Let Γu (T ) be a subspace of Γ∂ (T ) which is supplementing Γs (T ) in Γ∂ (T ). 3.14 F The operator L1 : ϕ 7→ χ of Theorem F extends to a bounded operator from C01 ( Iα (T )) to Γu (T ). For ϕ ∈ C01 ( Iα (T )), the function χ = L1 (ϕ) ∈ Γu (T ) is characterized by the property that the special Birkhoﬀ sums of ϕ − χ satisfy, for any τ > 0 kS(0, n)(ϕ − χ)kC 0 6 C(τ )kϕkC 1 kB(0, n)kτ . The inequality of the theorem for special Birkhoﬀ sums implies a similar inequality for general Birkhoﬀ sums: Corollary B.2. — For any T and ϕ as in Theorem B.1, for all τ > 0, we have k

N −1 X

(ϕ − χ) ◦ T j kC 0 6 C(τ )kϕkC 1 N τ .

j=0

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B.2. Higher differentiability. — To formulate a result allowing for smoother solutions of the cohomological equation which extend Theorem 3.22 in [38], we introduce the same finite-dimensional spaces than in [37] (although the notations are slightly diﬀerent). In the following definitions, T is an i.e.m., κ is a nonnegative integer, and r is a real number > 0. Definition B.3. F — We denote by F — Cκκ+r ( Iα (T )) the subspace of C κ+r ( Iα (T )) consisting of functions ϕ which satisfy ∂κi ϕ = 0 for 0 6 i 6 κ. F — Γ(κ, T ) the subspace of Cκ∞ ( Iα (T )) consisting of functions χ whose restriction to each [ui−1 (T ), ui (T )] is a polynomial of degree 6 κ. — Γs (κ, T ) the subspace of Γ(κ, T ) consisting of functions χ which can be written as χ = ψ◦T −ψ, for some function ψ of class C κ on the closure of [u0 (T ), ud (T )]. The proof given in [37] (p.1602) that if T is of restricted Roth type then the dimension of Γ(κ, T ) is equal to 2g + κ(2g − 1), and that the dimension of Γs (κ, T ) is equal to g + κ, also holds, essentially unmodified, under the assumption that T is of absolute restricted Roth type. Theorem B.4 (Generalization of Theorem 3.22 of [38]). — Let T be an i.e.m. of absolute restricted Roth type. Let κ be a nonnegative integer, and let r be a real number > 1. Let Γu (κ, T ) be a subspace of Γ(κ, T ) supplementing Γs (κ, T ). There F exist a real number δ¯ > 0, and bounded linear operators L0 : ϕ 7→ ψ from Cκκ+r ( Iα (T )) F ¯ to C κ+δ ([u0 (T ), ud (T )]) and L1 : ϕ 7→ χ from Cκκ+r ( Iα (T )) to Γu (κ, T ) such that F any ϕ ∈ Cκκ+r ( Iα (T )) satisfies ϕ = χ + ψ ◦ T − ψ.

The number δ¯ is the same than in Theorem 3.14. The derivation of Theorem 3.22 from Theorem 3.10 (the case κ = 0) is easy and done in Section 3.2 of [37] (see p.1602-1603). Following the same arguments, Theorem B.4 follows from Theorem 3.14. We hence will now focus on the adaptation of the proofs of Theorems 3.10 and 3.11, needed to prove Theorems 3.14 and B.1. B.3. Growth of B(0, nℓ ). — At many places in the proof of Theorems 3.10 and 3.11, one uses the following estimate, which we now prove under condition (a′ ) of absolute Roth type instead than condition (a) of Roth type. Proposition B.5. — Assume that condition (a′ ) of the definition of absolute Roth type is satisfied. Then, for any δ > 0, there exists a constant Cδ such that, for all k > 0, we have X kB(0, nℓ )k−δ 6 Cδ kB(0, nk )k−δ . ℓ>k

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Proof. — The desired estimate will be a consequence of the exponential growth of kB(0, nℓ )k, implied by the absolute Roth type condition, which will guarantee that the whole sum is controlled by its first term. Let v be a fixed vector in C (π(n0 )). The sequence |B(0, nℓ )v|1 (where | |1 is the ℓ1 -norm on RA ) is non decreasing and we will show that it grows fast enough to obtain the conclusion of the proposition. Let v(ℓ) := B(0, nℓ )v. By Lemma 3.12, one has max v(ℓ)α 6 CkB(nℓ−1 , nℓ )k min v(ℓ)α . α

α

We have therefore |v(ℓ + 1)|1 > |B(nℓ , nℓ + 1)v(ℓ)|1 > (1 + ckB(nℓ−1 , nℓ )k−1 )|v(ℓ)|1 . It follows that the smallest ℓ′ such that |v(ℓ′ )|1 > 2|v(ℓ)|1 satisfies, for any τ > 0 (and an appropriate constant Cτ ) ℓ′ 6 ℓ + Cτ kB(0, nℓ )kτ . Since for any fixed vector v ∈ C (π(n0 )) the sequence |v(ℓ)|1 doubles in at most Cτ kB(0, nℓ )kτ steps, the same holds for the norm kB(0, nℓ )k. Let ℓ0 = nk and define recursively ℓj+1 equal to the smallest ℓ′ such that kB(0, nℓ′ )k > 2kB(0, nℓj )k. The bound on ℓ′ − ℓ obtained above gives X X kB(0, nℓ )k−δ 6 kB(0, nk )k−δ Cτ kB(0, nℓj ) kτ 2−δ kB(0, nℓj )k−δ . j≥0

ℓ>k

Taking τ = 21 δ gives the required inequality. Along the same lines, one also have (with a very similar proof): Proposition B.6. — Assume that condition (a′ ) is satisfied. Then, for any δ > 0, there exists a constant Cδ such that, for all k > 0, we have X kB(0, nℓ )kδ 6 Cδ kB(0, nk )kδ . 06ℓ6k

B.4. Adaptation of the proof of Theorems 3.10 and 3.11 in [38]. — We here follow Sec(n) tion 3.5 of [38]. As mentioned above (Corollary 3.11), the bound from above for |Iα | is ′ still valid under condition (a ). The bound from below in Lemma 3.14 of [38] is not used in the proof of Theorem 3.11. The sequence (nk ) which appears in the proof of Theorem 3.11 (and was defined at the end of Section 2.5 of [38]) is the sequence that we have called (e nk ) in Section 3.3 of the present text. We have to replace it by the sequence (nk ) of condition (a′ ). The only property satisfied by (e nk ) under condition (a) which is not satisfied by (nk ) (n) under condition (a′ ) is the lower bound for |Iα | in Lemma 3.14.

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B.4.1. Adaptation of the proof of Lemma 3.15 in [38]. — In [38], we refer for the proof to an annex in [37]. The proof in this paper depends on a claim (Proposition B.7 below) which is still true, but the proof has to be modified. Proposition B.7. — Assume that the rotation number γ satisfies condition (a′ ). For every τ > 0, there exists C(τ ) > 0 such that, for any 0 6 ℓ 6 k, one has kB(n0 , nk )k 6 kB(n0 , nℓ )kkB(nℓ , nk )k 6 C(τ )kB(n0 , nk )k1+τ Proof. — The first inequality is trivial. To prove the second, choose v ∈ C (π (n0 ) ), v ′ ∈ C (π (nℓ ) ), and define w := B(n0 , nℓ ) · v. One has kB(n0 , nk )k > kvk−1 kB(n0 , nk )vk = kvk−1 kB(nℓ , nk )wk, kB(n0 , nℓ )k 6 C(π(n0 ))kvk−1 kB(n0 , nℓ )vk = C(π(n0 ))kvk−1 kwk, and therefore kB(n0 , nℓ )kkB(nℓ , nk )k kB(nℓ , nk )kkwk 6 C(π (n0 ) ) . kB(n0 , nk )k kB(nℓ , nk )wk From Lemma 3.12, one has max wα 6 CkB(nℓ−1 , nℓ )k min wα . α

α

and therefore kB(nℓ , nk )kkwk 6 C ′ kB(nℓ−1 , nℓ )k 6 Cτ kB(n0 , nk )kτ , kB(nℓ , nk )wk giving the required estimate. The proof of Lemma 3.16 in [38] is completely similar to the proof of Theorem 3.11 of [38] and the same remarks for its generalization apply. B.4.2. Space and Time decompositions. — Following Sections 3.7 and 3.8 in [38], one has just to remember that the sequence nk in the present setting is not the same than in [38]. Otherwise, the same formalism works and yields the same type of space and time decompositions. Section 3.9 also does not need any modification (except, as always, for the change of definition of the (nk )). B.4.3. General Hölder estimates. — We now follow Section 3.10 of [38]. The bound from below in Lemma 3.14 is used (for the first and last time) at the end of this section. However, Proposition 3.13 above is an adequate substitute. As in Section 3.10 in [38], we write any interval (x− , x+ ) as a countable union of intervals J which belong to some P (ℓ), ℓ > ℓmin (We use ℓmin instead of k because we are going to use Proposition 3.13). (n ) Let J ∈ P (ℓ) be one of these intervals. It is a translated copy of some Iα ℓ . We apply Proposition 3.13. It gives an integer k such that — J is the union of at most (s − 1) intervals of P (k); — |J| > Cτ−1 |I (0) | kB(0, nk )k−τ −1 .

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From Lemma 3.20 of [38] and the first item above, we have that (⋆)

|∆(J)| 6 CkB(0, nk )k−δ2 kϕkC r .

(where ∆(J) is the diﬀerence between the values of ψ at the endpoints of J). Let kmin be the minimal value of k when J runs among the intervals of the space decomposition of (x− , x+ ). We have then |x+ − x− | > Cτ−1 |I (0) | kB(0, nkmin )k−τ −1 . For ℓmin 6 m, there are at most kB(nm , nm+1 )k intervals in P (m) in the space decomposition of (x− , x+ ). For intervals J with m 6 kmin , we use (⋆). For the other intervals with m > kmin , we just use Lemma 3.20. We get X |ψ(x+ ) − ψ(x− )| 6 CkϕkC r {kB(0, nkmin )k−δ2 kB(nm , nm+1 )k ℓmin 6mkmin

6 C ′ kϕkC r

X

kB(0, nm ))k−δ2 /2

m>kmin

6 C kϕkC r kB(0, nkmin )k−δ2 /2 . ′′

We have used Propositions B.5 and B.6 (with δ = δ2 /2). The proof of the required inequality is complete. This concludes the modifications required to prove Theorem B.1 and Theorem 3.14, and hence also Theorem B.4.

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S. Marmi, Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126, Pisa, Italy E-mail : [email protected] C. Ulcigrai, Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland, and School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom • E-mail : [email protected] J.-C. Yoccoz

ASTÉRISQUE 416

Astérisque 416, 2020, p. 133–180 doi:10.24033/ast.1112

SMOOTH SIEGEL DISKS EVERYWHERE by Artur Avila, Xavier Buff & Arnaud Chéritat

Abstract. — We prove the existence of Siegel disks with smooth boundaries in most families of holomorphic maps fixing the origin. The method can also yield other types of regularity conditions for the boundary. The family is required to have an indiﬀerent fixed point at 0, to be parameterized by the rotation number α, to depend on α in a Lipschitz-continuous way, and to be non-degenerate. A degenerate family is one for which the set of non-linearizable maps is not dense. We give a characterization of degenerate families, which proves that they are quite exceptional. Résumé (De l’ubiquité des disques de Siegel à bord lisse). — Nous démontrons l’existence de disques de Siegel à bord lisse dans la plupart des familles de fonctions holomorphes fixant l’origine. La méthode peut également donner d’autres types de régularité pour le bord. On demande à la famille d’avoir un point fixe indiﬀérent en 0, d’être paramétrisée par le nombre de rotation α, de dépendre d’α de façon Lipschitzcontinue et d’être non-dégénérée. Une famille est dite dégénérée si l’ensemble de ses applications non-linéarisables n’est pas dense. Nous donnons une caractérisation des familles dégénérées, qui prouve qu’elles sont assez exceptionnelles.

Introduction In [24], Pérez-Marco was the first to prove the existence of univalent maps f : D → C having Siegel disks compactly contained in D and with smooth (C ∞ ) boundaries. The methods in [24] can in fact give any class of regularity below analytic, in particular quasi-analytic classes, but also maps that are C α for a prescribed α but for no bigger α, etc. However the maps produced in [24] do not a priori have an extension to an entire map, let alone polynomial. In [3], we were able to adapt some of these techniques and show the existence of quadratic polynomials having Siegel disks with smooth boundaries (for a simplification of the proof, and an extension to unisingular meromorphic maps, see [17]). In [11], two authors of the present article 2010 Mathematics Subject Classification. — 37F50; 37F25. Key words and phrases. — Rotation domains, linearization, sector renormalization.

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proved the existence of quadratic polynomials having Siegel disks whose boundaries have any prescribed regularity between C 0 and analytic (excluded). In this article, we generalize these results to most families of maps having a non persistent indiﬀerent cycle. Definition 1 (Non-degenerate families). — Assume I ⊂ R is an open interval. Consider a family of holomorphic maps fα : D → C parameterized by α ∈ I, with fα (z) = e2πiα z + O (z 2 ) and assume that fα depends continously (1) on α. We say that the family is degenerate if the set {α ∈ I ; fα is not linearizable} is not dense in I. Otherwise it is called nondegenerate. This definition is purely local so if we are given a holomorphic dynamical system on a Riemann surface, we can extend the definition above by working in a chart and restricting the map to a neighborhood of the fixed point. For example, if fα is a family of rational maps of the same degree d ≥ 2, then it is automatically non-degenerate. Indeed, a fixed point of a rational map of degree ≥ 2 whose multiplier is a root of unity is never linearizable. (2) In Appendix A we characterize degenerate families in the case where the dependence with respect to the parameter α is analytic. Notation 2. — Assume f : D → C is a holomorphic map having an indiﬀerent fixed point at 0. We write — K(f ) the set of points in D whose forward orbit remains in D and — ∆(f ) the connected component of the interior of K(f ) that contains 0; ∆(f ) = ∅ if there is no such component. Remark (Siegel disks). — If ∆(f ) 6= ∅ it is known that ∆(f ) is simply connected (3) and that the restriction f : ∆(f ) → ∆(f ) is analytically conjugate to a rotation via a conformal bijection between ∆(f ) and D sending 0 to 0, see Section 1.3. The set ∆(f ) is usually called a Siegel disk in the case α ∈ / Q and we will use the same terminology in this article for the case α ∈ Q, though subtleties arise. See Section 1.1 We prove here that the main theorem in [11] holds for a non-degenerate family under the assumption that the dependence α 7→ fα is Lipschitz. (4) We thus get in particular (see Appendix B for the general statement): (1)

This means: (α, z) 7→ fα (z) is continuous. This is a simple and classical fact, that seems diﬃcult to find in written form. If an iterate of f is conjugate on an open set U to a finite order rotation then a further iterate of f is the identity on U . Since f n is holomorphic on the Riemann sphere, it is the identity everywhere. This contradicts the fact that f n has degree dn > 1. (3) This is another classical fact. See Footnote 8 in Section 1.1. (4) By this we mean: (∃C > 0) (∀α ∈ I, α′ ∈ I, z ∈ D), |fα′ (z) − fα (z)| ≤ C|α′ − α|.

(2)

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Theorem 3. — Under the non-degeneracy assumptions of Definition 1, if moreover the dependence α 7→ fα is Lipschitz then — ∃α ∈ I \ Q such that ∆(fα ) is compactly contained in D and ∂∆(fα ) is a C ∞ Jordan curve; — ∃α ∈ I \Q such that ∆(fα ) is compactly contained in D and ∂∆(fα ) is a Jordan curve but is not a quasicircle; — ∀n ≥ 0, ∃α ∈ I \ Q, ∆(fα ) is compactly contained in D and ∂∆(fα ) is a Jordan curve which is C n but not C n+1 . A family satisfying the assumptions on the interval I also satisfies them on every sub-interval. It follows that the parameters α in the theorem above are in fact dense in I. Remark. — If fα is a restriction of another map gα and ∆(fα ) ⋐ D, then ∆(fα ) = ∆(gα ), see Section 1.1. So the result gives information on the Siegel disks not only of maps from D to C but in fact of any kind of analytic maps, for instance polynomials C → C, rational maps S → S, entire maps C → C, . . . Remark. — The main tool for Theorem 3 is Yoccoz’s sector renormalization as in several of our previous works (except [2] who uses Risler’s work instead [27]). In [2], [3] and [11] it was crucial to have a family for which it is known that fα is linearizable if and only if α is a Brjuno number. The progress here is to get rid of this assumption. (5) 1. Conformal radius, wild combs and the general construction. The method that Buﬀ and Chéritat first developped to get smooth Siegel disks is one of the oﬀsprings of a fine control, initiated in [14], on the periodic cycles that arise when one perturbs parabolic fixed points. Still today we can only make it work in specific contexts, which includes quadratic polynomials for instance. With the smooth Siegel disk objective in mind, Avila was able in [2] to identify essential suﬃcient properties so as to allow for a partial generalization, and also pointed to the bottleneck for a complete generalization. In this section, we essentially follow the presentation in [2]. We also mention a connection with continuum theory. In this whole section, except Section 1.1, we consider a non-empty open interval I ⊂ R and a continuous family of analytic maps fα : D → C parameterized by α ∈ I with fα (z) = e2πiα z + O (z 2 ). 1.1. Siegel disks and restrictions. — Given a one dimensional complex manifold S and a holomorphic map f :U →S (5)

Note that in [17], optimality of Brjuno’s condition was not required. However, it is assumed that f has a meromorphic extension to C that has only one non-zero critical or asymptotic value.

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defined on an open subset U of S, assume there is a neutral fixed point a ∈ U of multiplier e2πiα with α ∈ R. Call rotation domain any open set containing a on which the map is analytically conjugate to a rotation on a Euclidean disk or on the plane or on the Riemann sphere. (6) If α ∈ / Q then the rotation domains are totally ordered by inclusion. This is never the case if α ∈ Q. If α ∈ / Q there is a maximal element for inclusion, called the Siegel disk (7) of f at point a. If α ∈ Q existence of a maximal element may also fail, depending on the situation. If α ∈ / Q the Siegel disk of a restriction is automatically a subset of the original Siegel disk. If α ∈ Q this may fail. Remark. — The right approach in the general case is probably to use the Fatou set. Here is not the place for such a treatment, so we only give results specific to our situation In the sequel we assume S = C and U is bounded and simply connected. Following [32], Section 2.4, we let K be the set of points whose orbit is defined for all times. The set K ⊂ U is not necessarily closed in U . We let ∆(f ) = ∆ ⊂ U where ∆ is the connected component containing a of the interior K ◦ , or ∆ = ∅ if a∈ / K ◦ . Then ∆ is necessarily simply connected: this is one classical application of the maximum principle. (8) Any rotation domain for a is necessarily contained in K. It follows that any rotation domain for a is in fact contained in ∆. Moreover, let us prove that ∆ itself is a rotation domain: Proof. — First note that we have f (∆) ⊂ ∆. The set ∆ is conformally equivalent to the unit disk D. Conjugate f by a conformal map from ∆ to D sending a to 0. We get a holomorphic self-map of D with a neutral fixed point at its center. By the case of equality in Schwarz’s lemma this self-map is a rotation. Corollary 4. — (We do not make an assumption on α.) Let U ′ be an open subset of C. Let g : U ′ → C be holomorphic with a neutral fixed point a. Assume U is an open subset of U ′ containing a and let f be the restriction of g to U . Then ∆(f ) ⊂ ∆(g). If moreover U and U ′ are simply connected and if ∆(f ) is compactly contained in the domain of definition of f then ∆(g) = ∆(f ). Proof. — The first claim is immediate. For the second claim when ∆(f ) is compactly contained in U , consider the image of ∆(f ) by the uniformization (∆(g), 0) → (D, 0): we get a simply connected subset A of D, invariant by the rotation. In the case α ∈ /Q (6)

The last case is extremely specific, for we must have U = S isomorphic to the Riemann sphere and f is a rotation. (7) In the case of a rotation on the Riemann sphere this name is not appropriate since the disk is a sphere. . . (8) If ∆ would not be simply connected then there would exist a bounded closed set C 6= ∅ (not necessarily connected) such that ∆ ∩ C = ∅ and such that ∆′ := ∆(f ) ∪ C is open and connected (this is a theorem in planar topology). By the maximum principle, f k (∆′ ) ⊂ D for all k. Hence ∆′ would be an open subset of K(f ), contradicting the definition of ∆.

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this has to be a disk B(0, r) with r ≤ 1. If α ∈ Q, more sets are possible. In any case if A is not equal to D we can construct a connected invariant open subset of K(f ) that strictly contains ∆(f ), leading to a contradiction. In the case U = D our definition of ∆ coincides with Notation 2. 1.2. Properties of the conformal radius as a function of the angle. — Recall that we consider a non-empty open interval I ⊂ R and a continuous family of analytic maps fα : D → C parameterized by α ∈ I with fα (z) = e2πiα z + O (z 2 ). The set ∆(fα ) has been defined in Notation 2, and Section 1.1 gave a mild generalization and basic properties. Definition 5. — We let r(α) denote the conformal radius at 0 of ∆(fα ) if it is not empty. Otherwise we set r(α) = 0. Recall that the conformal radius of a simply connected open subset U of C at a point a ∈ U is defined as the unique r ∈ (0, +∞] such that there exists a conformal bijection ϕ : B(0, r) → U with ϕ(0) = a and ϕ′ (0) = 1 . Proposition 6. — Let B denote the set of Brjuno numbers. (9) The function α 7→ r(α) has the following properties: 1. It is upper semi-continuous: ∀α ∈ R, lim supx→α r(x) ≤ r(α). 2. It takes positive values at Brjuno numbers: α ∈ B =⇒ r(α) > 0. 3. It is weakly lower semi-continuous on the left and on the right at every Brjuno number: α ∈ B =⇒ lim supx→α− r(x) ≥ r(α) and lim supx→α+ r(x) ≥ r(α). Weak lower semi-continuity on each side can be rephrased as follows: there exists αn < α < αn′ with αn −→ α and αn′ −→ α such that lim r(αn ) ≥ r(α) and lim r(αn′ ) ≥ r(α). Since f is upper semi-continuous, these limits are in fact equal to r(α). The first property is classical, see Lemma 10 or Proposition 1, page 19 of [32]. The second is Brjuno’s theorem, [8, 9, 10, 28, 32]. The third follows from Risler’s work [27] or from a fine study of Yoccoz’s renormalization [3]. According to the method by Buﬀ and Chéritat explained in [3], to get lim inf r(αn ) ≥ r(α) it is enough to take a sequence αn such that Φ(αn ) → Φ(α) where Φ denotes Yoccoz’s variant of the Brjuno sum. Both references also imply: Complement 7. — In Item 3 above, one can take sequences αn and αn′ that are bounded type numbers (Diophantine of order 2). For instance if α = a0 + 1/(a1 + 1/ . . .) = [a0 ; a1 , . . .] is the continued fraction expansion of α ∈ B then the sequences (θ2n ) and (θ2n+1 ) work, where θn = [a0 ; a1 , . . . , an , 1 + an+1 , 1, 1, 1, . . .]. Indeed θn → α, alternating on each side (9)

See [32] for a definition.

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of α and one can check that Φ(θn ) → Φ(α), as follows for instance from the remark after Proposition 2 in [3]. Note that in the particular case of a family fα that depends on α in a Lipschitzcontinuous way, Complement 7 is also a corollary of the main lemma of the present article, Lemma 15. 1.3. Properties of the linearizing map. — Consider fα and recall that K(fα ) is defined as the set of points whose orbit stays in D. Assume that the interior of K(fα ) contains 0 and recall that ∆ = ∆(fα ) is defined as the connected component containing 0 of the interior of K(fα ). Recall that ∆ is simply connected. Any uniformization ϕ : rD → ∆ with ϕ(0) = 0 must linearize fα because: first f (∆) ⊂ ∆, second the conjugate is a self-map of rD that fixes 0 with multiplier of modulus one, so the case of equality of Schwarz’s lemma implies that it is a rotation. Notation 8 (Linearizing map). — We let ϕα : r(α)D → ∆(fα ) be the unique uniformization such that ϕα (0) = 0 and ϕ′α (0) = 1. We also write Rα (z) = e2πiα z. Now assume α is irrational. It is well known then that there is a unique formal power series Φ(X) with Φ(X) = X + O (X 2 ) and Φ ◦ Rα = fα ◦ Φ. It follows that when fα is linearizable, Φ is the power series expansion of ϕα . In particular the radius of convegence of Φ is greater or equal to r(α). (10) Consider any holomorphic map ψ satisfying ψ(0) = 0, ψ ′ (0) = 1 and such that ψ ◦ Rα = fα ◦ ψ holds near 0. Then f is linearizable and if α ∈ / Q then ψ must coincide with ϕα near 0: this can be seen either by comparing to Φ as above, or more directly: ϕ−1 α ◦ ψ has derivative 1 at 0, commutes with Rα and α is irrational so its power series expansion is reduced to a linear term only. Lemma 9 (Convergence of the linearizing maps). — Consider αn → α and let ρ = lim inf r(αn ). Then r(α) ≥ ρ. (11) Assume that ρ > 0. If α ∈ Q assume moreover that ρ ≥ r(α), i.e., ρ = r(α). Then ϕαn −→ ϕα uniformly on compact subsets of ρD. Proof. — If ρ > 0 then the restrictions of ϕαn to rn D with rn = min(ρ, r(αn )) −→ ρ take values in D thus form a normal family. Consider any extracted limit ϕ : ρD → C of these restrictions. Since ϕαn (0) = 0 and ϕ′αn (0) = 1 we have ϕ(0) = 0 and ϕ′ (0) = 1. Hence the limit is not constant and thus takes values in D. Passing to the limit in ϕαn ◦ Rαn = fαn ◦ ϕαn we get that ϕ ◦ Rα = fα ◦ ϕ on ρD. It follows that ϕ takes values in K(fα ) and since it is open, in the interior of K(fα ) hence in ∆(fα ). By ′ Schwarz’s inequality applied to ϕ−1 α ◦ ϕ we have |ϕ (0)| ≤ r(α)/ρ, i.e., r(α) ≥ ρ. If (10)

They do not have to be equal, as the maps fα may extend beyond D and the extension may well have a bigger Siegel disk. (11) It follows that r(α) ≥ lim sup r(αn ), see Lemma 10.

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ρ ≥ r(α) then we are in the case of equality of Schwarz’s lemma, and hence ϕ−1 α ◦ ϕ is the identity on ρD. Alternatively, if α ∈ / Q then by the uniqueness argument before the statement of the present lemma, ϕ must be equal to ϕα near 0. By analytic continuation of equalities, they coincide on all of ρD. Remark. — Note that the second claim sometimes fails if α ∈ Q if we do not assume ρ ≥ r(α): for instance to build a counterexample for α = 0 one may consider a vector field dz/dt = χ(z) which has a singularity at 0 (i.e., χ(0) = 0) with eigenvalue χ′ (0) = 2πi but is not linear and let ft be the restriction to D of the time-t map associated to this vector field. Then as t 6= 0 tends to 0, the complement of the interior of K(ft ) tends, in the sense of Hausdorﬀ, to the complement of the interior of the set K(χ) where K(χ) denotes the set of points in D whose forward orbit by the vector field is defined for all times and never leaves D. Moreover if t is irrationnal, then ∆(ft ) is in fact independent of t and equal to the component containing 0 of the interior of K(χ). On the other hand f0 = id hence K(f0 ) = D. Let us prove the first point in Proposition 6: Lemma 10 (Upper semi-continuity of r). — If αn −→ α then r(α) ≥ lim sup r(αn ). Proof. — Let ρ = lim sup r(αn ). For a subsequence an[k] we have r(αn[k] ) −→ ρ. The claim then follows from the first conclusion of Lemma 9 applied to the subsequence. 1.4. A remark on continuum theory. — (This section can be skipped as it is not necessary in the rest of the article.) A continuum is a non-empty, compact and connected metrizable topological space. In continuum theory, there is an object called the Lelek fan. It is a universal object in the sense that any continuum with some specific set of properties (see [12, 1]) is homeomorphic to the Lelek fan. A variant is the following, called straight one sided hairy arc in [1], that they abbreviate sosha, but we prefer to call it here a wild comb. The Lelek fan can be recovered from this continuum by contracting the base to a point. Definition 11 (The wild comb). — A straight one sided hairy arc is the sub-graph C = (x, y) ; 0 ≤ y ≤ f (x)

of a function f : [0, 1] → [0, +∞) such that: — f is upper semi-continuous, (12) — f is weakly lower semi-continuous on the left and on the right, — both x ∈ [0, 1] ; f (x) > 0 and f −1 (0) are dense in [0, 1], — f (0) = 0 = f (1). Its base is the segment (x, 0) ; x ∈ [0, 1] . (12)

See Proposition 6 for a definition.

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The first condition is equivalent to C being closed. In this article, we call it a wild comb or the wild comb to emphasize that it also possesses a form of topological uniqueness (see [1]). Under the condition of the above definition, it was proved in [1], Proposition 2.4, that the image by f of an interval [a, b] ⊂ [0, 1] with a < b is of the form [0, M ] for some M > 0. In particular: though f is highly discontinuous, it satisfies the intermediate value property. Also, C is the closure of the graph of f (Corollary 2.5 in [1]). The fact that the closure is contained in C follows from f being upper semi-continuous and non-negative. The fact that this closure contains C means that for any (x, y) with 0 ≤ y ≤ f (x) there exists a sequence xn → x such that f (xn ) → y. In fact if x 6= 0 or 1 then there is such a sequence satisfying xn < x and there is another satisfying xn > x. From the wild comb people have derived topological models for the Julia set of some exponential maps [1, 4], and for the hedgehogs (13) associated to non-linearizable fixed points of some polynomials [13]. We will also see here a wild comb, though not as a subset of some dynamical plane, but as the subgraph of the function α 7→ r(α) in the special case of Section 1.5. As a notable resurgence of smoothess, it was proved in [5] that some nonlinearizable holomorphic maps have hedgehogs that contain a Cantor set of smooth hairs. (14) 1.5. Special case: assuming Brjuno’s condition is optimal. — Assume here that we have a family for which we know that the Brjuno condition is optimal, in the sense that r(α) > 0 =⇒ α ∈ B where B denotes the set of Brjuno numbers. The first family for which optimality has been known is the family of degree two polynomials, thanks to the work of Yoccoz, see [32]. Lemma 12. — For all α ∈ I and all y with 0 ≤ y < r(α), the set r−1 (y) (is nonempty and) accumulates α on the left and on the right. In other words there exists αn < α < αn′ with αn → α and αn′ → α and such that r(αn ) = y = r(αn′ ). Proof. — (from [2]) Since R \ B is dense, there is a dense set on which r = 0. In particular the case y = 0 is trivial. Assume y > 0. Arbitrarily close to α, there are b ∈ I such that r(b) = 0. Assume b < α. Consider then K = x ∈ [b, α] ; r(x) ≥ y , which is non-empty because α ∈ K, and let c = inf K. By upper semi-continuity r(c) ≥ y. In particular r(c) 6= 0 hence c 6= b and c ∈ B by optimality assumption. If we had r(c) > y then by weak lower semi-continuity on the left at Brjuno numbers (Proposition 6), there would be some c′ ∈ (b, c) with r(c′ ) > y, contradicting the definition of c. The same holds if b > α using weak lower semi-continuity on the right. Finally r(c) = y 6= r(α) ensures that c 6= α. (13)

Pérez-Marco in [23] proved the existence of non-trivial totally invariant compact connected sets containing irrational non-linearizable fixed points that he named hedgehogs and developed their theory. He used them to provide a dictionary with the dynamics of analytic circle diﬀeomorphisms. (14) What is called a comb in [6] is diﬀerent from what we call a wild comb here.

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Remark. — Consider an interval [u, v] ⊂ I with u < v and r(u) = 0 = r(v) (there are plenty by hypothesis), and let C be the subgraph of r restricted to [u, v]. Then C is a wild comb as per Definition 11: we imposed r(u) = 0 = r(v) and all remaining conditions are satisfied according to Proposition 6. This implies the lemma above, by the discussion in Section 1.4, and in fact the proof of the intermediate value property for a general wild comb boils down to the same arguments as the proof of Lemma 12. Since we assumed optimality of Brjuno’s condition, the values αn in Lemma 12 belong to B. By Complement 7 we get: Corollary 13. — For all α ∈ I and all y with 0 ≤ y ≤ r(α), there exists αn < α < αn′ with αn → α and αn′ → α and such that: αn and αn′ are bounded type irrationals and r(αn ) and r(αn′ ) both tend to y. In other words we gain information on the arithmetic type of αn at the cost of weakening “r(αn ) = y” into “r(αn ) → y”. Note also that we gain the ability to reach y = r(α). 1.6. Smooth Siegel disks. — Recall the standing assumption that the family depends continuously on α. In [2, 3] is shown how to get smooth Siegel disks from Lemma 12, which assumes Brjuno’s condition is optimal. We adapt here the proof so that it does not use optimality directly but only depends on the following condition: (15) Condition 14. — For every α ∈ B and every ρ ∈ R with 0 < ρ < r(α), there exists a sequence of bounded type numbers αn −→ α such that r(αn ) −→ ρ. For a family on which the Brjuno condition is optimal, Corollary 13 implies that Condition 14 holds. Better: it provides a sequence on each side, though the construction below does not need it. Recall that the point of this article is to get rid of the hypothesis that the Brjuno condition is optimal: we will prove in Corollary 18 that Condition 14 holds whenever the family is Lipschitz-continuous with respect to the parameter, and non-degenerate as per Definition 1. We now begin the construction of a smooth Siegel disk assuming Condition 14. Recall that ϕα denotes the unique conformal bijection from r(α)D → ∆(fα ) such that ϕα (z) = z + O (z 2 ) and that ϕα linearizes fα .

(15) Other conditions are suﬃcient to apply the methods of [11]. For instance we can replace the assumption α ∈ B by α having bounded type. Also, bounded type numbers in the hypothesis and conclusion can be replaced by Herman numbers.

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Construction of a sequence θn . — Start from any θ0 ∈ B, so that r(θ0 ) > 0 and also θ0 ∈ / Q. Choose some target radius ρ ∈ (0, r(θ0 )). Choose also a strictly decreasing sequence ρn such that ρ0 = r(θ0 ) and ρn −→ ρ. Consider then a sequence αk → θ0 , provided by Corollary 13, such that r(αk ) −→ ρ1 and αk ∈ B. From the properties of linearizing maps (Lemma 9) and θ0 ∈ / Q it follows that ϕαk → ϕθ0 uniformly on every compact subset of ρ1 D. Since these are holomorphic maps, the same is true for their derivatives of all orders. We let θ1 = αk for a choice of k such that the restriction of ϕθ1 − ϕθ0 to the closure of ρD is less than 1/2 (for the sup norm) and such that its first derivative is less than 1/4. Since αk ∈ B it follows that θ1 ∈ / Q. Then we choose some open interval I1 of length ≤ 1/2, containing θ1 , and such that α ∈ I 1 =⇒ r(α) ≤ ρ0 (upper semi-continuity of r at θ1 ). We also ask that the the closure of I1 lies at positive distance from Z, which is possible since θ1 ∈ / Z. We then continue the inductive construction: given n ≥ 2, once θn−1 ∈ B and In−1 with θn−1 ∈ In−1 have been constructed we choose θn of the form αk where αk ∈ B is a sequence provided by Corollary 13 tending to θn−1 such that r(αk ) −→ ρn . The index k is chosen big enough so that: the restriction to the closure of ρD of ϕθn − ϕθn−1 is less than 1/2n , its first derivative less than 1/2n+1 and so on up to its n-th derivative less than 1/2n+n . It is also chosen big enough so that θn belongs to the interior of In−1 , a condition that we did not have for n = 1. Then we choose an open sub-interval In ⊂ In−1 of lenght ≤ 1/2n , containing θn , such that α ∈ I n =⇒ r(α) ≤ ρn and such that the closure of In lies at positive distance from n1 Z. And so on. . . Properties of the limit θ. — The intersection of the closures of the In is a singleton and θn tends to this value. Since r(θn ) = ρn −→ ρ, upper semi-continuity of r implies r(θ) ≥ ρ. By the defining properties of In , which contains θ, we get r(θ) ≤ ρn for all n so r(θ) ≤ ρ. Hence r(θ) = ρ. Also θ belongs to the closure of In for all n, so the distance from θ to n1 Z is positive for all n > 0 hence θ ∈ / Q. The conditions on ϕθn − ϕθn−1 implies the uniform convergence on the closure of ρD of the derivatives of all orders of ϕθn . Since θ ∈ / Q, Lemma 9 implies that ϕθn → ϕθ uniformly on compact subsets of ρD. (16) It follows that ϕθ has a C ∞ extension ϕ˜ to the closure of ρD. The image of the circle ρ∂D by this extension is the boundary of ∆(fθ ). By a straightforward modification of the construction above, we can ensure that the curve is compactly contained in D. Then ϕ˜ ◦ Rθ = fθ ◦ ϕ˜ also holds on the boundary circle. Hence the derivative of ϕ˜ on cannot vanish on this circle, for if it did, it would vanish on a dense subset using the equation above, hence everywhere on the circle, whence ϕ˜ would be constant by standard properties of holomorphic functions. The map ϕ˜ is also injective on the boundary circle (see [20], Lemma 18.7 page 193; it is stated for rational maps but is valid as soon as the rotation number is irrational and the Siegel disk compactly contained in the domain of the map). (16)

However even if we had θ ∈ Q, since r(θn ) → r(θ), we would still have ϕθn → ϕθ .

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Hence ∂∆(fθ ) is a C ∞ Jordan curve compactly contained in D. Remark. — An interesting feature of this construction is that we were able to prescribe the conformal radius of the Siegel disk. (17) 1.7. Other regularity classes for the boundary. — The construction in [11] of boundaries that are C n but not C n+1 , and of other examples (see Appendix B), is a refinement of the previous method. In the process we loose the ability to exactly prescribe the conformal radius. To apply the method of [11] and thus get Theorem 3, it is enough to have a continuous family of maps fα such that Condition 14 holds. In [11] there are two supplementary condition, but we can remove them: — The maps fα must be injective. But if the given family fα contains non-univalent maps, we first restrict to a sub-interval I and restrict f to a small enough disk εD. The Siegel disk ∆′ that it produces for the restriction to εD of f is compactly contained in εD and thus ∆(f ) = ∆′ by the end of Corollary 4. — The family must depend analytically on α. But it turns out that the proof given in [11] only uses continuity of the family and the fact that Condition 14 holds. Remark. — The proof that Condition 14 is enough is a bit elaborate so we refer the reader to [11]. Let us just mention that in the construction, to get obstructions to regularity we use as intermediate steps the existence of Siegel disks whose boundaries oscillate a lot. For this we use a theorem of Herman [19] (see also [23], statement B.3 (i) page 251): if α is a Herman number (this includes all bounded type numbers), (18) and if f is univalent then ∆(fα ) cannot (19) be compactly contained in D. Now if we have a sequence αn → α of Herman numbers such that r(αn ) → ρ ∈ (0, r(α)) then the sets ∂∆(fαn ) have a point in ∂D but also points close to the fα -invariant curve ϕα (ρ∂D) ⊂ ∆(α). See [11] for the rest of the argument. As an alternative to Herman’s theorem, we can use the following consequence of [18] (whose methods are quite diﬀerent from [19]): If α has bounded type then ∂∆ either meets ∂D or contains a critical point of f . Note that it is known only for bounded type rotation numbers. 1.8. General case. — Here we prove that Condition 14 holds for families that are non-degenerate (in the sense of Definition 1) and for which the dependence on α is Lipschitz continuous. This yields Theorem 3. (17)

By a linear change of variable z 7→ λ(α)z with λ continuous, we can thus prescribe the conformal radius to coincide with a continous function of α. (18) Herman numbers are defined as the set of irrational numbers α for which every orientation preserving analytic circle diﬀeomorphism of rotation number α has its Poincaré conjugacy to the rotation that is analytic too. Notably, Yoccoz gave a diophantine characterization of the set of Herman numbers, i.e., a complete determination in terms of simple manipulations of α, see [31]. (19) The Herman condition is optimal in this respect: Pérez-Marco proved that if α is not Herman, then there is a univalent map of rotation number α and whose Siegel disk is compactly contained in D, see statement B.3 (ii) page 251 in [23].

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This is based on the following perturbation lemma, to the proof of which we devote the whole of Section 2. Lemma 15 (Main lemma: perturbation of a rotation). — Let α ∈ R and let [a0 ; a1 , . . .] be its continued fraction expansion. If α ∈ Q then it has two such expansions (20) and both are finite : choose one, [a0 ; a1 , . . . , ak ]. √ 1. If α ∈ / Q let αn = [a0 ; a1 , . . . , an , 1 + an+1 , 1 + 2]. √ 2. If α ∈ Q let αn = [a0 ; a1 , . . . , ak , n + 1 + 2]. Assume that fn are holomorphic functions defined on D with fn (z) = e2πiαn z + O (z 2 ) and that fn tends to Rα in a Lipschitz way with respect to αn − α, i.e., |fn (z) − Rα (z)| ≤ K|αn − α| for some K ≥ 1.

(21)

Then

1. If α ∈ / Q then lim inf r(fn ) ≥ 1. 2. If α = p/q in irreducible form, then lim inf r(fn ) ≥ exp(−C(K, q)). Here C(K, q) > 0 is independent of α, of the sequence fn and of αn and satisfies the following: for all integers q ≥ 1, K 7→ C(K, q) is a continuous non-decreasing function of K ≥ 1; for all fixed K ≥ 1 we have C(K, q) −→ 0 as q → +∞. Remark 16. — If α ∈ B then Lemma 15 is already known: it follows from [11] or [27]. So the novelty is for non-Brjuno numbers and rational numbers. In Section 2 we prove that the following value of C(K, q) works: log q log K c1 + + q q q for some c1 > 0. This estimate may be non optimal. Note that αn → α. If α ∈ / Q then α2n < α < α2n+1 . If α ∈ Q then αn is on one side of α or the other depending√on which of the two continued fraction was chosen. Of course the choice √ of 1 + 2 is somewhat arbitraty and many other variants hold. Recall that 1 + 2 = [2; 2, 2, 2, . . .], so it is a close relative to the golden mean √ 1+ 5 = [1; 1, 1, 1, . . .], that we can use instead. Note that the class of αn mod Z only 2 depends on the class of α mod Z (and on n). The fact that we do not get lim inf r(fn ) = 1 when α ∈ Q is not just a limitation of our method. Indeed, as in the remark following Lemma 9, consider a vector field z˙ = χ(z) defined in a neighborhood of the closed unit disk and with (1)

(20)

C(K, q) =

See the end of Section 2.4. We necessarily have K ≥ 1 by Schwarz’s lemma comparing derivatives at the origin. To allow for smaller values of K we would compare fn to Rαn instead of Rα . However we are not interested in small values of K in this article. (21)

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χ(z) = 2πiz + O (z 2 ) and let ht be the associated time-t map. If the vector field is invariant by the rotation R1/q we may set ft = Rp/q ◦ ht−p/q . Then the family satisfies supD |ft − Rp/q | ∼t→p/q K|t − p/q| with K = supD |χ|. For t irrational, its Siegel disk is independent of t and coincides with the maximal linearization domain for χ, which is not the whole unit disk, except if χ(z) = 2πiz for all z. As a consequence of the main lemma we now prove: Lemma 17 (Perturbation lemma for Lipschitz families). — Let I be a non-empty interval and (fα )α∈I a family of functions D → C with expansion fα (z) = e2πiα z + O (z 2 ) at 0. Assume that the family is K-Lipschitz for some K ≥ 1, i.e., ∀α, β ∈ I and ∀z ∈ D, |fα (z) − fβ (z)| ≤ K|α − β|. Then for all α ∈ I, if we write αn an associated sequence like in Lemma 15, we have 1. If α ∈ / Q then lim inf r(fαn ) ≥ r(fα ). 2. If α = p/q in irreducible form, then lim inf r(fαn ) ≥ r(fα )/ exp(C ′ (K, q)). Similarly to the previous lemma, C ′ (K, q) is independent of the family (fα ), for each q the function K 7→ C ′ (K, q) is continuous, non-decreasing and for each K ≥ 1, C ′ (K, q) −→ 0 as q → +∞. Here we can get: log q log K c2 + + , q q q where c2 is a positive universal constant. As in eq. (1), this estimate may be nonoptimal. (2)

C ′ (K, q) = 4

Proof. — If r(α) = 0 the claim is trivial so we assume r(α) > 0. First, we can immediately improve the inequality |fα (z) − fβ (z)| ≤ K|α − β| by Schwarz’s inequality because fα − fβ maps 0 to 0: |fα (z) − fβ (z)| ≤ K|α − β| · |z|. Consider the linearizing map ϕα . Let gβ = ϕ−1 α ◦ fβ ◦ ϕα . Then gα is the restriction of Rα to r(α)D. The maps gβ are defined on subsets dom gβ of the disk r(α)D that tend to this disk in the following sense: ∀r < r(α), ∃η > 0 such that |β − α| < η =⇒ rD ⊂ dom gβ . Fix for a moment a value r < r(α). Write ε = 1 − r/r(α) so that r = (1 − ε)r(α). Consider the family f˜β = r−1 gβ (rz) restricted to D and to values β such that |β − α| < η where η is as above, so that f˜β is indeed defined on the whole of D. We show that for β close enough to α the family f˜β is K ′ -Lipschitz for a constant K ′ that we determine.

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We will use the following two property of univalent maps, see [26], Theorem 1.3 page 9: if ϕ : D → C is holomorphic, injective and satisfies ϕ(z) = z + O (z 2 ) at 0 then |ϕ(z)| ≤ |z|/(1 − |z|)2 and |ϕ′ (z)| ≥ (1 − |z|)/(1 + |z|)3 . These bound are optimal because the Koebe function f (z) = z/(1 − z)2 reaches them. Transferring them to the function ϕα by letting ϕ(z) = r(α)−1 ϕα (r(α)z) this implies r 1 − r(α) |z| ′ ∀z ∈ rD, |ϕα (z)| ≤ |ϕα (z)| ≥ , r r (1 − r(α) (1 + r(α) )2 )3 which implies |z| ε |ϕ′α (z)| ≥ . ε2 8 From the lower bound on ϕ′α it follows that the family gβ satisfies the following estimate: ∀z ∈ rD, |gβ (z) − gα (z)| ≤ K(β)|β − α| · |ϕα (z)| ∀z ∈ rD,

|ϕα (z)| ≤

with K(β) −→ 8K/ε as β → α: this can be proved for instance by contradiction and extracted subsequences for |z| ∈ [r/2, r] and then by the maximum principle it extends to |z| < r. Transferring to f˜β and using the upper bound on ϕα (z) we get ∀z ∈ rD,

|f˜β (z) − f˜α (z)| ≤ K ′ (β)|β − α| · |z|

with K ′ (β) −→ 8K/ε3 so we can take a uniform K ′ = 9K/ε3 by requiring β to be close enough to α. We now apply Lemma 15 to the family f˜β . If α ∈ / Q we get that lim inf r(f˜αn ) ≥ 1 hence lim inf r(fαn ) ≥ r. Since this is true for all r < r(α) this implies lim inf r(fαn ) ≥ r(fα ). If α ∈ Q we get lim inf r(f˜αn ) ≥ exp(−C(K ′ , q)) with K ′ = 9K/ε3 . Recall that r = (1 − ε)r(α) hence lim inf r(αn ) ≥ r exp(−C(K ′ , q)) = r(α) exp(−Q) with Q = − log(1 − ε) + C(9K/ε3 , q). Since this is true for all ε ∈ (0, 1) we get lim inf r(αn ) ≥ r(α) exp(−C ′ (K, q)) with C ′ (K, q) := inf − log(1 − ε) + C(9K/ε3 , q) ; ε ∈ (0, 1) .

The map K 7→ C ′ (K, q) is continuous. One argument to prove this claim goes as follows: by Lemma 15 the map K 7→ C(K, q) is continuous and has a limit as K → +∞ because it is monotonous. Hence the expression Q extends to a continuous function of (ε, K) from [0, 1] × (0, +∞) to [0, +∞] where the topology is extended to include infinity in the range. This is a suﬃcient condition for the function K 7→ inf ε∈(0,1) Q(ε, K, q) to be continuous. Increasing K while fixing q and ε does not decrease Q hence K 7→ C ′ (K, q) is non-decreasing. For each q, fixing K and ε, we have Q −→ − log(1 − ε) when q → +∞ and this quantity can be made close to 0 by choosing ε small. Hence C ′ (K, q) −→ 0 when K is fixed and q → +∞.

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Proof of Equation (2) from eq. (1). — Using the notation of the proof above, we must derive an upper bound for the infimum over ε ∈ (0, 1) of − log(1 − ε) + c1 /q + 1q log(9Kq/ε3 ). This is a function of ε whose derivative has the following 1 3 − qε . So the function is strictly convex with infinite limits simple expression 1−ε at ε = 0 and ε = 1, and a unique minimum at ε = 1/(1 + q/3). We get C ′ (K, a) = log(1 + 3/q) + c1 /q + log(9Kq)/q + 3 log(1 + q/3)/q ≤ log(K)/q + 4 log(q)/q + c2 /q. Corollary 18. — If the family (fα ) is non-degenerate in the sense of Definition 1 and Lipschitz-continuous with respect to α then Condition 14 holds. Recall that Condition 14 is stated as follows: For every Brjuno number α and ρ ∈ R with 0 < ρ < r(α), there exists a sequence of bounded type numbers αn −→ α such that r(αn ) −→ ρ. Here we moreover get that there is such a sequence on each side of α. Proof. — We adapt the proof of Lemma 12. Let α ∈ B and ρ ∈ R with 0 < ρ < r(α). By the non-degeneracy assumption, arbitrarily close to α there are b ∈ I such that r(b) = 0. Choose one and assume for simplic ity that b < α (the other case is similar). Consider then K = x ∈ [b, α] ; r(x) ≥ ρ , which is non-empty because α ∈ K, and let c = inf K. By upper semi-continuity r(c) ≥ ρ. In particular r(c) 6= 0, hence c 6= b. Here, compared to Lemma 12, we cannot anymore deduce that c ∈ B because we do not assume optimality. Instead, we use Lemma 17 with c in place of α. It provides some special sequence αn −→ c of bounded type numbers. If c ∈ / Q we let cn = α2n < c. If c ∈ Q the sequence αn is either below or above c, depending which continued fraction of c was chosen among the two possible, so we choose it so that αn < c and let cn = αn . For all n big enough we have cn ∈ (b, c). Then by definition of c, we have r(cn ) < ρ. Now there are two cases. — Either c ∈ / Q. Then Lemma 17 states that lim inf r(cn ) ≥ r(c) so lim r(cn ) = ρ. We can thus choose n so that r(cn ) is arbitrarily close to ρ. — Or c ∈ Q. Then Lemma 17 states that lim inf r(cn ) ≥ r(c)/ exp(C ′ (K, q)) where q is the denominator of c = p/q written in irreducible form. So lim inf r(cn ) ∈ [ρ, ρ/ exp(C ′ (K, q))]. The number c ∈ (b, α) above depends on the choice of b. If we now let b tend to α then c tends to α and in particular: whenever c is rational its denominator q tends to +∞, so C ′ (K, q) tends to 0. As mentionned in Section 1.7, Condition 14 is all that is needed to get the results of [11]. Hence by the corollary above the results of [11] extend to all families that are Lipschitz and non-degenerate. In particular we have Theorem 3.

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2. Proof of the main lemma We will use a construction due to Douady and Ghys which has been made rigorous and quantified by Yoccoz (see [15, 32, 22, 27]), and is called sector renormalization. It has been treated in many articles and books since, so we will not motivate its construction here. 2.1. Lifts 2.1.1. — — — —

Definitions. — Let H denote the upper half plane, T : Z 7→ Z + 1. Tα : Z 7→ Z + α. For α ∈ R, let S (α) be the space of univalent (i.e., injective holomorphic) maps F : H → C such that F ◦ T = T ◦ F holds on H and such that F (Z) − Z −→ α as Im(Z) → +∞. We call α the rotation number of F even though it is rather a translation that F is compared to, and we write it α(F ). A map F ∈ S (α) satisfies the property: F (Z) = Z + α + h(e2πiz ) for a holomorphic map h : D → C with h(0) = 0. The map E(z) = e2πiz is a universal cover from C to C∗ = C \ {0} and its restriction to H is a universal cover from H to D∗ = D \ {0}. For any univalent map f : D → C which fixes 0 with derivative e2iπα , a lift is a holomorphic map F such that f ◦ E = E ◦ F . Then F ∈ S (α′ ) for some α′ ≡ α mod Z. Lifts exist and are unique if we require α′ = α. Conversely every F ∈ S (α) arises as the lift of a (unique) univalent map f as above. Given α ∈ R, the space S (α) is compact for the topology of uniform convergence on compact subsets of H. If F ∈ Sα , we let K(F ) be the set of points Z ∈ H whose orbit under iteration of F remains in H. For the corresponding f , we have K(f ) = {0} ∪ E(K(F )) and ∆(f ) = ∅ if ∆(F ) = ∅, otherwise ∆(f ) = {0} ∪ E(∆(f )). We say that F is linearizable whenever the corresponding map f is linearizable. This happens if and only if K(F ) contains an upper-half plane and we write h(F ) = inf h > 0 ; K(F ) contains “ Im Z > h” . We set h(F ) = +∞ otherwise. We then have

r(f ) ≥ e−2πh(F ) .

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2.1.2. Transfer of the Lipschitz condition to the lifts. — Consider the lifts Tα , Fn ∈ S (αn ) of Rα , fn . We can factor fn as follows: fn (z) = Rα (z)gn (z) with gn (0) = e2πi(αn −α) 6= 0. Then from |fn (z) − Rα (z)| ≤ K|αn − α| on D we get |gn (z) − 1| ≤ K|αn − α| by a form of the maximum principle. In particular for n big enough we have that kgn − 1k∞ ≤ 1/2. Then Fn (Z) − Tα (Z) = log gn (E(z)) for the principal branch of log. (For this we also have to take n big enough so that |αn − α| < 1, but note that with the special sequence αn under consideration, it already holds for all n ≥ 0.) Since the derivative of log has modulus less than 2 on B(1, 1/2) we get ∀Z ∈ H, |Fn (Z) − Z − α| ≤ 2K|αn − α|, ∀Z ∈ H, |Fn (Z) − Z − αn | ≤ (2K + 1)|αn − α|. In the rest of Section 2 we will prove the following version of the main lemma (Lemma 15): Lemma 19. — There exists a continuous function C(K) such that for all α, if we define αn as in Lemma 15 (we repeat the definition below for convenience) then for all K ≥ 1 and all sequence Fn ∈ S (αn ) such that |Fn (Z) − Z − αn | ≤ K|αn − α| : 1. If α is irrational then lim sup h(Fn ) ≤ 0. 2. If α = p/q in irreducible form, then lim sup h(Fn ) ≤ C ′′ (K, q) =

log(Kq) c3 + , 2πq q

for some universal constant c3 > 0. It implies Lemma 15, with the constant C(K, q) = 2πC ′′ (2K+1, q) ≤ log(Kq)/q+c1 /q for some universal constant c1 > 0. Reminder. — For convenience, we repeat here the definition of αn given in Lemma 15: let [a0 ; a1 , . . .] be the continued fraction expansion of α. If α ∈ Q then it has two such expansions (22) and both are finite : we choose one, [a0 ; a1 , . . . , ak ]. √ 1. If α ∈ / Q we let αn = [a0 ; a1 , . . . , an , 1 + an+1 , 1 + 2]. √ 2. If α ∈ Q we let αn = [a0 ; a1 , . . . , ak , n + 1 + 2]. The proof of Lemma 19 is based on renormalization and on the following: √ Lemma 20. — There exists C√2 > 0 such that ∀F ∈ S ( 2), h(F ) ≤ C√2 . (22)

See the end of Section 2.4.

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It is a consequence of Brjuno’s or Siegel’s theorems, and can also be proved using renormalization (Yoccoz), see [8, 9, 10, 28, 30, 32]. Note that for the proof of Lemma 19 it is enough to assume α ∈ [0, 1) because h(T −a ◦ F ) = h(F ) for any a ∈ Z, and shifting α by an integer shifts by the same amount the special sequences αn defined in Lemma 15. 2.2. Gluing. — Yoccoz rernormalization uses gluings, described below. We present here first a simplified version and an associated basic estimate. The next section will transpose this construction to a class of lifts. Let ℓ = i(0, +∞), i.e., half of the imaginary axis, endpoint excluded. Consider a holomorphic map F defined in a neighborhood (23) of ℓ and such that: (3)

(∀W ∈ ℓ) |F (W ) − W − 1| ≤ 1/10 and

|F ′ (W ) − 1| ≤ 1/10.

We do not assume here that F belongs to some S (α). The curve ℓ ∪ [0, F (0)] ∪ F (ℓ) bounds an open strip U in C. See Figure 1. Gluing the boundaries ℓ and F (ℓ) of U via F , we obtain a surface with boundary that we write U . Its “interior” U = U \ ∂ U is a Riemann surface for the complex structure inherited from a neighborhood of ℓ ∪ U ∪ F (ℓ); this includes ℓ (the gluing is analytic). The Riemann surface U is biholomorphic to the punctured disk D∗ or equivalently to the half-infinite cylinder H/Z: it was proved by Yoccoz, see [32]. It also follows from the existence of a quasiconformal homeomorphism that we build below (the construction of the quasiconformal homeomorphism was not invented by us, it can be found in [29] for instance). For an introduction to quasiconformal maps we recommend the following reference: [7]. Write W = X + iY . Define a homeomorphism G : [0, 1] × (0, +∞) → ℓ ∪ U ∪ F (ℓ) as follows: on each horizontal G is a linear interpolation between iY and F (iY ): G(X + iY ) = (1 − X)iY + XF (iY ). Because of the hypothesis eq. (3), we get that G extends to a neighborhood of its domain to a quasiconformal map that commutes with F , see [29] for details. (24) Lemma 21. — The map G descends to a quasiconformal homeomorphism G from H/Z to U : G // U ′ B H/Z (23)

G

// U

Since ℓ does not contain its endpoint, it means that the inner radius of such a neighborhood V may shrink near this point. Soon we will consider the case V = H. (24) They use the constant 1/4 instead of 1/10.

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commutes where B = [0, 1] × (0, +∞), the vertical arrows are passing modulo Z and modulo F and U ′ = ℓ ∪ U ∪ F (ℓ) = U \ [0, F (0)] (so that U = U ′ /F ). Proof. — There is a unique map G satisfying the diagram: the only place in H/Z where the projection to H/Z has not a unique preimage is the imaginary axis. There, an element has two antecedents: iY and iY +1 for some Y > 0. But then G is uniquely defined there because G(iY + 1) = F (G(iY )). In the domain, on can use (0, 1) × (0, +∞) as a fist chart for (a subset of) H/Z and U as a chart in the range. In this chart, G is a C 1 diﬀeomorphism with Beltrami derivative of norm at most a := 1/9 so it is K-qc with K = (1 + a)/(1 − a) = 5/4. A neighborhood of H ∩ iR can be used as a second chart in the domain, and we use a neighborhood of ℓ in the range. There, G is given by two C 1 diﬀeomorphisms patched along iR: G on the right of iR and T ◦ F −1 ◦ G on the left. These two diﬀeos extend slightly across iR and coincide there, and are both 5/4-quasiconformal. By classical quasiconformal gluing lemmas, (25) G is 5/4-quasiconformal in this second chart too. Since U is quasiconformally equivalent to H/Z it is also conformally equivalent to H/Z. The composition of this isomorphism with the natural projection ℓ ∪ U ∪ F (ℓ) → U ′ has a lift by the natural projection H → H/Z. Call it L : ℓ ∪ U ∪ F (ℓ) → H. It has a holomorphic extension to a neighborhood V of ℓ ∪ U ∪ F (ℓ) such that L(F (W )) = L(W ) + 1 holds in a neighborhood of ℓ. We can assume that V is simply connected by taking a restriction if necessary, but we do not assume that it contains [0, F (0)]. However, by Caratheodory’s theorem the map L indeed has a continuous extension to [0, F (0)] ¯ By adding a real constant to L, we can furthermore assume that that we call L. ¯ L(0) = 0. Lemma 22. — Assume that eq. (3) holds. Then ∀ W, W ′ ∈ ℓ ∪ U | Im(W − W ′ )| − C1 ≤ | Im (L(W ) − L(W ′ )) | ≤ A| Im(W − W ′ )| + C1 A for two universal constants A > 1, C1 > 0. If | Im(L(W ) − L(W ′ ))| > C1 then Im(L(W ) − L(W ′ )) and Im(W − W ′ ) have the same sign. Lemma 23. — Let δ ≤ 1/10 and assume F is a function as above such that (4)

(∀W ∈ ℓ)

|F (W ) − W − 1| ≤ δ

and

|F ′ (W ) − 1| ≤ δ.

(25) See for instance Rickman’s lemma, Lemma 1.20 in [7] with Φ = the patched map, ϕ = G and C = iR or the closed right half plane intersected with an open neighborhood of iR. In this particular case where we glue along a straight line, there are simpler proofs.

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≈1

1

U

L

ℓ

T F F (0) 0 Figure 1. Gluing.

Then for all M > 0, |L(W ) − W | ≤ B(M )δ

sup | Im Z| 1, C > 0, and the same estimate holds with H because it is also quasiconformal with the same supremum of Beltrami diﬀerential. Hence (6)

′ ˜ ˜ (| Im(W − W ′ )| − C)/A ≤ | Im(H(W ) − H(W ))| ≤ A| Im(W − W ′ )| + C.

˜ it satisfies the same inequalities. Using this The map H being a restriction of H, and the bound |G(W ) − W | ≤ 1/10, the first claim of the lemma follows with C1 = C + 2A/10. Moreover, given W , the image of the horizontal closed line in C/Z through W is a closed curve winding around C/Z and of total height at most C. It follows that if W ′ is another point such that Im W ′ > Im W , then Im H(W ′ ) > Im H(W ) − C and if Im W ′ < Im W then Im H(W ′ ) < Im H(W ) + C. So if | Im(H(W ) − H(W ′ ))| > C then Im(H(W ) − H(W ′ )) and Im(W ′ − W ) have the same sign. A similar argument proves that if | Im(G(W )−G(W ′ ))| > 2/10 then Im(G(W )−G(W ′ )) and Im(W ′ −W ) have the same sign. Now if | Im(L(W ) − L(W ′ ))| > C + 2A/10 then by the right hand side of eq. (6) we get | Im(G−1 (W ) − G−1 (W ′ ))| > 2/10 and by the discussion above Im(L(W ) − L(W ′ )) has the same sign as Im(G(W ) − G(W ′ )) which has the same sign as Im(W ′ − W ). We proved the last claim of Lemma 22. To prove Lemma 23, note that the estimate on G is global, so we have |G(W ) − W | ≤ δ for all W ∈ dom G and |G−1 (W ) − W | ≤ δ for all W in dom G−1 . We have L = H ◦ G−1 (first case) and L−1 = G ◦ H −1 (second case) so we now look for an estimate on H that is valid on the cylinder 0 < Im W < M for H −1 in the second case and on the cylinder 0 < Im W < M + 1/10 for H in the first case. We can for instance proceed as follows. Normalize H by adding a real constant so that it fixes 0: i.e., we consider the map H −H(0). It can be embeded in a holomorphic motion on C/Z by straightening t × BG, with t a complex number of modulus small 9 enough so that the essential supremum of t × BG is < 1. By the study above, |t| < 10δ is enough. We normalize the motion by requiring that 0 stays fixed. Consider the hyperbolic distance d1 on the Riemann surface C/Z \ {0} and d0 on the Riemann surface D. The holomorphic motion implies d1 (W, H(W )) ≤ d0 (0, 10 9 δt). The result follows. We are now going to give two statements under the supplementary assumption that F extends to H into a holomorphic funtion satisfying eq. (3) for all W ∈ H: (7)

(∀W ∈ H) |F (W ) − W − 1| ≤ 1/10 and

|F ′ (W ) − 1| ≤ 1/10.

Since H is convex, the condition on F ′ implies that F is injective on H.

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Equation (7) has a visual consequence: a portion of orbit F (W ), . . . , F k (W ) belongs to a cone of apex W , with a central axis which is horizontal, opening to the right and with half opening angle θ = arcsin 1/10. Lemma 24. — Assume F satisfies eq. (7). Then an F -orbit can pass at most once in ℓ ∪ U . Proof. — Consider γ0 = iR and let us extend F (ℓ) by a vertical half line going down ′ and stemming from F (0), into a curve that we call γ1 . Since |F − 1| ≤ 1/10, it follows that γ1 can be parameterized by the imaginary part: γ1 = g(Y ) + iY ; Y ∈ R for some continuous function g : R → R with at most one non-smooth point, corresponding to the corner F (0). √ This function is constant below this point. Above, it satisfies |g ′ (Y )| ≤ tan θ = 1/ 99. The set V : 0 ≤ X < g(Y ) is well-defined and contains ℓ ∪ U . It is disjoint from the set V+ of equation X ≥ g(Y ). It is enough to check that F (V+ ) ⊂ V+ and F (V ) ⊂ V+ . The first inclusion follows from the cone property mentionned above and the bound on |g ′ |. For the second inclusion, first note that if Z∈ / H then F (Z) is not defined, so we now assume that Z ∈ V ∩ H. Link Z ∈ U with the unique point Z ′ ∈ ℓ of same imaginary part by the horizontal segment [Z ′ , Z]. Then [Z ′ , Z] ⊂ H and the image by F of [Z ′ , Z] will not deviate from the horizontal 1 and links a point of F (ℓ) with F (Z). We conclude using the by more than arcsin 10 ′ bound on |g |. In the next lemma we use one of Koebe’s distortion theorems, which we copy here from [26] (Theorem 1.3 page 9, equation (15)): for a univalent map f from D to C: |f ′ (0)|

|z| |z| ≤ |f (z) − f (0)| ≤ |f ′ (0)| . 2 (1 + |z|) (1 − |z|)2

Consequence: for a univalent map f : B(a, R) → C: 2 2 |f (z) − f (a)| |z − a| |z − a| |f (z) − f (a)| ′ (8) 1− 1+ ≤ |f (a)| ≤ . z−a R z−a R Lemma 25. — Assume that F extends to H into a holomorphic funtion satisfying eq. (7), and that moreover F (W ) − W − 1 −→ 0 as Im W → +∞. Then L′ (W ) −→ 1 as Im W → +∞ within ℓ ∪ U . Proof. — The method is from [32], pages 28–29, S simplified here for our setting. It is a standard trick in this field to extend L to F k (ℓ ∪ U ), k ∈ Z so that the relation L ◦ F (W ) = T ◦ L(W ) holds whenever both sides are defined. The extension is welldefined because of Lemma 24 and is holomorphic. Since the set ℓ ∪ U contains the set defined by the equations Im W > 1/10 and 0 ≤ Re(W ) < 9/10, it follows by the cone property that the domain of definition of the extension contains the set of equation 2 9 2 2 and − (Im W − ) tan θ < Re W < + (Im W − ) tan θ, Im W > 10 10 10 10 where θ = arcsin 1/10. The margin 2/10 is here to ensure that when F k (W ) ∈ U , we have Im F k (W ) > 1/10 and hence F k (W ) lies above the lower segment [0, F (0)] of ∂U .

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The extension is injective: indeed if L(W1 ) = L(W2 ) then consider k1 , k2 ∈ Z such that W1′ := F k1 (W1 ) ∈ ℓ ∪ U and W2′ := F k2 (W2 ) ∈ ℓ ∪ U . Then L(W1 ) = L(W1′ ) − k1 and L(W2 ) = L(W2′ )−k2 . The points L(W1′ ) and L(W2′ ) both belong to L(ℓ∪U ) which is a fundamental domain for the action of T on H hence k1 = k2 and thus L(W1′ ) = L(W2′ ). So W1′ = W2′ i.e., F k1 (W1 ) = F k2 (W2 ) and using k1 = k2 again and the injectivity of F we get W1 = W2 . Now for W ∈ ℓ ∪ U with Im W big, the extension L is defined on a big disk centered on W , injective and satisfies L(W ′ ) = L(W ) + 1 where W ′ := F (W ) is close to W + 1, by hypothesis. The conclusion then follows using eq. (8) with a = W and z = W ′ . Weaker assumptions are enough and stronger conclusions hold. We only proved here statements that are suﬃcient to get the main lemma. 2.3. Iterations and rescalings. — Let Tα (Z) = Z + α and T = T1 . For a holomorphic map F commuting with T , defined on a domain containing an upper half plane and satisfying F (Z) = Z + α + o(1) as Im Z → +∞ we call α its rotation number and let it be denoted by α(F ). The following properties are elementary and stated without proof. Let α ∈ R and assume that kF − Tα(F ) k∞ < K|α(F ) − α|. — Then for k > 0: α(F k ) = kα(F ) and kF k − Tα(F k ) k∞ < K|α(F k ) − kα|. — Let a ∈ C, b ∈ R with b > 0, write λ(Z) = bZ + a and G = λ ◦ F ◦ λ−1 . Then α(G) = bα(F ) and kG − Tα(G) k∞ < K|α(G) − bα|. Assume instead that kF ′ − 1k∞ < exp(K|α(F ) − α|) − 1. — Then k(F k )′ − 1k∞ < exp(K|α(F k ) − kα|) − 1, — and kG′ − 1k∞ = kF ′ − 1k∞ . 2.4. Reminder on continued fractions. — We state a few classical properties for reference. Let a0 ∈ Z and an ∈ N∗ for n > 0. The notation [a0 ; a1 , . . . , an ] = a0 + 1/(a1 + 1/(. . . + 1/an )) is often used with integers only but we will use it too with the last entry being a real number: [a0 ; a1 , . . . , an , x] = a0 + 1/(a1 + 1/(. . . + 1/(an + 1/x))). If we write pn /qn = [a0 ; a1 , . . . , an ] in lowest terms (with p−1 /q−1 = 1/0) then pn x + pn−1 α := [a0 ; a1 , . . . , an , x] = qn x + qn−1

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and conversely x=−

qn−1 α − pn−1 . q n α − pn

Also, pn−1 qn − pn qn−1 = (−1)n , q n α − pn =

(−1)n qn x + qn−1

and qn−1 α − pn−1 =

(−1)n−1 qn + qn−1 x−1

from which we can get the following classical inequality (shifting the index n): |qn α − pn | ≤

1 qn+1

.

If β = [a0 ; a1 , . . . , an , y] then β − α = (−1)n+1

y−x . (qn x + qn−1 )(qn y + qn−1 )

Also there are the famous induction relations, for n ≥ 1: pn = an pn−1 + pn−2 , qn = an qn−1 + qn−2 . In this article we call continued fraction expansion the notation [a0 ; a1 , . . .] where the sequence an is finite or infinite and where a0 ∈ Z and an ≥ 1 for n ≥ 1. If α ∈ R\Q then it has only one continued fraction expansion and it is infinite. If α ∈ Q then it has exactly two continued fraction expansions: [a0 ; a1 , . . . , as , 1] and [a0 ; a1 , . . . , as + 1]. 2.5. Renormalization 2.5.1. Foreword. — The renormalization procedure we describe here is a variant of what is usually done. Consider a map F ∈ S (α). Usually a fundamental domain U is defined, bounded by ℓ∪[iy0 , F (iy0 )]∪F (ℓ) where ℓ is the vertical half line from iy0 to +i∞ and y0 > 0 is a real chosen big enough to ensure good behavior of the construction. Then a “return” map from T −1 (ℓ ∪ U ) to ℓ ∪ U is defined. Conjugacy through the gluing basically defines the renormalization. Usually this procedure is iterated a great number of times to give information on high iterates of the original map. Here, proximity to a rotation allows to bypass this and apply a one-step renormalization procedure directly to the high iterates.

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2.5.2. Construction. — Consider F ∈ S (α). Let pk−1 /qk−1 , pk /qk and pk+1 /qk+1 be three successive convergents of α with k ≥ 0. Implicitly α 6= pk /qk . The construction described below depends on the choice of k and will also depend on the choice of a positive constant y0 . In this article we call k + 1 the order of the renormalization. Write J = T −pk−1 ◦ F qk−1 , H = T −pk ◦ F qk . The symbol J refers to a jump and H to a hop: indeed when Im Z is big enough, J moves points by a bigger (26) amount than H. Let β ′ = α(J) = qk−1 α − pk−1 , β = α(H) = qk α − pk . By the theory of continued fractions, 1/2 ≤ qk+1 |β| ≤ 1, hence β 6= 0 and β and β ′ both tend to 0 as k → +∞. Moreover the sign of β alternates: it coincides with the sign of (−1)k and β ′ has the opposite sign. Remark 26. — In the particular case k = 0, we have p0 = a0 , q0 = 1 and by convention p−1 = 1, q−1 = 0 so J = T −1 and H = T −a0 ◦ F . Then the construction is essentially the classical renormalization of [32], and we call it order 1 renormalization according to the convention above. Assume that we have identified a height y0 such that the following statements hold: — the domain of definition of H contains Im Z > y0 ; as a consequence the domain of J also does; — ∀Z with Im Z > y0 : |H(Z) − Z − β| ≤ |β|/10,

|H ′ (Z) − 1| ≤ 1/10,

|J(Z) − Z − β ′ | ≤ |β|/10, (27)

|J ′ (Z) − 1| ≤ 1/10.

In particular if β > 0 we have Re H(Z) > Re Z and Re J(z) < Re Z, and if β < 0 then it is the opposite: Re H(Z) < Re Z and Re J(z) > Re Z. Remark 27. — There always exists such a height y0 , (28) and part of the work in further sections will be to get some control over it. Notation 28. — Since the construction involves many changes of variables we adopt the following notation: if Z 7→ λ(Z) is a change of variable then instead of denoting the new variable λ(Z) or Z ′ or W we may choose the notation Z λ . We speak of the (26) There is one exception: α = m = [m − 1; 1] for some m ∈ Z and we choose k = 0. We then get β ′ = −1 and β = 1. This case is not necessary for our main result but all we state here holds for it too, except the claim that |β| < |β ′ |. (27) This is not a typographic mistake: we want β ′ on the left hand side of the inequality and β on the right hand side. (28) Maps in S (α) are close to Z 7→ Z + α when Im Z is big, see for instance [32] page 26.

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Z λ -space, instead of the W -space or such. If a map acts on Z λ -space we may choose to use the notation F λ . Similarly a set in Z λ -space may be denoted by S λ . We now define the following change of variable λ: (9)

If β > 0 let λ(Z) = (Z − iy0 )/β. If β < 0 let λ(Z) = (Z − iy0 )/β.

By Section 2.3 the map H λ = λ◦H ◦λ−1 then satisfies eq. (3) stated in Section 2.2, with H λ in place of F . The sets ℓ and U constructed there will be denoted here by ℓλ and U λ because they live in Z λ -space, so that we can call ℓ and U their images by λ−1 , which live in Z-space. Then ℓ = i(y0 , +∞) and ∂U = ℓ ∪ [iy0 , H(iy0 )] ∪ H(ℓ). If β < 0 then U sits on the left of ℓ, otherwise it is on the right. A portion of H-orbit Z, H(Z), . . . , H n (Z) that stays above y0 (except maybe at the last iteration) can hit ℓ ∪ U at most once: this follows from Lemma 24 applied to the restriction to H of H λ . We now define a return map R, defined on a subset of ℓ ∪ U and taking values on ℓ ∪ U . For Z ∈ ℓ ∪ U : — If there is some n ≥ 0 such that Z and J(Z), H(J(Z)), H 2 (J(Z)), . . . , H n−1 (J(Z)) are all above y0 and H n (J(Z)) ∈ ℓ ∪ U then we let R(Z) = H n (J(Z)). Below we temporarily write n(Z) this unique value of n. — Otherwise we let R be undefined at Z. Recall that U refers to the quotient U /H. It is a surface with boundary and its “interior” is denoted by U and is a Riemann surface. We have U = (ℓ ∪ U ∪ H(ℓ))/H and the canonical projection is a bijection from ℓ ∪ U to U . Lemma 29. — The map R is injective. Passing to the quotient U (i.e., conjugating by the canonical projection ℓ ∪ U → U ), R becomes continuous and better: is the restriction of a holomorphic map to dom R. (29) Proof. — Injectivity: Assume that R(Z1 ) = R(Z2 ) for Z1 , Z2 ∈ ℓ ∪ U with R(Z1 ) = H n1 (J(Z1 )) and R(Z2 ) = H n2 (J(Z2 )). Since F is injective, it follows that J and H are injective too. Hence J(Z1 ) and J(Z2 ) belong to the same H-orbit. Up to permuting them, we can assume J(Z1 ) = H k (J(Z2 )) for some k ≥ 0. Now H k (J(Z2 )) = J(H k (Z2 )), whence Z1 = H k (Z2 ) by injectivity of J. Using Lemma 24 we get k = 0.

(29) By definition, holomorphic maps are defined on open sets. The domain of R may fail to be open near points of ℓ in the quotient for subtle reasons in the definition of R.

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Continuity and holomorphy: (30) Let us use two charts for the analytic structure on U . The first chart is the union of U and of a small enough connected open neighborhood V of ℓ. Recall that ℓ does not contain its endpoint, so we can take a neighborhood of size that shrinks to 0 near y0 . The second chart is U ∪ H(V ). They can be glued along V using H to give a complex dimension one manifold W/H where W = V ∪ U ∪ H(V ), and this manifold is canonically isomorphic to U . We choose V small enough so that V and H(V ) are disjoint, so that V ⊂ U ∪ ℓ ∪ H −1 (U ) and so that orbits of the restriction of H to Im Z > y0 intersect W in exactly one point or in exactly two in consecutive iterates, one in V, the other in H(V ), and finally so that the image of W ∩ dom J by J does not intersect H(V ). Then for any representative Z of a point of (dom R)/H, for any n ≥ 0 such that H n (J(Z)) ∈ W , then H n (J(Z)) is a representative of R(Z). The result follows. Recall that in Section 2.2 we associated a map L from ℓλ ∪ U λ to H via gluing, uniformization, then unfolding. The composition Lλ goes from ℓ ∪ U to H. We omit the symbol “◦” in L ◦ λ for more compact expressions. Lemma 30. — The domain dom R contains every point in ℓ ∪ U of high enough imaginary part. The set Z + Lλ(dom R) contains some upper half plane. Proof. — We assume β > 0, the other case being similar. The first time a portion of H-orbit passes from the left (strictly) to the right of the imaginary axis (inclusive), then a suﬃcient condition for the point to belong to ℓ ∪ U , is that its imaginary part be > max(Im(iy0 ), Im H(iy0 )). Now for Z ∈ ℓ ∪ U with high enough imaginary part, J(Z) is defined, lies on the left of iR and | Re(J(Z))| is bounded, for instance by |β ′ | + β/10. Applying H to a point above height y0 increases the real part by at least 9β/10 while the imaginary part changes by at most β/10. From there the details are left to the reader. For the second claim consider a point W L ∈ H and let us translate it by an integer so that W L ∈ L(ℓλ ∪ U λ ), which is possible since L(ℓλ ∪ U λ ) is a fundamental domain for H/Z. Let us apply the right hand side of Lemma 22, and the sign comparison claim of the same lemma, to W := L−1 (W L ) and W ′ := ε and let ε → 0. Noting that Im L(W ′ ) −→ 0 we get Im W L > C1 =⇒ Im W ≥ (Im W L − C1 )/A =⇒ Im(Lλ)−1 (W L ) ≥ y0 + (Im W L − C1 )β/A. The second claim follows from this and the first claim. Let H0 = inf H > 0 ; Im Z > H =⇒ Z ∈ Z + Lλ(dom R) (30) Let us give a heuristic justification. The map J has an essentially well-defined action on the orbits of the restriction of H to Im Z > y0 because these two maps commute; the quotient of the gluing can be seen as a subset of the space of orbits and its analytic structure is such that Z 7→ orbit(Z) is holomorphic. Now there are some problems in this approach since the space of orbits is not that well defined, or does not have such a nice topological structure, and the action of J is not so well defined.

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and let (10)

Λ(Z) = Lλ(Z) − iH0 . Λ

Given Z ∈ H there is a unique k = k(Z Λ ) ∈ Z such that Z Λ + k ∈ Λ(ℓ ∪ U ). Let Z = Λ−1 (Z Λ + k). By definition of H0 , we have Z ∈ dom R, in particular there exists some (unique) m = m(Z Λ ) ≥ 0 such that H m ◦ J(Z) = R(Z) ∈ ℓ ∪ U . Let (11)

R F (Z

Λ

) = Λ(R(Z)) − m − k.

Λ

If Im(Z ) ∈ (−H0 , 0] we choose to declare R F undefined at Z Λ , even though the procedure above may reach fruition. The map Λ conjugates (a restriction of) R to R F mod Z. Claim. — The map R F , which we defined on H and takes values in “Im Z Λ > −H0 ”, is continuous and better: holomorphic. ˜ of L to a neighborhood of Proof. — Indeed consider a holomorphic extension L ′ ′ ′λ ˜ ˜ λ )+1 for Z λ in U := λ(U ) with U = ℓ∪U ∪H(ℓ), which satisfies L(H λ (Z λ )) = L(Z λ ˜ = Lλ ˜ − iH0 . a neighborhood of ℓ := λ(ℓ) (see the beginning of Section 2.2) and let Λ Then ˜ ˜ (12) Λ(H(Z)) = Λ(Z) +1 holds in a neighborhood of ℓ. Consider k and m as in eq. (11). It is enough to check that in a neighborhood of any point Z0Λ ∈ H, the formula R F (Z

Λ

˜ ◦ Hm ◦ J ◦ Λ ˜ −1 (Z Λ + k) − m0 − k0 )=Λ

is locally valid, with m0 = m(Z0Λ ) and k0 = (Z0Λ ). If nearby Z Λ have a diﬀerent value of k in eq. (11), this means the initial Z Λ belongs to ℓΛ := Λ(ℓ), m0 > 0 and the nearby values of Z Λ have a value of k that equals k0 or k0 + 1. In the latter case we ˜ −1 (Z Λ + k0 )). In both cases the can use eq. (12) and get Λ−1 (Z Λ + k(Z Λ )) = H(Λ ˜ −1 (Z Λ + k0 ) − m − k, which following holds locally: R F (Z Λ ) = Λ ◦ H m+k−k0 ◦ J ◦ Λ we rewrite R F (Z

Λ

˜ −1 (Z Λ + k0 ) − m0 − k0 − δ ) = Λ ◦ H m0 +δ ◦ J ◦ Λ

with δ = (m − m0 ) + (k − k0 ). Similarly if local values of δ diﬀer, then H m0 ◦ J(Λ−1 (Z0λ + k0 )) ∈ ℓ and δ = 0 or 1. Again, we can use eq. (12). The map R F commutes with T and satisfies (13)

R F (Z

Λ

) = Z Λ + α′ + o(1)

as Im Z Λ → +∞, where α′ = α(R F ) =

β′ qk−1 α − pk−1 = −[ak+1 ; ak+2 , ak+3 , . . .]. = β q k α − pk

Equation (13) follows from L′ (W ) tending to 1 as Im W → +∞ by Lemma 25, while the domain U of L has a bounded projection to the real axis. Indeed J moves

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points by essentially β ′ , λ is a translation followed by a rescaling (31) by 1/β and in the definition of R , we compensated the eﬀect of k and m. See also [32] where a finer estimate on L is given. As an alternative one can use the invariance of the rotation number of holomorphic maps by homeomorphisms ([21, 16]). Lemma 31. — Assume that Z ∈ ℓ ∪ U and that Λ(Z) ∈ K(R F ). Then Z ∈ K(F ). Proof. — First note that the hypothesis implies that some Z + n, n ∈ Z, can be iterated infinitely many times by R. Since F and T commute, and by definition of R, we have R(Z + n) = T k F m (Z) for some k ∈ Z and m ∈ N with m > 0 or m = 0 in some rare exceptional cases that may happen if we do order 1 renormalization. But in this case R(Z) = Z − 1 so this cannot happen twice in a row, because T −2 (ℓ ∪ U ) is disjoint from ℓ ∪ U . 2.6. Proof of Lemma 19 for α ≡ 0 mod Z. — As already noted, we can assume α = 0. The number α has the following two continued fraction expansions: 0 = [0] = [−1; 1]. √ √ √ They respectively give √ αn = [0; n+1+ 2] = 1/(n+1+ 2) or αn = [−1; 1, n+1+ 2] = · · · = −1/(n + 2 + 2). If αn < 0 we can conjugate the sequence Fn by the reﬂection of vertical axis: X + iY 7→ −X + iY , and proceed then exactly as below, so we assume now that αn > 0. We will apply order 1 renormalization, i.e., proceed to the construction of Section 2.5 with k = 0 and F = Fn Then p0 = a0 = 0, q1 = 1, β = αn , H = Fn , p−1 = 1, q−1 = 0, β ′ = −1 and J = T −1 . By assumption on Lemma 19: (∀Z ∈ H)

|Fn (Z) − Z − αn | ≤ Kαn

By the Schwarz-Pick inequality this implies: |Fn′ (Z) − 1| ≤

Kαn . 2 Im Z

So there exists εn −→ 0 such that sup |Fn′ (Z) − 1| ≤ 1/10. Im Z>εn

As explained in Section 2.1.1 one can also write F (Z) = Z + α + h(e2πiZ ) with h a holomorphic function mapping 0 to itself. Schwarz’s inequality thus implies |Fn (Z) − Z − αn | ≤ Kαn e−2π Im Z . We take y0 = max(εn , log(10K)/2π) (31)

Whatever the sign of β is, we have Re λ(Z) = (Re Z)/β and Im λ(Z) = (Im Z − y0 )/|β|.

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∼α

1

Fn (Z) ∈ B(Z + α, α/10) Z U Z +α ℓ

Z

Fn (ℓ)

Fn

Im = y0

iy0

Fn (iy0 )

Figure 2. The orbit of Z remains in a cone, at least as long it stays above height y0 . In this picture we exagerated the angle of the cone.

so that Im Z > y0 =⇒ |Fn (Z) − Z − αn | ≤ αn /10 and

|Fn′ (Z) − 1| < 1/10.

In the notation y0 and many of the notations that follow we omit the index n for better readability. The construction yields two sets ℓ = i(y0 , +∞) and U , a map Lλ : U → H where Z − iy0 λ(Z) = , αn and a return map R from ℓ ∪ U to itself. It also introduces: a constant H0 defined as the smallest H ≥ 0 such that Z + Lλ(dom R) contains “ Im Z > H”; the map Λ(Z) = Lλ(Z) − iH0 ; and finally the renormalized map R Fn , which is a modification of the restriction to H of the conjugate of R by Λ. By the properties stated in Section 2.5, including eq. (13), √ we have R Fn ∈ S(α′ ) with α′ = ± 2. It follows that h(R Fn ) ≤ C√2 , see Lemma 20. For any Z with Im Z > y0 , the point Fn (Z) lies in a horizontal cone of apex Z and with half opening angle θ = arcsin(1/10). The orbit will thus stay in that cone as long as the previous iterates all lie above y0 , see Figure 2. Let us give a more explicit version of Lemma 30:

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Lemma. — Every point in the strip Re Z ∈ [−1, 0[ and Im Z > y0 + tan θ has an orbit by Fn that eventually passes the imaginary axis, i.e., Re Fnk (Z) ≥ 0. The first time it does, Im Fnk (Z) ≥ Im Z − αn /10 − tan θ. Before, it stays above y0 . Proof. — By the cone condition, it follows by induction on i that Fni (Z) stays above y0 as long as it belongs to the strip. By assumption when we iterate a point above y0 , the real part increases by at least 9/10αn so we know the orbit will eventually pass the imaginary axis. Just before it was above Im Z − tan θ and at the next iterate the imaginary part decreases at most by αn /10. If the first iterate Fnk (Z) passing the imaginary axis in the lemma above satisfies Im Fnk (Z) > max(y0 , Im Fn (y0 )) then (32) Fnk (Z) ∈ ℓ ∪ U . Now Im Fn (y0 ) ≤ y0 + αn /10. By the lemma above, dom R contains every point in ℓ ∪ U of imaginary part strictly larger than y1 with y1 = y0 + 2αn /10 + tan θ. We will apply Lemma 22 to L. It introduced constants A > 1 and C1 > 0. Lemma. — For Z ∈ ℓ ∪ U and Im Z > y2 with y2 = y1 + αn (

1 + C1 + A max(C1 , C√2 )) 10

then Z ∈ K(Fn ). Proof. — Consider such a Z. We apply Lemma 22 to specific values of W and W ′ : consider a point V0 in the boundary of Z + Lλ(dom R) and maximizing the imaginary part, so Im V0 = H0 . By adding a (possibly negative) integer, we may assume that V0 ∈ Lλ(dom R). Let Z0 = (Lλ)−1 (V0 ) ∈ ℓ∪U . If Z0 ∈ U then Im Z0 ≤ y1 otherwise it would be in the interior of dom R. If Z0 ∈ ℓ then Im Z0 ≤ y1 +αn /10 otherwise Lλ(Z0 ) belongs to the interior of Z + Lλ(dom R). In all cases, Im Z0 ≤ y1 + αn /10. We take W = λ(Z) and W ′ = λ−1 (V0 ). Note that Im W − Im W ′ > (y2 − y1 − αn /10)/αn > 0. By the left hand inequality in the Lemma 22: | Im L(W ) − Im L(W ′ )| ≥ (| Im W − Im W ′ | − C1 )/A ≥ max(C1 , C√2 ) ≥ C1 . By the second claim in Lemma 22 we get that Im L(W ) > Im L(W ′ ). Now Im L(W ) > Im L(W ′ ) + C√2 i.e., Im Λ(Z) > C√2 hence Λ(Z) ∈ K(R Fn ), hence Z ∈ K(Fn ) by Lemma 31. Now this construction could have been carried out on the conjugate of Fn by any horizontal translation Z 7→ Z +x, which amounts to replace the origin iy0 of the line ℓ (32) For a justification of this claim, consider the horizontal segment from Fnk−1 (Z) to ℓ. Its image is a curve with tangent deviating less that θ < π/2 from the horizontal, whereas F (ℓ) has a tangent that deviates less than θ from the vertical, so F (ℓ) is contained in z ∈ C ; | arg(z − Fnk (Z))| < π/2 + θ . It follows that Fnk (Z), can be linked to ℓ by a horizontal segment going to the left and that does not cross the other boundary lines of U .

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by x + iy0 . In particular every point with imaginary part ≥ y0 is contained in the set ℓ ∪ U associated to an appropriate choice of x. Hence h(Fn ) ≤ y2 . Putting everything together, we have proved that h(Fn ) ≤ αn (1/10 + C1 + A max(C1 , C√2 )) + 2αn /10 + tan θ + max(εn , log(10K)/2π), where θ = arcsin 1/10. Since αn −→ 0 this gives: (33) lim sup h(Fn ) ≤ C(K) := C0 + n→+∞

log K 2π

for some universal constant C0 > 0. 2.7. Improvement through renormalization for maps tending to a non-zero rotation. — Consider α ∈ R with α ∈ / Z and αn ∈ R with αn −→ α. Consider a sequence Fn ∈ S (αn ) and assume Fn −→ Tα uniformly on H when n → +∞ where Tα (Z) = Z + α. The first statement we give does not need a Lipschitz type assumption on how fast this convergence occurs. Consider a renormalization as per Section 2.5: it involves the choice of k such that α has a continued fraction (34) of which [a0 ; a1 , . . . , ak+1 ] is an inital segment. We let k be constant, i.e., independent of n. To proceed with the construction of the renormalization, we need to choose y0 such that the conditions of Section 2.5 are satisfied. We will use the notation H and J as in that section, i.e., without the index n: J = T −pk−1 ◦ Fnqk−1 , H = T −pk ◦ Fnqk . Let βn = α(H) = qk αn − pk , βn′ = α(J) = qk−1 αn − pk−1 be their respective rotation numbers and let β = q k α − pk β ′ = qk−1 α − pk−1 be their respective limits. We have 0 < |β| < |β ′ | ≤ 1. βn −→ β 6= 0 and βn′ −→ β ′ 6= 0. n→∞

(33) (34)

In fact, αn ≤ 1 is enough. α has one or two c.f. expansions, see Section 2.4.

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n→∞

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Note that, as n → +∞, H −→ Tβ and J −→ Tβ ′ hence kH − Tβn k∞ −→ 0 and kJ − Tβn′ k∞ −→ 0. n→∞

n→∞

Since βn −→ β 6= 0 it follows that for n big enough we have ∀Z ∈ dom H, |H(Z) − Z − βn | < |βn |/10 and ∀Z ∈ dom J, |J(Z) − Z − βn′ | < |βn |/10. Also, for all ε > 0, H ′ −→ 1 and J ′ −→ 1 as n → +∞ both uniformly on the subset of H defined by the equation Re Z > ε. Whence the existence of εn −→ 0 and δn −→ 0 with δn < 1/10 such that for n big enough, Im Z > εn =⇒ Z ∈ dom H (hence Z ∈ dom J) and |H ′ (Z) − 1| < δn , |J ′ (Z) − 1| < δn . Thus we can take y0 = εn , assuming n big enough. Note that y0 −→ 0 as n → +∞. The exact value of y0 is not so important, what matters is that it tends to zero: y0 −→ 0 n→∞

Then Section 2.5 associates an order k + 1 renormalization R Fn to the pair of maps H, J, via a return map R and a straightening Lλ mod Z of a Riemann surface U = (ℓ ∪ U ∪ H(ℓ))/H, where λ is a change of variable that takes the form λ(Z) = (Z − iy0 )/βn or (Z − iy0 )/βn (it depends on k). Lemma 32. — We have lim sup h(Fn ) ≤ |β| lim sup h(R Fn ). The same statement holds with lim sup replaced by lim inf. Proof. — It is enough for both statements to prove that if h0 ≥ 0 and if we have a subsequence n ∈ I ⊂ N and points Zn ∈ H with Zn ∈ / K(Fn ) but Im Zn ≥ h0 then lim inf n∈I h(R Fn ) ≥ h0 /|β|. From now on all limits are taken for n ∈ I. We can conjugate Fn by a real translation and assume that Re(Zn ) = 0. In Section 2.5 is defined a constant H0 ≥ 0, the infimum of heights of upper half planes contained in Z + Lλ dom R. Is also defined the map Λ = Lλ − iH0 . Let H0′ = inf h > 0 ; (Z ∈ ℓ ∪ U and Im Z > h) =⇒ Z ∈ dom R . We claim that H0′ ≤ y1 where y1 = y0 + kJ − Tβn′ k∞ + u + kH − Tβn k∞ with u = (|βn |+kJ −Tβn′ k∞ ) tan arcsin(kH −Tβn k∞ /|βn |). The arguments are similar to Section 2.6, when we controlled the domain of R via a constant also called y1 : for Im Z > y1 , the point J(Z) is defined and has a forward iterate Z ′ = H m(Z) ◦ J(Z) by H which hits ℓ ∪ U while staying above y0 + kH − Tβn k∞ . By definition R(Z) = Z ′ . In particular H0′ −→ 0 as n → +∞. This implies that H0 −→ 0 as n → +∞: indeed, λ tends to the linear map Z 7→ Z/β or Z 7→ Z/β and L tends to the identity on the set of points Z with Im Z ≤ 1 by Lemma 23.

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The quantity y1 tends to 0 when n → +∞. Recall that Im Zn ≥ h0 which does not depend on n. Hence for n big enough we have Im Zn > y1 . So Zn′ = R(Zn ) is defined. Note that Zn′ ∈ / K(Fn ) for otherwise Zn would belong to K(Fn ) too. By Lemma 31 we have Λ(Zn′ ) ∈ / K(R Fn ). Note that lim inf Im λ(Zn ) ≥ h0 /|β|. We apply Lemma 23 to L. Noting H λ = λ ◦ H ◦ λ−1 , we have |(H λ )′ − 1| < δn on H and H λ −→ T1 uniformly on H, so the hypotheses of Lemma 23 are satisfied, with a value of δ that depends on δn and tends to 0. The conclusions of this lemma with M = 1 + h0 /|β| then imply that lim inf Im Lλ(Zn′ ) ≥ h0 /|β|. (Indeed let y = 1/2 + h0 /|β|. For one thing the image of the segment [iy, H λ (iy)] by L followed by the projection C → C/Z is a closed curve that tends to a horizontal curve as n → +∞. Points on or above the segment are mapped by L to point whose imaginary part is at least the infimum of Im Z over this closed curve, and this infimum tends to y. Whereas points Z below the segment satisfy | Im L(Z) − Z| < B(M )δ and recall that δ −→ 0 as n → +∞ and that lim inf Im λ(Zn ) ≥ h0 /|β|.) Since Lλ(Zn′ ) − iH0 ∈ / K(R Fn ) and H0 −→ 0 we get lim inf h(R Fn ) ≥ h0 /|β|. We complement this lemma with the following one, which requires a Lipschitz-type assumption on αn 7→ Fαn and also on αn 7→ Fα′ n . Lemma 33. — Assume |Fn (Z) − Z − αn | ≤ K|αn − α| and a new assumption: |Fn′ (Z) − 1| ≤ K|αn − α|. Then for all n big enough we have sup |R Fn (Z) − Tα′n (Z)| ≤ DK|αn′ − α′ |,

Im Z>0

where αn′ is the rotation number of R Fn and α′ is its limit. Here D > 1 is a universal constant. Proof. — According to Section 2.5 have αn′ = βn′ /βn and α′ = β ′ /β. An elementary computation yields (−1)k (αn − α). αn′ − α′ = ββn We want to apply Lemma 23 with M = 1 to get information on L. For this we need to estimate H λ := λ◦H ◦λ−1 . Note that βn is the rotation number of H = T −pk ◦Fnqk . From the first Lipschitz assumption we get that (14)

|H(Z) − Z − βn | ≤ K|βn − β|, |J(Z) − Z − βn′ | ≤ K|βn′ − β ′ |,

see Section 2.3. From the second that |H ′ (Z) − 1| ≤ dn := (1 + K|αn − α|)qk − 1 ∼ K|βn − β|

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when n → +∞ (there is a similar estimate for J ′ but we will not use it). The rotation number of H λ is 1 and |H λ (Z) − Z − 1| ≤ K|βn − β|/|βn |, |(H λ )′ − 1| ≤ dn ∼ K|βn − β|. The bound on the derivative of H λ is better than the bound on H λ . However we will apply Lemma 23 which only uses a common bound, i.e., here: K|βn − β|/|βn |, since for n big enough, we have |βn | < 1. By this lemma applied to M = 1 we get (15)

Im W ≤ 1 =⇒ |L(W ) − W | ≤ B(1)K|βn − β|/|βn |

and (16)

Im W ≤ 1 =⇒ |L−1 (W ) − W | ≤ B(1)K|βn − β|/|βn |.

Note that, as n → +∞: |βn − β|/|βn | ∼ qk |αn − α|/|β|. Now for Z ∈ dom R the return map is R(Z) = H m(Z) ◦ J(Z) for some m(Z) ∈ N. Let βn′′ (Z) = βn′ + m(Z)βn , β ′′ (Z) = β ′ + m(Z)β. Then |R(Z) − (Z + βn′′ (Z))| ≤ K|βn′ − β ′ | + Km(Z)|βn − β| =∗ K|βn′ − β ′ + m(Z)(βn − β)| = K|βn′′ (Z) − β ′′ (Z)|, in short (17)

|R(Z) − (Z + βn′′ (Z))| ≤ K|βn′′ (Z) − β ′′ (Z)|,

where equality (*) comes from the fact that βn′ − β ′ and βn − β have the same sign (and m(Z) ≥ 0), which may sound surprising since βn′ and βn have opposite signs (β ′ and β too), so we justify this by the following explicit computation: βn′ − β ′ = (qk−1 αn − pk−1 ) − (qk−1 α − pk−1 ) = qk−1 (αn − α), and βn − β = (qk αn − pk ) − (qk α − pk ) = qk (αn − α). Also: βn′′ (Z) − β ′′ (Z) = (qk−1 + m(Z)qk )(αn − α). We will need a (rough) bound on m(Z): we treat the case β > 0, the other one is symmetric and yields the same bound. Consider all n big enough so that |βn | − |βn − β| < |β|/2 and |βn′ | − |βn′ − β ′ | < |β ′ |/2. For such an n, by eq. (14), for Z ∈ ℓ ∪ U with Z ∈ dom J, the map J shifts the real part of Z in the negative direction by at most 3β ′ /2 and the map H of at least β/2 in the positive direction and at most 3β/2. Since Z ∈ ℓ ∪ U we get Re(Z) ∈ [0, 3β/2]. Since m(Z) is the first

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m ≥ 0 such that Re(H m J(Z)) ∈ ℓ ∪ U , it follows that Re(H m J(Z)) < 3β/2, and by the above remarks Re(H m J(Z)) > mβ/2 + 3β ′ /2 (recall that β ′ < 0) so m = m(Z) < 3 − 3β ′ /β = 3(1 + |β ′ /β|). In particular : |βn′′ (Z) − β ′′ (Z)| −→ 0 uniformly w.r.t. Z as n → +∞. Now let Z Λ ∈ H and, as in Section 2.5, let k = k(Z Λ ) ∈ Z bet the unique integer such that Z Λ + k ∈ Λ(ℓ ∪ U ) and define Z = Λ−1 (Z Λ + k). Recall that we defined there m = m(Z) ∈ N such that R(Z) = H m ◦ J(Z) and that R Fn (Z

Λ

) = ΛR(Z) − k − m.

Recall also that Λ = Lλ − iH0 . We now proceed to the estimate: R Fn (Z

Λ

) = LλR(Z) − iH0 − k − m = LλR(Z) − λR(Z) + λR(Z) − λ(Z + β ′′ (Z)) + λ(Z + βn′′ (Z)) − iH0 − k − m.

And since λ(X + iY ) = X/βn + iY /|βn | − iy0 we get λ(Z + βn′′ (Z)) = λ(Z) + βn′′ (Z)/βn = λ(Z) + βn′ /βn + m = λ(Z) + αn′ + m. From this and Z Λ + k = Λ(Z) = Lλ(Z) − iH0 we get Λ Λ ′ ′′ R Fn (Z ) − (Z + αn ) = LλR(Z) − λR(Z) + λR(Z) − λ(Z + β (Z)) + λ(Z) − Lλ(Z) .

Using the estimates above we get:

If Im λ(Z) ≤ 1 and Im λR(Z) ≤ 1 then 1 |R Fn (Z Λ ) − (Z Λ + βn′ /βn )| ≤ B(1)K|βn − β|/|βn | + K|βn′′ (Z) − β ′′ (Z)| |βn | + B(1)K|βn − β|/|βn | i.e., |R Fn (Z Λ ) − Z Λ + αn′ )| ≤ un with un :=

ASTÉRISQUE 416

K 2B(1)|βn − β| + |βn′′ (Z) − β ′′ (Z)| . |βn |

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Let us now estimate un , using the equivalents mentioned in the present proof. We use an . bn on non-negative sequences to mean ∃cn ≥ 0 such that for n big enough, an ≤ cn and cn ∼ bn . K un ∼ (2B(1)qk + qk−1 + m(Z)qk )|αn − α| |β| K . (2B(1)qk + qk−1 + 3(1 + |β ′ /β|)qk )|αn − α|. |β| And using the comparison between αn′ − α′ and αn − α given at the beginning of the present proof: 1 |αn − α| |αn′ − α′ | = |ββn | so using |β| ≤ 1/qk+1 and |β ′ | ≤ 1/qk : un . K|β| (qk−1 + 2B(1)qk + 3(1 + |β ′ /β|)qk ) ′ |αn − α′ | qk qk−1 + 2B(1)qk + 3K( + 1) .K qk+1 qk+1 . (2B(1) + 7)K. Finally: we proved that ∃cn such that for n big enough, then for all Z Λ ∈ H satisfying (35) (18)

Im λ(Z) ≤ 1 and Im λR(Z) ≤ 1,

we have

|R Fn (Z Λ ) − (Z Λ + αn′ )| ≤ cn ∼ (2B(1) + 7)K. |αn′ − α′ | We claim that the inequality above extends to all values of Z Λ ∈ H by the maximum principle. Indeed the diﬀerence R Fn (Z Λ )−Z Λ −αn′ is Z-periodic and is bounded as Im Z Λ → +∞ because it tends to 0, so it is enough to prove that A contains the intersection of H with a neighborhood of R in C, where A denotes the set of Z Λ for which eq. (18) is satisfied. Recall that Z = Λ−1 (Z Λ + k), i.e., λ(Z) = L−1 (Z Λ + k + iH0 ). Since H0 −→ 0 as n → +∞ we can assume that H0 < 1/4. By eq. (16), for n big enough we have |L−1 (W ) − W | < 1/4 for all W ∈ dom L such that Im(W ) < 1. So for Im Z Λ < 1/4 we get Im(Z Λ + k + iH0 ) ≤ 2/4 thus we can apply the estimate on L, so Im L−1 (Z Λ + k + iH0 ) ≤ 3/4, i.e., Im λ(Z) ≤ 3/4. Now from eq. (17) we get Im R(Z) ≤ Im Z + K|βn′′ (Z) − β ′′ (Z)| whence Im λR(Z) ≤ Im λZ + K|βn′′ (Z) − β ′′ (Z)|/|βn |, so Im λR(Z) ≤ 3/4 + |βn′′ (Z) − β ′′ (Z)|/|βn | and we have already seen that the right hand side of the sum tends to 0 uniformly w.r.t. Z as n → +∞. Thus for n big enough we have Im λR(Z) ≤ 1.

(35)

Where Z depends on Z Λ in the way described earlier in the present proof.

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The first lemma above implies that we gain a factor |β| in estimates of the size of the linearization domain, but by the second lemma we loose a universal factor D in the Lipschitz constant for α 7→ Fα . Moreover this second lemma requires an assumption on the Lipschitz constant for α 7→ Fα′ . See later for how we deal with this. 2.8. Proof of Lemma 19 for α = p/q. — Here we use the results of Section 2.7 to transfer the case α = 0 covered in Section 2.6 to the case α = p/q. In the process, the estimate will improve for big values of q. Let us consider one of the two continued fraction expansions of p/q and write it as follows: p/q = [a0 ; a1 , . . . , ak+1 ]. We have k ≥ 0 since p/q ∈ / Z. Let p′ /q ′ = [a0 ; a1 , . . . , ak ] be its last convergent before p/q itself. Then p′ q − pq ′ = (−1)k+1 . In Lemma √ 19, which we are proving, is defined the sequence αn = [a0 ; a1 , . . . , ak+1 , n + 1 + 2] (note that we shifted the p + p′ xn index k by one, to match with the notation of Section 2.5). We have αn = q + q ′ xn √ with xn = 1/(n + 1 + 2). It is important to note that, though n → +∞, the numbers q and q ′ remain fixed here. We now proceed to the order k+1 direct renormalization as described in Section 2.5. This yields maps R Fn . Let β = qk α − pk be the quantity introduced in Section 2.7. ′ ′ q Here β = q ′ α − p′ = q p−p hence q β = (−1)k /q. To apply Lemma 33 we need to control not only the distance from Fn to the rotation but also the distance from Fn′ to the constant function 1. For this we just apply the Schwarz-Pick inequality: sup |Fn − Tαn | K ≤ |αn − α|. 2 Im Z 2 Im Z We restrict Fn to Im z > ε for some ε ∈ (0, 1/2) and then conjugate by the translation by −iε to make the domain equal to H. This yields maps Fen . We can apply Lemma 33 to Fen with the constant K replaced by K/2ε because the control on Fn′ is not as good as the control on Fn . The lemma gives that the maps R Fen satisfy the hypotheses of Lemma 19 with a Lipschitz constant of DK/2ε where D > 1 is a universal constant. Their rotation number αn′ is equal to αn′ = xn , which tends to α′ = 0. So by the case α = 0 of Lemma 19 covered in Section 2.6 we get that 1 DK lim sup h(R Fen ) ≤ C(DK/2ε) = C0 + log 2π 2ε for some C0 > 0. By Lemma 32 1 lim sup h(Fen ) ≤ lim sup h(R Fen ) q and since lim sup h(Fn ) ≤ ε + lim sup h(Fen ) |Fn′ (Z) − 1| ≤

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we get lim sup h(Fn ) ≤ ε +

1 DK C0 + log . q 2πq 2ε

Optimizing the choice of ε ∈ (0, 1/2) we get lim sup h(Fn ) ≤

1 C0 + (1 + log(DKπq)) . q 2πq

In this proof we iterated renormalization: we did an order k + 1 direct renormalization followed by an implicit order 1 renormalization when using the result of Section 2.6. In fact, order k +1 direct renormalization and iterating k +1 times order 1 renormalization are closely related procedures, so morally we could consider we did k + 2 renormalizations. 2.9. Proof of Lemma 19 for α ∈ R \ Q. — It will be carried out in two steps. Recall that, denoting α = [a0 ; a1 , a2 , . . .] we defined αn = [a0 ; a1 , . . . , an , 1 + an+1 , 1 +

√

2].

1. We first prove a weak version of the lemma. For this we use a first direct renormalization at order n + 1 for αn , which brings the rotation number αn of Fn √ to 2 mod Z for R Fn . 2. Then, if necessary, we enhance the weak version using a prior renormalization of the type of Section 2.7, at some fixed but high order. So let us apply order n + 1 direct renormalization to Fn as described in Section 2.5. Be careful with the notations: what is called α in that section is called αn here, and the integer k in that section is so that k = n. A pair of maps is introduced, which we recall: J = T −pn−1 ◦ F qn−1 , H = T −pn ◦ F qn . Their respective rotation numbers are βn′ = qn−1 αn − pn−1 , βn = qn αn − pn . Then a height y0 must be provided satisfying conditions that we recall too: — the domain of definition of H contains Im Z > y0 , and hence the domain of J also does; — ∀Z with Im Z > y0 : |H(Z) − Z − βn | ≤ |βn |/10,

|H ′ (Z) − 1| ≤ 1/10,

|J(Z) − Z − βn′ | ≤ |βn |/10,

|J ′ (Z) − 1| ≤ 1/10.

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Let us proceed to some estimates on rotation numbers. According to Section 2.4 and some expression manipulation βn =

(−1)n √ . qn+1 + qn 2

Also, αn − α = (−1)n+1

√ 2 − xn √ , (qn+1 + qn 2)(qn+1 + qn xn )

where xn := [0; an+2 , an+3 , . . .] ∈ (0, 1). Let M = qn K|αn − α| < K

qn √ 2 2 qn+1

and note that M −→ 0 as n → +∞. The map H is defined at least on Im Z > M and satisfies there that H diﬀers from Tβn by at most M . However the inequality M ≤ |βn |/10 does not necessarily hold. By the above √ qn ( 2 − xn ) M/|βn | = K , qn+1 + qn xn so √ Kqn √ 2−1 Kqn 2. · ≤ M/|βn | ≤ qn+1 2 qn+1 The quotient qn /qn+1 = qn /(an+1 qn + qn−1 ) is less than one and can be very small if an+1 is big, but it can also be very close to 1 if an+1 = 1, depending on the continued fraction expansion of αn . Now since H commutes with T1 and H(Z) − Z tends to βn as Im Z −→ +∞, we can improve the estimate on H as follows: Im Z > M =⇒ |H(Z) − Tβn (Z)| ≤ e−2π(Im Z−M ) M. Hence in all cases the inequality |H(Z) − Tβn (Z)| ≤ |βn |/10 will hold if Im Z > y0 with 10M 1 log+ y0 = M + 2π |βn | denoting log+ x = max(0, log x). We have y0 ≤ M +

√ 1 log(10K 2). 2π

A similar analysis holds for J with better estimates, so we can just take the same constants as above.

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√ √ The rotation number of R Fn is −(an+1 + 2) ≡ − 2 mod Z. By Lemma 20, K(R Fn ) contains Im Z > C√2 . We claim that the return map R, see Section 2.5, is defined on (ℓ ∪ U ) ∩ “ Im Z > y1 ” with 3 1 y1 := y0 + |βn | + (|βn′ | + |βn |) tan θ, 10 10 where θ = arcsin(1/10). Proof. — We justify it in the case βn > 0, the other case being completely similar. If βn > 0 then βn′ < 0. We have Re J(Z) ≥ βn′ − βn /10 = −(|βn′ | + |βn |/10). By the cone property (see the paragraph between eq. (7) and Lemma 24), the H-orbit stays in a cone of apex J(Z) and half opening angle θ and central axis J(Z) + R+ , as long as it remains in Im Z > y0 . The condition Im Z > y1 ensures that Im J(z) > y1 − |βn |/10 and that the orbit will stay above y0 as long as it has not passed the imaginary axis. 2 1 |βn |) tan θ = y0 + 10 |βn |. Before passing it it stays above height y1 −|βn |/10−(|βn′ |+ 10 It will pass the imaginary axis (because the real part increases by a definite amount) and when it does, the imaginary part will be at least y0 + |βn |/10, which ensures that it will belong to ℓ ∪ U (the argument is similar to the paragraph marked (*) on page 163). Let λ, L, H0 and Λ be as in Section 2.5. We have λ(X +iY ) = X/βn +i(Y −y0 )/|βn | and Λ = Lλ − iH0 . We claim that every point Z in (ℓ ∪ U ) ∩ “ Im Z > y2 ” with 1 y2 := y1 + (A max(C√2 , C1 ) + C1 + )|βn | 10 is mapped by Λ to a point of imaginary part > C√2 , where A and C1 are the constants in Lemma 22. Proof. — Consider the set A = Z + Lλ(dom R) ⊂ H: it follows from the definition of H0 that H0 = supZ∈∂ A Im Z where the boundary is relative to C. Let w′ ∈ ∂ A with Im w′ = H0 . By subtracting an integer to w′ we can assume that w′ ∈ Lλ(ℓ ∪ U ). Assume now that Im Z > y2 defined above. We will apply Lemma 22 to W = λ(Z) and W ′ = L−1 (w′ ) ∈ λ(ℓ ∪ U ). Let Z ′ = λ−1 W ′ ∈ ℓ ∪ U . If Z ′ ∈ U then we have Im λ−1 W ′ ≤ y1 for otherwise Z ′ would have a neighborhood V contained in dom R hence its image by Lλ would belong to the interior of A , contradicting the definition of w′ . For similar reasons, if Z ′ ∈ ℓ then we have Im Z ′ ≤ y1 + |βn |/10: recall that the map L extends to a neighborhood of ℓλ = λ(ℓ) and a neighborhood of H λ (ℓλ ) and satisfies L ◦ H λ = L + 1 near ℓλ . The margin |βn |/10 is there to ensure that both V and H(V ) are above y1 for a small enough neighborhood V of any Z with Im Z > y1 +|βn |/10. Hence Im Z ′ ≤ y1 +|βn |/10 in all cases. Hence Im W −Im W ′ > (y2 −y1 −|βn |/10)/|βn | = A max(C√2 , C1 )+C1 . Lemma 22 gives | Im L(W ) − Im L(W ′ )| ≥ (| Im W − Im W ′ | − C1 )/A > max(C1 , C√2 ). In particular | Im L(W ) − Im L(W ′ )| > C1 so by the second part of Lemma 22, the quantities Im L(W ) − Im L(W ′ ) and Im W − Im W ′ have the same sign. We thus get that Im Λ(Z) = Im Lλ(Z) − H0 = Im L(W ) − Im L(W ′ ) > C√2 .

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Under these conditions on Z, it follows that Λ(Z) ∈ K(R Fn ). Hence Z ∈ K(Fn ). The same analysis can be applied to the conjugate of Fn by a horizontal translation Tx (Z) = Z + x. Write ℓx and Ux the sets constructed from Tx−1 ◦ Fn ◦ Tx instead of Fn . (It turns out that ℓx = (iy0 , +i∞) = ℓ is independent of x.) For any point Z ∈ H with Im Z > y2 , there is a translation Tx so that Tx−1 Z ∈ ℓx ∪ Ux (in fact take x = Re Z, then Tx−1 Z ∈ ℓ = ℓx ). Hence Tx−1 Z ∈ K(Tx−1 ◦ Fn ◦ Tx ), which is equivalent to the statement Z ∈ K(Fn ). It follows from this analysis that h(Fn ) ≤ y2 . The quantity y2 depends √ on n and we have y2 − y0 −→ 0 hence lim sup y2 = 1 lim sup y0 ≤ 2π log(10K 2). As a consequence we have proved the following (weak) asymptotic estimate √ 1 (19) lim sup h(Fn ) ≤ log(10K 2). 2π n→+∞ √ The√constant 2 here has nothing to do with our choice of rotation numbers involving 2. We now enhance this estimate by a prior renormalization of fixed—yet high— order. More precisely we temporarily fix some k ≥ 0 and ε > 0. Let Feε be the map obtained by conjugating Fε by the translation by −iε and then restricting to H. Then h(Fn ) ≤ ε + h(Fen ). By the Schwarz-Pick inequality we have |Fn′ (Z) − 1| ≤

K|αn − α| , 2 Im Z

sup |Fen′ − 1| ≤

K αn − α . 2ε

and this implies H

Now for n > k let R Fn be the order k + 1 direct renormalization of Fen provided by Lemma 32. According to this lemma, lim sup h(Fen ) ≤ |β| lim sup h(R Fen ),

where β = β(k) = qk α − pk (recall that for fixed k, the first k convergents of α and α′ coincide for large enough n). Now by Lemma 33 with K replaced by K/2ε (we assume ε < 1/2), the sequence R Fen satisfies sup |R F˜n (Z) − Tα′n (Z)| ≤

Im Z>0

DK ′ |α − α′ | 2ε n

where αn′ and α′ are the respective rotation numbers of R Fen and√of its limit. We have α′ = −[ak+1 , ak+2 , . . .] and αn′ = −[ak+1 , . . . , an , 1 + an+1 , 1 + 2]. We are thus in the situation of Lemma 19 for −α′ in place of α and s ◦ (R F˜n ) ◦ s in place of Fn , where s(X + iY ) = −X + iY . By the weak estimate above Equation (19) we thus have √ 1 5DK 2 e lim sup h(R Fn ) ≤ log 2π ε n→+∞ ASTÉRISQUE 416

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and thus

√ 1 5DK 2 lim sup h(Fn ) ≤ ε + |β(k)| log . 2π ε n→+∞ Now this is valid for all k > 0 and since β(k) −→ 0 as k → +∞ and neither D, K, nor ε depend on k, we get lim sup h(Fn ) ≤ ε. n→+∞

Since this is valid for all ε ∈ (0, 1/2) we conclude: lim sup h(Fn ) ≤ 0. n→+∞

This ends the proof of Lemma 19. For this case, as in the case α = p/q, we used a direct renormalization of a direct renormalization, though in a more subtle way. Appendix A Analytic degenerate families An obvious way of obtaining degenerate families is to conjugate the family of rigid rotations Rα (z) = e2πiα by a family of varying analytic diﬀeomorphisms. The next lemma shows that in the case of families depending analytically on α, this is the only way. Proposition 34. — Let I be an open subset of R. Assume {fα : D → C}α∈I is an R-analytic family of maps which fix 0 with multiplier e2iπα . The following are equivalent: 1. the family {fα }α∈I is degenerate; 2. there exist an open interval J ⊂ I, a real δ > 0 and an analytic map ϕ : J × B(0, δ) → C such that for all α ∈ J, ϕα (z) = z + O (z 2 ) and for all z ∈ B(0, δ), fα = ϕα ◦ Rα ◦ ϕ−1 α (with ϕα = ϕ(α, ·)). Proof. — (2) =⇒ (1). Obvious. (1) =⇒ (2). Let U be a domain intersecting R in an interval J contained in I such that fα is defined for all α ∈ U and is linearizable for every α ∈ J ∩ Q. Let ϕα , α ∈ U \Q be the (uniquely defined) formal linearization of fα , so ϕα is a formal power series ∞ X an (α)z n ϕα (z) = z + n=2

satisfying (20)

ϕa ◦ Rα = fα ◦ ϕa

formally. The an can be found recursively from the power series expansion of fα (see [25] or below), and from the formula one obtains it follows that they are, in general,

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meromorphic functions of α ∈ U , with possible poles when the multiplier is a root of unity of order ≤ n, i.e., when α = p/q with 1 ≤ q ≤ n. Recall that we assumed that fα is linearizable for all α ∈ J ∩ Q. Let us prove that this implies that the functions an (α) have no poles. Lemma 35. — The function α 7→ an (α) has a holomorphic extension to U . Proof. — We will proceed by induction on n. We have a1 (α) = 1, which initializes the recurrence. Let n > 1 and assume that for all k < n, the map α 7→ ak (α) has a holomorphic extension ρ(α) = ei2πα be the multiplier and let bn (α) such P to U . Let m that fα = ρ(α) + m≥2 bm (α)z . Equation (20) then reads X X X X ai (α)ρ(α)i z i = ρ(α) ai (α)z i + bm ( ai (α)z i )m i≥1

i≥1

m≥2

i≥1

In particular for the coeﬃcient in z n :

ρ(α)n an (α) = ρ(α)an (α) +

n X

m=2

bm (α)

X

ai1 (α) · · · aim (α),

...

P where the ... is over all the m-uplets of positive integers whose sum are equal to n. We rewrite the last line as follows: (ρ(α)n − ρ(α))an (α) = Pb(α),n (a2 (α), . . . , an−1 (α)) Pn P where Pb(α),n (x2 , . . . , xn−1 ) = m=2 bm (α) ... xi1 · · · xim . By the induction hypothesis, the right hand side of eq. (21) is holomorphic, hence the function an (α) has at most simple poles, situated at α = p/(n − 1), p ∈ Z. If we prove that the right hand side of eq. (21) vanishes for α = p/(n − 1), then we will have completed the induction. Now we must be careful: by assumption for every p/q there is a solution ϕ to ϕ ◦ Rp/q = fp/q ◦ ϕ; however this solution is not unique. (21)

Sublemma 36. — If ζ is a formal power series such that ζ ◦ Rp/q − Rp/q ◦ ζ = O (z m ) and if q|(m − 1) then ζ ◦ Rp/q − Rp/q ◦ ζ = O (z m+1 ). Proof. — By a straightforward computation, for any formal power series, all the coeﬃcients of ζ ◦ Rp/q − Rp/q ◦ ζ with order in 1 + qZ vanish. P Sublemma 37. — Let f (z) = e2πip/q z + n≥2 bn z n be a holomorphic or formal power P ˜n z n solution series, and assume that there is a formal power series ϕ˜ = z + n≥2 a P of ϕ˜ ◦ Rp/q = f ◦ ϕ˜ and another formal power series ϕ = z + n≥2 an z n such that ϕ ◦ Rp/q − f ◦ ϕ = O (z m ) for some m ≥ 2. This depends only on a2 , . . . , am−1 ; fix these values and consider the equation ϕ ◦ Rp/q − f ◦ ϕ = O (z m+1 ) with unknown am . Assume that p/q is in its lowest terms. Then 1. if m − 1 is not a multiple of q, there is a unique solution am ; 2. if m − 1 is a multiple of q, all am ∈ C are solutions: in other words Pb,m (a2 , . . . , am−1 ) = 0.

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Proof. — Case (1) is immediate in view of eq. (21). Assume we are in case (2). The formal power series ζ = ϕ−1 ◦ ϕ˜ commutes with Rp/q up to order m − 1 included, and thus by Sublemma 36 up to order m included. It follows that ϕ ◦ Rp/q − f ◦ ϕ = O (z m+1 ). It follows from Case (2) of the above lemma applied to the reduced form of p/(n−1) that Pb(α),n (a2 (α), . . . , an−1 (α)) = 0. This cancels the possible simple pole to an (α) at α = p/(n − 1) and proves heredity of the induction hypothesis. Lemma 35 follows. P If α = p/q ∈ Q, we let ϕa = z + n≥2 an (α)z n for the holomorphic extension of the functions an (α) at a = p/q. For α ∈ U not necessarily real, let R(α) ∈ [0, +∞] be the radius of convergence of ϕa , s(α) ≤ R(a) be the radius of the maximal disk centered at 0 around 0 where ϕα takes values in D, and r(α) ≤ s(α) the maximal radius disk of such a disk for which moreover ϕa is injective. It is easy to see that s(α) is locally bounded away from zero in U \ J. (36) Lemma 38. — If for α0 ∈ J we have r(α0 ) > 0, then lim inf s(α0 + iε) > 0. ε→0

Proof. — Set gε = ϕ−1 ◦ fα0 +iε ◦ ϕ, where ϕ denotes the restriction of ϕα0 to r(α0 )D. The domain of gε tends to r(α0 )D as ε → 0 and gε −→ Rα0 uniformly locally. Also, gε (z) = e−2πε+2iπα0 z + O (εz 2 ). In particular, |gε (z)| = (1 − 2πε)|z| + O (εz 2 ). So, there exists 0 < r0 < r(α0 ) so that — when |z| < r0 and ε > 0, |gε (z)| < |z| and — when |z| < r0 and ε < 0, |gε (z)| > |z|. For ε suﬃciently close to 0, gε is univalent on B(0, r0 ). So, there is a univalent map ψα : B(0, r0 ) → C which conjugates gε to Rα0 +iε . The map ϕ˜ := ϕα0 ◦ ψα−1 satisfies the equation ϕ˜ ◦ Rα = f ◦ ϕ˜ near 0 so by uniqueness has the same expansion as ϕα . It follows from the Koebe One Quarter Theorem applied to ψα that as ε → 0, lim inf r(α0 + iε) ≥ r0 /4. Since s ≥ r, the lemma follows. Consider two Brjuno numbers α0 < α1 in J (so that r(α0 ) > 0 and r(α1 ) > 0) and y > 0 small enough so that the box U ′ of equation α0 < Re z < α1 and | Im z| < y is compactly contained in U . By Lemma 38 we have s| ′ ≥ δ > 0. By Cauchy’s formula ∂U applied to ϕa in the disk s(α)D, we see that an (α) ≤ δ −n holds for all α ∈ ∂U ′ and thus by the maximum principle holds for all α ∈ U ′ . It follows that (α, z) → ϕα (z) is defined (36)

This also holds for r(α).

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and holomorphic U ′ ×B(0, δ) → D. It satisfies ϕα (z) = z + O (z 2 ) and ϕα ◦Rα = fα ◦ϕα by analytic continuation. This proves claim (2) for J = (α0 , α1 ). Appendix B General statement We recall here the main statement in [11], adapted it to our situation. Notation 39. — Let X and Y be topological spaces and X ⊂ Y . We write X ⊂0 Y if the canonical injection X ֒→ Y is continuous. If moreover X is a normed vector space and Y a Fréchet space, (37) we write X ⊂c Y if every bounded set in X has compact closure in Y . In the theorem below we assume, as in most of the present article, that I ⊂ R is an open interval and that fα : D → C is a family of analytic maps that depends continously on α ∈ I, with fα (z) = e2πiα z + O (z 2 ). Below we use r(α) from Notation 8 in the present article, and ϕα from Definition 5. The notation C 0 refers to the set of holomorphic maps on D that have a continuous extension to D, endowed with the sup-norm. The notation C ω refers to the set of holomorphic maps on D that have a holomorphic extension to a neighborhood of D in C. Theorem 40. — Let F be any Fréchet space such that C ω ⊂ F ⊂0 C 0 , and let B ⊂c F be a Banach space. If the family (fα ) is non-degenerate (see Definition 1) and the dependence on α is Lipschitz then there exists a Brjuno number α such that — ∂∆α is compactly contained in D, — the map z 7→ ϕα (r(α)z) belongs to F but not to B. Equivalently, one can replace the Banach space B ⊂c F by a countable union of Banach spaces Bn ⊂c F or by a countable union of compact sets Kn ⊂ F . See Section 1 of [11] to see how one deduces Theorem 3 from Theorem 40. References [1] J. M. Aarts & L. G. Oversteegen – “The geometry of Julia sets”, Trans. Amer. Math. Soc. 338 (1993), p. 897–918. [2] A. Avila – “Smooth siegel disks via semi-continuity: a remark on a theorem by Buﬀ and Chéritat”, manuscript. (37)

We do not assume that the norm on X and the distance on Y are related.

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[3] A. Avila, X. Buff & A. Chéritat – “Siegel disks with smooth boundaries”, Acta Math. 193 (2004), p. 1–30. [4] K. Barański, X. Jarque & L. Rempe – “Brushing the hairs of transcendental entire functions”, Topology and its Applications 159 (2012), p. 2102–2114. [5] K. Biswas – “Smooth combs inside hedgehogs”, Discrete Contin. Dyn. Syst. 12 (2005), p. 853–880. [6]

, “Hedgehogs of Hausdorﬀ dimension one”, Ergodic Theory Dynam. Systems 28 (2008), p. 1713–1727.

[7] B. Branner & N. Fagella – Quasiconformal surgery in holomorphic dynamics, Cambridge Studies in Advanced Math., vol. 141, Cambridge Univ. Press, 2014. [8] A. D. Brjuno – “Power asymptotics of solutions of non-linear systems”, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), p. 329–364. [9] A. D. Brjuno – “Analytical form of diﬀerential equations.”, Trans. Mosc. Math. Soc. 25 (1973), p. 131–288. [10]

, “The analytic form of diﬀerential equations. II.”, Trans. Mosc. Math. Soc. 26 (1974), p. 199–239.

[11] X. Buff & A. Chéritat – “How regular can the boundary of a quadratic Siegel disk be?”, Proc. Amer. Math. Soc. 135 (2007), p. 1073–1080. [12] W. o. J. Charatonik – “The Lelek fan is unique”, Houston J. Math. 15 (1989), p. 27– 34. [13] D. Cheraghi – arXiv:1706.02678.

“Topology

of

irrationally

indiﬀerent

attractors”,

preprint

[14] A. Chéritat – “Recherche d’ensembles de Julia de mesure de Lebesgue positive”, Ph.D. Thesis, Université Paris-Sud, Orsay, 2001. [15] A. Douady – “Disques de Siegel et anneaux de Herman”, Séminaire Bourbaki, vol. 1986/1987, exposé no 677, Astérisque 152-153 (1987), p. 151–172. [16] J.-M. Gambaudo, P. Le Calvez & E. Pécou – “Une généralisation d’un théorème de Naishul”, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), p. 397–402. [17] L. Geyer – “Smooth Siegel disks without number theory”, Math. Proc. Cambridge Philos. Soc. 144 (2008), p. 439–442. [18] J. Graczyk & G. Światek – “Siegel disks with critical points in their boundaries”, Duke Math. J. 119 (2003), p. 189–196. [19] M.-R. Herman – “Are there critical points on the boundaries of singular domains?”, Comm. Math. Phys. 99 (1985), p. 593–612. [20] J. Milnor – Dynamics in one complex variable, third ed., Annals of Math. Studies, vol. 160, Princeton Univ. Press, 2006. [21] V. A. Na˘ıšul′ – “Topological invariants of germs of analytic mappings and area preserving mappings, and their application to analytic diﬀerential equations in CP 2 ”, Funktsional. Anal. i Prilozhen. 14 (1980), p. 73–74. [22] R. Pérez-Marco – “Sur les dynamiques holomorphes non linéarisables et une conjecture de V. I. Arnol′ d”, Ann. Sci. École Norm. Sup. 26 (1993), p. 565–644. [23]

, “Fixed points and circle maps”, Acta Math. 179 (1997), p. 243–294.

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[24]

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, “Siegel disks with quasi-analytic boundary”, Preprint, Université Paris-Sud, 97-52, 1997.

[25] G. A. Pfeiffer – “On the conformal mapping of curvilinear angles. The functional equation ϕ[f (x)] = a1 ϕ(x)”, Trans. Amer. Math. Soc. 18 (1917), p. 185–198. [26] C. Pommerenke – Boundary behaviour of conformal maps, Grundl. math. Wiss., vol. 299, Springer, 1992. [27] E. Risler – “Linéarisation des perturbations holomorphes des rotations et applications”, Mém. Soc. Math. Fr. (N.S.) 77 (1999), p. 102. [28] H. Rüssmann – “über die Iteration analytischer Funktionen”, J. Math. Mech. 17 (1967), p. 523–532. [29] M. Shishikura – “Bifurcation of parabolic fixed points”, in The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, 2000, p. 325–363. [30] C. L. Siegel – “Iteration of analytic functions”, Ann. of Math. 43 (1942), p. 607–612. [31] J.-C. Yoccoz – “An introduction to small divisors problems”, in From number theory to physics (Les Houches, 1989), Springer, 1992, p. 659–679. [32]

, “Théorème de Siegel, nombres de Bruno et polynômes quadratiques”, Astérisque 231 (1995), p. 3–88.

A. Avila, Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland, & IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brazil X. Buff, Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118, route de Narbonne, 31062 Toulouse Cedex, France A. Chéritat, Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118, route de Narbonne, 31062 Toulouse Cedex, France

ASTÉRISQUE 416

Astérisque 416, 2020, p. 181–191 doi:10.24033/ast.1113

ON QUASI-INVARIANT CURVES by Ricardo Pérez-Marco

Abstract. — Quasi-invariant curves are the fundamental tool for the study of hedgehog’s dynamics. The Denjoy-Yoccoz lemma is the preliminary step for Yoccoz’s complex renormalization techniques for the study of linearization of analytic circle diffeomorphisms. We give a new geometric interpretation of the Denjoy-Yoccoz lemma using the hyperbolic metric that gives a new direct construction of quasi-invariant curves without renormalization theory as in the original construction. Résumé (Sur les courbes quasi-invariantes). — Les courbes quasi-invariantes sont un outil fondamental dans l’étude de la dynamique des hérissons. Le lemme de Denjoy-Yoccoz est le premier pas dans la théorie de renormalisation de Yoccoz des diﬀéomorphismes analytiques du cercle et l’étude de sa linéarisation. On donne une nouvelle version du lemme de Denjoy-Yoccoz en termes de métrique hyperbolique, ce qui fournit une nouvelle construction directe des courbes quasi-invariantes sans utiliser la renormalisation comme dans la construction originelle.

1. Introduction Quasi-invariant curves and their properties were announced in 1995 in a Note to the Comptes Rendus of the Académie des Sciences [6] presented by J.-Ch. Yoccoz. Their construction using renormalization techniques was carried out in the unpublished manuscript [8]. The goal of the present article is to present a short and direct construction of quasi-invariant curves without renormalization theory. In particular, this is the first complete published construction of quasi-invariant curves. Theorem 1 (Quasi-invariant curves). — Let g be an analytic circle diﬀeomorphism with irrational rotation number α. Let (pn /qn )n≥0 be the sequence of convergents of α given by the continued fraction algorithm. 2010 Mathematics Subject Classification. — 37 F 50, 37 F 25. Key words and phrases. — Complex dynamics, indiﬀerent fixed points, hedgehogs, analytic circle diffeomorphisms, small divisors, centralizers, renormalization.

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182

Given C0 > 0 there is n0 ≥ 0 large enough such that there is a sequence of Jordan curves (γn )n≥n0 , homotopic to S1 and exterior to D, such that all the iterates g j , 0 ≤ j ≤ qn , are defined on a neighborhood of the closure of the annulus Un bounded by S1 and γn , and we have j DP (g (γn ), γn ) ≤ C0 , where DP is the Hausdorﬀ distance between compact sets associated to dP , the Poincaré distance in C − D. We also have for any z ∈ γn , dP (g qn (z), z) ≤ C0 , that is, ||g qn − id||CP0 (γn ) ≤ C0 . The curves γn are called quasi-invariant curves for g. The delicate, and useful, part of the construction of quasi-invariant curves is to obtain the estimates for the Poincaré metric, which is much harder and stronger than the estimates for the euclidean metric since the curves γn are close to S1 . This is also what is needed for the application to hedgehog’s dynamics. Hedgehogs are totally invariant continua associated to indiﬀerent irrational non-linearizable fixed points discovered by the author in [7]. The dynamics in a neighborhood outside a hedgehog K is conjugated to the dynamics of an analytic circle diﬀeomorphism by the dictionary construction presented in [7]. Quasi-invariant curves with their Poincaré metric estimates can then be transported to osculating curves around the hedgehog and provide the tool to analyze the dynamics on the hedgehog. This is only possible thanks to the Poincaré estimates. g = h−1 ◦ f ◦ h

f K h

D S1

Figure 1. Dictionary of fixed points and circle maps using hedgehog K.

The new construction of quasi-invariant curves without renormalization is based on the key observation that the Denjoy-Yoccoz Lemma from [13] (Proposition 4.4 in Section 4.4) has a natural hyperbolic interpretation. First we carry out the construction in the situation where we assume that the non-linearity of g is small, that is ||D log Dg||C 0 small. This case is enough for most of the applications, in particular for the solution in [9] of Briot and Bouquet problem from 1856 (see [1]). The general case reduces to this situation by the same arguments as in Section 3.6 of [13] that reduces

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by a sectorial return map the dynamics of a general analytic circle diﬀeomorphism to the dynamics of one with arbitrarily small non-linearity. The proof of the DenjoyYoccoz Lemma relies on real estimates for the iterates of circle diﬀeomorphisms that follow from classical work by M. Herman ([4]) and J.-Ch. Yoccoz ([11]). 2. Analytic circle diffeomorphisms 2.1. Notations. — We refer to the Thesis of M. Herman [4] for the classical theory and background on circle diﬀeomorphisms. We denote by T = R/Z the abstract circle, and S1 = E(T) its embedding in the complex plane C given by the exponential mapping E(x) = e2πix . We study analytic diﬀeomorphisms of the circle, but we prefer to work at the level of the universal covering, the real line, with its standard embedding R ⊂ C. We denote by Dω (T) the space of increasing analytic diﬀeomorphisms g of the real line such that, for any x ∈ R, g(x + 1) = g(x) + 1, i.e g commutes with T (x) = x + 1, the generator of the group of deck transformations of the universal covering. Thanks to H. Poincaré [10], we know that an element of the space Dω (T) has a well defined rotation number ρ(g) ∈ R which is the constant uniform limit of n1 (g n − id) when n → +∞. Thanks to A. Denjoy [2], we know that the order preserving diﬀeomorphism g is indeed conjugated to the rigid translation Tρ(g) : x 7→ x + ρ(g), by an orientation preserving homeomorphism h : R → R, such that h(x + 1) = h(x) + 1. For ∆ > 0, we note B∆ = {z ∈ C; |ℑz| < ∆}, and A∆ = E(B∆ ). The subspace Dω (T, ∆) ⊂ Dω (T) is composed by the elements of Dω (T) which extend analytically to a holomorphic diﬀeomorphism, denoted again by g, such that g and g −1 are defined on B∆ . 2.2. Real estimates. — We refer to J.-Ch. Yoccoz article [13] for the results on this section. We assume that the orientation preserving circle diﬀeomorphism g is C 3 and that the rotation number α = ρ(g) is irrational. We consider the convergents (pn /qn )n≥0 of α obtained by the continued fraction algorithm (see [3] for notations and basic properties of continued fractions). For n ≥ 0, we define the map gn (x) = g qn (x) − pn and the intervals In (x) = [x, gn (x)], Jn (x) = In (x)∪In (gn−1 (x)) = [gn−1 (x), gn (x)]. Let mn (x) = g qn (x)−x−pn = ±|In (x)|, Mn = supR |mn (x)|, and mn = minR |mn (x)|. Topological linearization obviously implies limn→+∞ Mn = 0, since this holds for a rigid rotation, and is equivalent to this condition since then any orbit is dense modulo 1 and determines uniquely h modulo 1. This is always true for analytic diﬀeomorphisms by Denjoy’s Theorem, that also holds for C 1 diﬀeomorphisms such that log Dg has bounded variation. Since g is topologically linearizable, the combinatorics of the irrational translation, or the continued fraction algorithm, shows (see Lemma 3.7 [13]): Lemma 2. — Let x ∈ R, 0 ≤ j < qn+1 and k ∈ Z. The intervals g j ◦ T k (In (x)) have disjoint interiors, and the intervals g j ◦ T k (Jn (x)) cover R at most twice.

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We have the following fundamental estimate (see [4], [11] and, more precisely, Corollary 3.16 in [13]) on the Schwarzian derivatives of the iterates of f , for 0 ≤ j ≤ qn+1 , j Mn e2V S Sg (x) ≤ , |In (x)|2

with S = ||Sg||C 0 (R) and V = Var log Dg. These estimates imply a control of the non-linearity of the iterates (see Corollary 3.18 in [13]): Proposition 3. — For 0 ≤ j ≤ 2qn+1 , c =

√

2SeV , we have 1/2

||D log Dg j ||C 0 (R) ≤ c

Mn . mn

These estmates on the iterates of g give estimates on gn . More precisely, we have (Corollary 3.20 in [13]): Proposition 4. — For some constant C > 0, we have || log Dgn ||C 0 (R) ≤ CMn1/2 . Corollary 5. — For any ǫ > 0, there exists n0 ≥ 1 such that for n ≥ n0 , we have ||Dgn − 1||C 0 (R) ≤ ǫ. 1/2

Proof. — Take n0 ≥ 1 large enough so that for n ≥ n0 , CMn use Proposition 4 and the estimate |ew − 1| ≤

< min( 32 ǫ, 12 ), then

3 |w| 2

for |w| < 1/2. Corollary 6. — For any ǫ > 0, there exists n0 ≥ 1 such that for n ≥ n0 , for any x ∈ R and y ∈ In (x) we have 1−ǫ≤

mn (y) ≤ 1 + ǫ. mn (x)

Proof. — We have Dmn (x) = Dgn (x) − 1, and |mn (y) − mn (x)| ≤ ||Dmn ||C 0 (R) |y − x| ≤ ||Dgn − 1||C 0 (R) |mn (x)|. We conclude using Corollary 5.

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3. Denjoy-Yoccoz Lemma Once we have these real estimates, and, more precisely, a control on the nonlinearity, we can use them in a complex neighborhood. Using the notations introduced in the previous section, the raw form of the Denjoy-Yoccoz Lemma (see [12]) is the following: Lemma 7 (Denjoy-Yoccoz Lemma). — Let ∆ > 0 and g ∈ Dω (T, ∆) holomorphic and continuous on B∆ . We assume that 1 τ = ||D log Dg||C 0 (B∆ ) < , 16 and that for some n ≥ 0, ∆ Mn ≤ , 2D0 1 . where 4 < D0 < 4τ Let z ∈ C, we write z0 = x0 + imn (x0 )y0 , y0 ∈ C, and we assume that |y0 | ≤ D0 . Then for 0 ≤ j ≤ qn+1 , we have g j (z0 ) = f j (x0 ) + imn (g j (x0 )) yj , with |yj − y0 | ≤ 3D0 τ |y0 |. Proof. — Let zj = g j (z0 ) and xj = g j (x0 ). We prove the lemma by induction on j ≥ 0. For j = 0 the result is obvious. Assume the result for i ≤ j − 1, and using the induction hypothesis we have |yi | ≤ 4|y70 | ≤ 2D0 − 1, since 3D0 τ ≤ 3/4 and D0 > 4. By the chain rule we have j

log Dg (z0 ) =

j−1 X

log Dg(zl ),

l=0

so

j−1 X log Dg j (z0 ) − log Dg j (x0 ) ≤ |log Dg(zl ) − log Dg(xl )| l=0

≤τ

j−1 X

|zl − xl |

l=0

≤ τ (2D0 − 1)

j−1 X

|mn (xl )|.

l=0

Considering the j-iterate of g on the interval ]x0 , g qn (x0 ) − pn [, we obtain a point ζ ∈ ]x0 , g qn (x0 ) − pn [ such that, Dg j (ζ) =

mn (xj ) . mn (x0 )

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By a similar argument as before for g j (z0 ) applied to g j (ζ) we have j−1 X log Dg j (ζ) − log Dg j (x0 ) ≤ τ |mn (xl )|. l=0

Adding the two previous inequalities, we have j−1 X log Dg j (z0 ) − log mn (xj ) ≤ 2D0 τ |mn (xl )|. mn (x0 ) l=0

The intervals ]xl , g (xl ) − pn [, 0 ≤ l < qn+1 being disjoint modulo 1, we have qn

qn+1 −1

X

|mn (xl )| < 1.

l=0

So we obtain

log Dg j (z0 ) − log mn (xj ) ≤ 2D0 τ, mn (x0 )

and taking the exponential (using |ew − 1| ≤ 3/2|w|, for |w| < 1/2, since 2Dτ < 1/2), j Dg (z0 ) − mn (xj ) ≤ 3D0 τ mn (xj ) . mn (x0 ) mn (x0 )

This last estimate holds for any point zt in the rectilinear segment [x0 , z0 ]. Integrating along this segment we get the definitive estimate, j g (z0 ) − g j (x0 ) − iy0 mn (xj ) ≤ 3D0 τ |y0 ||mn (xj )|. 4. Hyperbolic Denjoy-Yoccoz Lemma 4.1. Flow interpolation in R. — Since g is analytic, from Denjoy’s Theorem we know that g|R is topologically linearizable, i.e. there exists an increasing homeomorphism h : R → R, such that for x ∈ R, h(x + 1) = h(x) + 1, and h−1 ◦ g ◦ h = Tα ,

where Tα : R → R, x 7→ x + α. We can embed g into a topological ﬂow on the real line (ϕt )t∈R defined, for t ∈ R, ϕt = h ◦ Ttα ◦ h−1 (thus ϕ1 = g and ϕα−1 = T ). In general, when g is not analytically linearizable (i.e. h is not analytic), the maps ϕt are only homeomorphism of the real line, although for t ∈ Z, ϕt is analytic since for these values they are iterates of g. In some cases for other values of t, ϕt is an analytic diﬀeomorphism in the analytic centralizer of g (see [5] for more information on analytic centralizers). Now (ϕt )t∈[0,1] is an isotopy from the identity to g. The ﬂow (ϕt )t∈R is a one parameter subgroup of homeomorphisms of the real line commuting to the translation by 1.

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4.2. Flow interpolation in C. — There are diﬀerent complex extensions of the ﬂow (ϕt )t∈R suitable for our purposes. For n ≥ 0, we can extend this topological ﬂow to a topological ﬂow in C by defining, for z0 = x0 + i |mn (x0 )|y0 ∈ C, with x0 , y0 ∈ R, (n)

ϕt (z0 ) = z0 (t) = ϕt (x0 ) + i |mn (ϕt (x0 ))|y0 . (n)

We denote Φz0 the ﬂow line passing through z0 , (n)

Φ(n) z0 = (ϕt (z0 ))t∈R .

4.3. Hyperbolic Denjoy-Yoccoz Lemma. — We are now ready to give a geometric version of the Denjoy-Yoccoz Lemma. We denote by dP the Poincaré distance in the upper half plane. Lemma 8 (Hyperbolic Denjoy-Yoccoz Lemma). — There exists ǫ0 > 0 small enough universal constant such that the following holds. Let 4 < D0 < 4ǫ10 . Let ∆ > 0 and g ∈ Dω (T, ∆) holomorphic and continuous on B∆ such that ||D log Dg||C 0 (B∆ ) < ǫ0 . Then there exists n0 ≥ 1 and a universal constant C0 > 0, such that for n > n0 , for any z0 ∈ B∆ , ℑz0 > 0, z0 = x0 + imn (x0 )y0 , with 0 < y0 < D0 , and 0 ≤ j ≤ qn+1 , we have (n)

dP (g j (z0 ), ϕj (z0 )) ≤ C0 . Proof. — Since Mn → 0, we choose n0 ≥ 1 big enough so that for n ≥ n0 we have Mn ≤

∆ , 2D0

so we can use the Denjoy-Yoccoz lemma in the previous section. The Poincaré metric in the upper half plane is given by |ds| =

|dξ| . ℑξ

Therefore (n) dP (zj , ϕj (z0 ))

≤

Z

(n) [zj ,ϕj (z0 )]

|dξ| 1 ≤ |mn (xj )| |yj − y0 | ℑξ inf ξ∈[zj ,ϕ(n) (z0 )] ℑξ j

4 ≤ |mn (xj )| |yj − y0 | |mn (xj )| y0 |yj − y0 | ≤4 ≤ 3 = C0 , y0 where in the second line we used that ℜyj ≥ 14 y0 which follow from |yj − y0 | ≤ 43 y0 that we also used in the last inequality.

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5. Quasi-invariant curves Using now the notations of the previous section, we prove now that the ﬂow (n) lines Φz0 , with 21 < y0 < 1 and n ≥ n0 for n0 ≥ 1 large enough, are quasi-invariant curves satisfying Theorem 1. These ﬂow lines are graphs over R. Proposition 9. — There exists a constant C0 > 0 and n0 ≥ 0 large enough, such that (n−1) for some z0 with 1/2 < ℑz0 = y0 < 1, we have, for for n ≥ n0 and γn = Φz0 0 ≤ j ≤ qn , DP (g

j

(γn ), γn ) ≤ C0

Proof. — We prove this proposition for n+1 instead of n (the proposition is stated to match n in Theorem 1). It follows from the hyperbolic Denjoy-Yoccoz Lemma 8 that for n0 ≥ 1 large enough, for k ∈ Z and 0 ≤ j ≤ qn+1 the iterate g j ◦ T k is C0 -close (n) in the Poincaré metric to the ﬂow map ϕj+kα−1 . The last map leaves invariant γn , hence the result. This proves the first property of Theorem 1. For the second property, we consider (n) ﬂow lines Φz0 with 12 < y0 < 1. Given an interval I ⊂ R, we label I˜(n) the piece of (n) Φz0 over I. z3

z4 z5

z2 (n) I˜n

z0

z1

In x0

x1

x2

x3

x4

x5

...

R

Figure 2. A quasi-invariant curve

Lemma 10. — We assume that 12 < y0 < 1. There is n0 ≥ 1 such that for n ≥ n0 and (n) for any x ∈ R, the piece I˜n (x) has bounded Poincaré diameter. (n) Proof. — Let z = u + i |mn (u)|y0 be the current point in I˜n (x). We have

dz = (1 ± i (Dgn (u) − 1)y0 ) du.

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For any ǫ0 > 0, choosing n0 ≥ 1 large enough, for n ≥ n0 , according to Corollary 5 we have dz du − 1 ≤ ǫ0 . Therefore, we have Z Z 1 1 |dz| ≤ (1 + ǫ0 ) du. lP (I˜n(n) (x0 )) = (n) |m (u)| y |m (u)| y0 ˜ n 0 n In (x0 ) In (x0 ) Now using Corollary 6 with ǫ = ǫ0 and increasing n0 if necessary, we have Z 1 1 + ǫ0 1 1 + ǫ0 1 + ǫ0 lP (I˜n(n) (x)) ≤ du ≤ ≤2 ≤ C. y0 1 − ǫ0 1 − ǫ0 In (x0 ) |mn (x0 )| y0 1 − ǫ0 We assume n ≥ n0 from now on. Lemma 11. — We assume that 21 < y0 < 1. For 0 ≤ j < qn+1 and any x ∈ R, the (n) (n) pieces (g j ◦T k (J˜n (x)))0≤j≤qn+1 ,k∈Z have bounded Poincaré diameter and cover Φz0 . (n)

Proof. — From Lemma 10 any I˜n (x) has bounded Poincaré diameter, thus also any (n) (n) (n) J˜n (x) = I˜n (x) ∪ I˜n (gn−1 (x)). Moreover, we have g j ◦ T k (Jn (x)) = Jn (g j ◦ T k (x)), (n) j k and all J˜n (g ◦ T (x)) have also bounded Poincaré diameter. From Lemma 2 these (n) pieces cover Φz0 . (n)

Corollary 12. — For some C0 > 0, the ﬂow orbit (ϕj+kα−1 (z0 ))0≤j 0 if necessary): Proposition 13. — For any z0 ∈ Φ(n) , we have dP (z0 , g qn+1 (z0 )) ≤ C0 . From Lemma 11 we also get the property that the hyperbolic balls (n) (n) BP (ϕj+kα−1 (z0 ), C0 ) cover Φz0 . Proposition 14. — We have that Un =

[

(n)

BP (ϕj+kα−1 (z0 ), C0 )

0≤j 0 there exists n large enough, such that the new base (gn , gn+1 ) for the action can be conjugated by an analytic diﬀeomorphism hn : R → R such that hn ◦ gn ◦ h−1 n =T hn ◦ gn+1 ◦ h−1 n = Gn with ||D log DGn ||C 0 (R) ≤ ǫ0 . Note that in this construction the diﬀeomorphism Gn is analytic with a small nonlinearity, but we don’t have control on ∆ > 0 such that Gn is holomorphic in B∆ . This is acceptable because taking g = Gn we can apply the Denjoy-Yoccoz lemma from Section 3 (or the hyperbolic version from Section 4) where there is no restriction on how small ∆ > 0 can be. If ∆ > 0 is very small we compensate by taking n large enough which makes Mn as small as we like. Therefore, using the previous results in this article, we can construct quasi-invariant curves for Gn . We only need to lift with estimates these quasi-invariant curves into quasi-invariant curves of the original circle diﬀeomorphism g which defines the original Z2 -action. The curves can be lifted to translation arcs that invariant by the dynamics and are C-close to being Z-periodic in the Poincaré metric. This is proved observing that because hn is an analytic diﬀeomorphism it is Lipschitz in the Poincaré metric of the upper half plane in a fundamental region with bounded real part. So this translation arcs can be perturbed to Z-periodic curves that are quasi-invariant curves (i.e. having the properties of Theorem 1). In the original construction from [8] this

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lift construction was iterated through multiple sectorial renormalizations. Having to carry out the estimates, this made the construction long and technical. To be able to lift the curves only once, and skip the iterative renormalization part, gives the important simplification of the construction that we present in this article. Acknowledgements. — I thank the referee for his extreme careful reading. References [1] C. Briot & J.-C. Bouquet – “Recherches sur les proprietés des équations diﬀérentielles”, J. École impériale polytechnique 21:36 (1856), p. 133–198. [2] A. Denjoy – “Sur les courbes définies par les équations diﬀérentielles à la surface du tore”, J. Math. Pures et Appl. 11 (1932), p. 333–375. [3] G. H. Hardy & E. M. Wright – An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. [4] M.-R. Herman – “Sur la conjugaison diﬀérentiable des diﬀéomorphismes du cercle à des rotations”, Inst. Hautes Études Sci. Publ. Math. 49 (1979), p. 5–233. [5] R. Pérez Marco – “Nonlinearizable holomorphic dynamics having an uncountable number of symmetries”, Invent. math. 119 (1995), p. 67–127. [6]

, “Sur une question de Dulac et Fatou”, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), p. 1045–1048.

[7] R. Pérez-Marco – “Fixed points and circle maps”, Acta Math. 179 (1997), p. 243–294. [8] [9]

, “Hedgehog dynamics”, manuscript, 1998. , “Solution to Briot and Bouquet problem on singularities of diﬀerential equations”, preprint arXiv:1802.03630.

[10] H. Poincaré – “Sur les courbes définies par les équations diﬀérentielles”, J. Math. Pures et Appl. 1 (1885), p. 167–244. [11] J.-C. Yoccoz – “Conjugaison diﬀérentiable des diﬀéomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne”, Ann. Sci. École Norm. Sup. 17 (1984), p. 333–359. [12]

, “Théorème de Siegel, nombres de Bruno et polynômes quadratiques”, Astérisque 231 (1995), p. 3–88.

[13]

, “Analytic linearization of circle diﬀeomorphisms”, in Dynamical systems and small divisors (Cetraro, 1998), Lecture Notes in Math., vol. 1784, Springer, 2002, p. 125– 173.

R. Pérez-Marco, CNRS, IMJ-PRG, Paris 7, Boîte courrier 7012, 75005 Paris Cedex 13, France E-mail : [email protected]

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2020

Astérisque 416, 2020, p. 193–211 doi:10.24033/ast.1114

HEDGEHOGS FOR NEUTRAL DISSIPATIVE GERMS OF HOLOMORPHIC DIFFEOMORPHISMS OF (C2 , 0) by Tanya Firsova, Mikhail Lyubich, Remus Radu & Raluca Tanase

Abstract. — We prove the existence of hedgehogs for germs of complex analytic diﬀeomorphisms of (C2 , 0) with a semi-neutral fixed point at the origin, using topological techniques. This approach also provides an alternative proof of a theorem of Pérez-Marco on the existence of hedgehogs for germs of univalent holomorphic maps of (C, 0) with a neutral fixed point. Résumé (Hérissons pour les germes dissipatifs neutres des difféomorphismes holomorphes de (C2 , 0)) Nous montrons l’existence de hérissons pour les germes de diﬀéomorphismes holomorphes de (C2 , 0) ayant un point fixe semi-neutre à l’origine, en utilisant uniquement des techniques topologiques. Cette approche donne également une preuve alternative d’un théorème de Pérez-Marco sur l’existence de hérissons pour les germes de diﬀéomorphismes holomorphes de (C, 0) ayant un point fixe neutre.

1. Introduction Let α ∈ R\Q and let pn /qn be the convergents of α given by the continued fraction algorithm. We say that α satisfies the Brjuno condition if P log qn+1 < ∞. (1) qn n≥0

Brjuno [1] and Rüssmann [20] showed that if α satisfies Bjruno’s condition, then any holomorphic germ with a fixed point with indiﬀerent multiplier λ = e2πiα is linearizable. The linearization is the irrational rotation with rotation number α. Yoccoz [27] proved that Brjuno’s condition is the optimal arithmetic condition that guarantees linearizability. If α does not verify inequality (1), then there exists a holomorphic germ f (z) = λz + O (z 2 ) which is non-linearizable around the origin, that is f is not

2010 Mathematics Subject Classification. — 37D30, 37E30, 32A10, 54H20. Key words and phrases. — Hedgehogs, holomorphic germs in C2 , partial hyperbolicity, center manifold, invariant petals, Brower Plane Translation Theorem.

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conjugate to the linear map z 7→ λz via a holomorphic change of coordinates. The origin is called a Cremer fixed point. The local dynamics of a non-linearizable map with a Cremer fixed point is complex and hard to visualize. In the ’90s, Pérez-Marco [13] proved the existence of interesting invariant compact sets near the Cremer fixed point, called hedgehogs. Using deep results from the theory of analytic circle diﬀeomorphisms developed by Yoccoz [26], Pérez-Marco [12, 14, 11, 15, 16] showed that even if the map on a neighborhood of the origin is not conjugate to an irrational rotation, the points of the hedgehog are recurrent and still move under the inﬂuence of the rotation. Cheraghi [4] built some models for the local dynamics near Cremer points for specific cases of quadratic polynomials with high type rotation numbers using the near-parabolic renormalization of Inou and Shishikura [8]. In this paper we show the existence of non-trivial compact invariant sets for germs of diﬀeomorphisms of (C2 , 0) with semi-indiﬀerent fixed points. The proof is purely topological and also provides an alternative proof for the existence of hedgehogs in dimension one. A fixed point x of a holomorphic germ f of (C2 , 0) is semi-indiﬀerent (or semineutral) if the eigenvalues λ and µ of the linear part of f at x satisfy |λ| = 1 and |µ| < 1. In analogy with the one-dimensional dynamics, a semi-indiﬀerent fixed point can be semi-parabolic, semi-Siegel or semi-Cremer, which essentially depends on the arithmetic properties of the neutral eigenvalue λ. We say that an isolated fixed point x is semi-parabolic if λ = e2πiα and the angle α = p/q is rational. If α is irrational and there exists an injective holomorphic map ϕ : D → C2 such that f (ϕ(ξ)) = ϕ(λξ), for ξ ∈ D, we call the fixed point semi-Siegel. Finally, if α is irrational and there does not exist an invariant disk on which the map is analytically conjugate to an irrational rotation, then the fixed point is called semi-Cremer. Note that in the latter case α does not satisfy the Brjuno condition (1). Let E s and E c denote the eigenspaces of df0 corresponding to the dissipative eigenvalue µ and respectively to the neutral eigenvalue λ. Let B ′ be a neighborhood of 0 and let Exs and Exc be not necessarily invariant continuous distributions such that E0s = E s , E0c = E c , and Tx B ′ = Exs ⊕ Exc for all x ∈ B ′ . We define the vertical cone Cxv to be the set of vectors in the tangent space at x that make an angle less than or equal to α with Exs , for some α > 0. The horizontal cone Cxh is defined in the same way, with respect to Exc . The map f is partially hyperbolic (1) on B ′ (see Pesin [17]) if there exist two real numbers µ and λ such that 0 < |µ| < µ < λ < 1 and a family of invariant cone fields C h/v on B ′ , (2)

(1)

v v dfx (Cxh ) ⊂ Int Cfh(x) ∪ {0}, dff−1 (x) (Cf (x) ) ⊂ Int Cx ∪ {0},

In this defintion, adapted to our purposes, there is no unstable bundle.

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such that for every x ∈ B ′ we have (3) (4)

λ kvk ≤ kdfx (v)k ≤ 1/λ kvk, for v ∈ Cxh , kdfx (v)k ≤ µ kvk, for v ∈ Cxv ,

for some Riemannian metric k · k. If f is partially hyperbolic, then the rate of contraction along Exs dominates the behavior of dfx along the complementary direction Exc . This domination ensures the c existence of local center manifolds Wloc (0) relative to a neighborhood B ′ of 0 as graphs c ′ s of functions ϕf : E ∩ B → E , as discussed in Section 2. Theorem A. — Let f be a germ of a holomorphic diﬀeomorphism of (C2 , 0) with a semi-indiﬀerent fixed point at 0 with eigenvalues λ and µ, where |λ| = 1 and |µ| < 1. There exists a neighborhood B ′ ⊂ C2 of 0 on which f is partially hyperbolic such that for any open ball B ⋐ B ′ centered at 0 there exists a set H ⊂ B with the following properties: c c a) H ⋐ Wloc (0), where Wloc (0) is any local center manifold of the fixed point 0 constructed relative to B ′ . b) H is compact, connected, completely invariant and full. c) 0 ∈ H , H ∩ ∂B 6= ∅. ss d) Every point x ∈ H has a well defined local strong stable manifold Wloc (x), consisting of points from B whose orbits converge exponentially fast to the orbit of x. The strong stable set of H is laminated by vertical-like holomorphic disks. We say that H is completely invariant if f (H ) ⊂ H and f −1 (H ) ⊂ H . The c set H is full if its complement in Wloc (0) is connected. The local strong stable manifold ss Wloc (x) of a point x ∈ H is defined as the set {y ∈ B : f n (y) ∈ B ∀n ≥ 1, lim dist(f n (y), f n (x))/µn = 0}, n→∞

where µ is the constant of partial hyperbolicity from (4). We call the set H from Theorem A a hedgehog. The most intriguing case is when the argument of λ is irrational and H is not contained in the closure of a linearization domain. This happens for instance, when the origin is semi-Cremer. Theorem A is applicable to the local study of dissipative polynomial automorphisms of C2 with a semi-indiﬀerent fixed point. The theorem generalizes directly to the case of germs of holomorphic diﬀeomorphisms of (Cn , 0), for n > 2, which have a fixed point at the origin with exactly one eigenvalue on the unit circle and n − 1 eigenvalues inside the unit disk. One interesting fact is that in this article we give a topological proof for the existence of hedgehogs in all dimensions, without using tools such as the Uniformization Theorem which are applicable only in the one-dimensional case. This approach encourages to look for topological methods that can be applied to germs with more than one neutral eigenvalue. In a second article [10] we use tools from complex diﬀerential geometry and quasiconformal theory (e.g., the Measurable Riemann Mapping Theorem) to study fine properties of the hedgehog. In principle, we could do everything by

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the latter methods, but it is instructive to know when such powerful tools are actually needed and when they can be bypassed by softer arguments. Acknowledgement. — We are grateful to Romain Dujardin for reading an earlier version of the paper and for providing valuable comments. The first author was supported by NSF grant DMS-1505342. The second author was supported by NSF grants DMS-1301602 and DMS-1600519.

2. Center manifolds of the semi-indifferent fixed point Let f : (C2 , 0) → (C2 , 0), f (x, y) = (λx + f1 (x, y), µy + f2 (x, y)) be a holomorphic germ with a semi-indiﬀerent fixed point at the origin. We also refer to f as a neutral dissipative germ of (C2 , 0). The semi-indiﬀerent fixed point has a well-defined unique analytic strong stable manifold W ss (0) corresponding to the dissipative eigenvalue µ. It consists of points that are attracted to 0 exponentially fast, and defined as (5)

W ss (0) := {x ∈ C2 : lim dist(f n (x), 0)/µn = const.}. n→∞

c (0) The semi-indiﬀerent fixed point also has a (non-unique) center manifold Wloc k c of class C for some integer k ≥ 1, tangent at 0 to the eigenspace E of the neutral eigenvalue λ. There exists a ball Bδ (where the size of δ depends on k) centered at the origin in which the center manifold is locally the graph of a C k function ϕf : E c → E s and has the following properties: c c a) Local Invariance: f (Wloc (0)) ∩ Bδ ⊂ Wloc (0). −n c (0). Thus center b) Weak Uniqueness: If f (x) ∈ Bδ for all n ∈ N, then x ∈ Wloc manifolds may diﬀer only on trajectories that leave the neighborhood Bδ under backward iterations. c) Shadowing: Given any point x such that f n (x) → 0 as n → ∞, there exists a c positive constant k and a point y ∈ Wloc (0) such that kf n (x) − f n (y)k < kµn as n → ∞. In other words, every orbit which converges to the origin can be described as an exponentially small perturbation of some orbit on the center manifold.

Consider the space of holomorphic germs g of (Bδ , 0) which are C k -close to f such that g has a semi-indiﬀerent fixed point at the origin. We will later consider a sequence of germs with a semi-parabolic fixed point which converges uniformly to a germ with a semi-Cremer fixed point. Proposition 2.1 shows that even if the center manifold of g is not unique, it may be chosen to depend continuously on g for the C k topology. Let E c (δ) = E c ∩ Bδ . Proposition 2.1. — The map g has a C k center manifold defined as the graph of a C k function ϕg : E c (δ) → E s (δ) such that the map (g, x) 7→ ϕg (x) is C k with respect to g.

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We refer to [21] (Chapter 5, Appendix III) and [7] for the theory of stable and center manifolds. For a proof of Proposition 2.1 see Theorem 5.1 and §5A in [7]. Assume that the map f is partially hyperbolic on a ball Bδ . By using a cut-oﬀ function we can consider a C k -smooth extension f˜ of f to C2 such that f˜(x) = f (x) on Bδ , f˜(x) = df0 · x on the complement of B2δ in C2 , and kf˜ − df0 kC 1 < ǫ for some small ǫ, which depends on the constants of partial hyperbolicity µ and λ from (3) c and (4). We can construct a local center manifold Wloc (0) for f satisfying the three properties above as the restriction to Bδ of the global center manifold W c (0) for f˜ (see [19], [25]). The proof of the existence of the center manifold for the modified function follows the usual contracting argument on the space of graphs of Lipschitz maps h : E c → E s . The fact that the strong contraction along E s (and in vertical cones Cxv ) dominates the behavior of df along E c (and in horizontal cones Cxh ) ensures that the action of f˜ on the space of graphs is a contraction, hence it has a unique fixed point, which is the center manifold W c (0). This is globally defined and homeomorphic to R2 , but clearly non-unique for the initial function f as it depends on the choice of the extension f˜. To obtain the C k -smoothness of the center manifold, it suﬃces to assume that the constants µ and λ satisfy µλ−j < 1 for 1 ≤ j ≤ k on Bδ , condition which is true if δ is small, since |µ| < 1 and |λ| = 1. We will only use center manifolds of class C 1 , so the condition 0 < µ < λ < 1 in the definition of partial hyperbolicity suﬃces for our purposes. We can take the neighborhood B ′ from Theorem A to be a ball Bδ as above.

3. Semi-parabolic germs In this section we discuss the local structure of a germ f of holomorphic diﬀeomorphism of (C2 , 0) with a semi-parabolic fixed point at the origin and we show the existence of big invariant petals. For simplicity, we call f a semi-parabolic germ. Throughout the section, denote the eigenvalues of df0 by λ = e2πip/q and µ, where |µ| < 1 and p/q is a rational number with gcd(p, q) = 1. The following result is Proposition 3.3 from [18]. Proposition 3.1. — Let f be a semi-parabolic germ of transformation of (C2 , 0), with eigenvalues λ and µ, with λ = e2πip/q and |µ| < 1. There exists a neighborhood U of 0 and local coordinates (x, y) on U in which f has the form f (x, y) = (x1 , y1 ), with ( x1 = λ(x + xνq+1 + Cx2νq+1 + a2νq+2 (y)x2νq+2 + · · · ) (6) y1 = µy + xh(x, y) where C is a constant and aj (·) and h(·, ·) are germs of holomorphic maps from (C, 0) to C, respectively from (C2 , 0) to C, with h(0, 0) = 0. The multiplicity of the fixed point as a solution of the equation f q (x, y) = (x, y) is νq + 1. We call ν the semi-parabolic multiplicity of the fixed point.

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Let f be a germ defined on a domain U ∋ 0 as in Proposition 3.1. A point z ∈ U is uniformly convergent for f if there is a neighborhood W ⊂ U of z such that the sequence of forward iterates (f n |W )n≥1 converges uniformly to the origin. An open subset D of U is called a local base of uniform convergence for f if f (D) ⊂ D, every point in D is uniformly convergent to the origin, and for every uniformly convergent point z ∈ U there exists a positive integer n such that f n (z) ∈ D. In this section, we will only deal with semi-parabolic multiplicity ν = 1. Otherwise we would have ν cycles of q petals, invariant under f q . Using the results of Ueda [23, 24] and Hakim [6], we can describe the local dynamics of semi-parabolic germs as follows: Theorem 3.2. — Let f be a semi-parabolic germ of transformation of (C2 , 0), with eigenvalues λ and µ, with λ = e2πip/q and |µ| < 1. Assume that the semi-parabolic multiplicity is 1. Let U be the normalizing neighborhood from Proposition 3.1. Inside U , there exist q attracting petals Patt,j and q repelling petals Prep,j for 1 ≤ j ≤ q. The attracting petals are two-dimensional and their union is a local base of uniform convergence for f . The repelling petals are one-dimensional and their union is a local base of uniform convergence for f −1 . There are several ways to define local attractive and repelling petals. We will define fat attractive petals as in [18, Section 4]. Consider the sets (7)

2

2

2 ∆± r = {x ∈ C : (Re(x) ± r) + (|Im(x)| − r) < 2r }.

and the coordinate transformation T√: (x, y) 7→ (xq , y). Geometrically, ∆+ r is the union of two complex disks of radius 2r centered at −r ± ir. Let Patt,j , 1 ≤ j ≤ q, ′ ′ be the preimages under T of the set {x ∈ ∆+ r , |y| < r }, for some r, r > 0 suﬃciently small. A construction of repelling petals is done by Ueda in [24] for λ = 1. Ueda shows that the local repelling petal is a smooth graph {y = ψ(x), |x−r| < r}, for some r > 0 suﬃciently small. However, using the same techniques from [24] we can construct a larger local repelling petal which is a smooth graph {y = ψ(x), x ∈ ∆− r } for some r > 0 suﬃciently small. Similarly, when λ = e2πip/q , there are q fat repelling petals Prep,j , 1 ≤ j ≤ q, which can be defined as the preimages under T of a smooth graph {y = ψ(x), x ∈ ∆− r }, for r > 0 suﬃciently small. c Let Wloc (0) be any local center manifold of 0. We remark that, by the weak uniquec ness property, for r small enough, the local repelling petals are contained in Wloc (0). Let Patt and Prep denote the union of the q attractive, and respectively of the q repelling petals. We have f (Patt ) ⊂ Patt ∪ W ss (0) and all points in Patt are attracted to the semi-parabolic fixed point at the origin in forward time. Similarly f −1 (Prep ) ⊂ Prep ∪ {0} and all points in Prep are attracted to the origin in backward time. Slicing an attracting or repelling petal by a center manifold yields an open set in the center manifold. The repelling and attracting petals, sliced by a center manifold,

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alternate as we turn around the origin. By construction, the intersection of an attracting petal with the two adjacent repelling petals consists of two open disjoint disks. More precisely, let Dj1 and Dj2 be the two connected components of Patt,j ∩ Prep . Then [ k (8) Pinv,j = f nq (Djk ), for k = 1, 2 and 1 ≤ j ≤ q n∈Z

are 2q completely invariant local petals. They are contained in the union of Prep and We view these invariant petals as small, because they are a priori defined only on the neighborhood U where the map f is conjugate to the normal form (6). In Section 4 we prove the existence of hedgehogs for germs of holomorphic diffeomorphisms of (C2 , 0) with a semi-neutral fixed point at the origin. Following the original strategy of Pérez-Marco, we construct the hedgehog as a Hausdorﬀ limit of completely invariant petals for approximating germs with a semi-parabolic fixed point at 0. However, the normalizing domains on which the semi-parabolic germs can be conjugate to their corresponding normal forms given in Proposition 3.1 might shrink to 0 as the sequence converges to a semi-Cremer germ, because the semi-Cremer germ is non-linearizable. Therefore we first need to construct big invariant petals before applying the construction in Section 4. Let B ⋐ B ′ be as in Theorem A such that f is partially hyperbolic on B ′ . There exist a horizontal cone field C h which is forward invariant and a vertical cone field C v which is backward invariant on B ′ , as in Equation (2). We say that a C 1 curve γ is vertical-like/horizontal-like if for any point y on γ, the tangent space to γ at y is ss contained in the vertical/horizontal cone at y. Let Wloc (0) be the local stable manifold of the semi-indiﬀerent fixed point, i.e., the connected component of W ss (0) ∩ B which ss contains 0. Partial hyperbolicity implies that Wloc (0) is vertical-like. We show that the invariant petals are big with respect to the ball B (a natural way to express this is to ask that they touch the boundary of B). In dimension one, PérezMarco achieves this using the Uniformization Theorem and the theory of analytic circle diﬀeomorphisms, tools which are not readily available in the two-dimensional setting. The main obstruction is that, while each local invariant petal is contained in a holomorphic curve (the asymptotic curve Σ introduced below), the union of the 2q petals belongs to a (non-unique) center manifold of class C k for some k ≥ 1, which is not complex or real analytic (see e.g., [22]). Instead of complex methods we will use some topological tools: Brouwer’s Plane Translation Theorem 3.7 and covering space theory. Using Theorem 3.2, we define the asymptotic curve(s) Σ to be the set of points in the domain of f , diﬀerent from 0, which are attracted to 0 under backward iterations of f . The set Σ has q connected components which contain 0 in the boundary, and each of them is an f q -invariant Riemann surface immersed in C2 . If f is a global diﬀeomorphism of C2 , Ueda [24] showed that these are biholomorphic to C. Patt .

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Proposition 3.3. — The set ΣB = {x ∈ Σ : f −n (x) ∈ B for all n ≥ 0} is horizontallike. Proof. — Let x ∈ ΣB . There exists m > 0 such that y = f −m (x) ∈ Prep . The repelling petal Prep is horizontal-like from [24] and the construction above. Therefore any tangent vector to Σ at y belongs to the horizontal cone at y, and all forward iterates f i (y), 0 ≤ i ≤ m, remain in B, so Tx Σ is contained in the horizontal cone at x. Let P be the union of the 2q connected components of the set (9)

{x ∈ B \ {0} : f n (x) ∈ B ∀n ∈ Z and f n (x) → 0 as n → ±∞}

which contain 0 in their boundaries. We refer to P as the set of maximal invariant petals relative to the ball B. By definition P ⊂ ΣB , hence it is horizontal-like by Proposition 3.3. Each component of P contains a local invariant petal as defined by Equation (8) and Theorem 3.2 and is invariant by f q (see Figure 1).

Figure 1. Maximal completely invariant petals for q = 3, relative to the ball B. Some petals touch the boundary of B.

Let P and ∂ P denote the closure, respectively the boundary of the set P in C2 . In the following two propositions we collect a couple of elementary results about P . Proposition 3.4. — The set P is open rel Σ and its connected components are simply connected. Proof. — Let us first notice that the only point of intersection of the vertical-like local ss (0) and the horizontal-like set P is 0, by transversality. If strong stable manifold Wloc ss ss x ∈ W (0) ∩ P then there exists a positive integer m such that f m (x) ∈ Wloc (0) ∩ P , which implies that x is the fixed point 0, which does not belong to P . Therefore W ss (0) ∩ P = ∅.

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We now show that P is open relative to Σ. Let Bpar (0) =

[

201

f −n (Patt ) be the

n≥0

basin of the semi-parabolic fixed point 0. If x ∈ P then x ∈ Bpar (0), which is open in C2 . Moreover, since Patt and Prep are bases of convergence for f on Bpar (0) and respectively for f −1 on Σ, there exists a first iterate n such that f n (x) ∈ Patt and a first iterate m such that f −m (x) ∈ Prep . There exists a neighborhood U ⊂ Σ of x \ such that f n (U ) ⊂ Patt , f −m (U ) ⊂ Prep and f −i (U ) ⊂ B. Hence U is an −m 0, h,α

Cx

= {v ∈ Tx B ′ , ∠(v, Ex1 ) ≤ α},

where the angle of a vector v and a subspace E is simply the angle between v and its projection prE v on the subspace E. The vertical cone Cxv,α is defined in the same way, with respect to Ex2 . We will suppress the angles α from the notation of the cones, whenever there is no danger of confusion. The map f is partially hyperbolic on B ′ if there exist two real numbers µ1 and λ1 such that 0 < |µ| < µ1 < λ1 < 1 and a family of invariant cone fields C h/v (3)

dfx (Cxh ) ⊂ Int Cfh(x) ∪ {0},

v v dff−1 (x) (Cf (x) ) ⊂ Int Cx ∪ {0},

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such that for some Riemannian metric we have strong contraction in the vertical cones, whereas in the horizontal cones we may have contraction or expansion, but with smaller rates: (4)

λ1 kvk ≤ kdfx (v)k ≤ λ−1 1 kvk, kdfx (v)k ≤ µ1 kvk,

for v ∈ Cxh

for v ∈ Cxv .

Let B be a domain in C2 containing the origin such that B ⊂ B ′ and f (B) ⊂ B ′ . Remark 3.1. — Since B is compactly contained in B ′ , the angle between dfx (Cxh ) and ∂ Cfh(x) is uniformly bounded independently of x. This implies that there exists 0 < ρ < 1 such that for every x ∈ B, the angle opening of the cone dfx (Cxh ) is ρα, a fraction of the angle opening of the cone Cfh(x) . c c The semi-indiﬀerent fixed point has a local center manifold Wloc := Wloc (0) of 1 c class C , tangent at 0 to the eigenspace E0 corresponding to the neutral eigenvalue λ. c Throughout the paper, Wloc will denote the local center manifold of 0. The local center manifold is the graph of a C 1 function ϕf : E0c ∩ B ′ → E0s and has the following properties: c c a) Local invariance f (Wloc ) ∩ B ′ ⊂ Wloc . −n ′ c b) Weak uniqueness: If f (x) ∈ B for all n ∈ N, then x ∈ Wloc . n c) Shadowing: Given any point x such that f (x) → 0 as n → ∞, there exists c a positive constant k and a point y ∈ Wloc such that kf n (x) − f n (y)k < kµn1 as n → ∞. In other words, every orbit which converges to the origin can be described as an exponentially small perturbation of some orbit on the center manifold. c We can choose a neighborhood B ′ of the origin so that Wloc is horizontal-like: the c h c tangent space Tx Wloc is a subset of the horizontal cone Cx , for any point x ∈ Wloc ∩B ′ . The center manifold is generally not unique. However, the formal Taylor expansion at the origin is the same for all center manifolds. The center manifold is unique in some cases, for instance when f is complex linearizable at the origin. For uniqueness, existence, and regularity properties of center manifolds, we refer the reader to [14], [31], [30], and [36]. It is also worth mentioning the following reduction principle for center manifolds: the map f is locally topologically semi-conjugate to a function on the center manifold given by u 7→ λu + f1 (u, ϕf (u)). In this article, we will not make use of the reduction principle, as it only gives a topological semi-conjugacy to a model map which is as regular as the center manifold, hence not analytic. The assumption of partial hyperbolicity implies that any point x with f n (x) ∈ B ss (x), given by for all n ≥ 0 has a well defined local strong stable manifold Wloc ss Wloc (x) = {y ∈ B : dist(f n (x), f n (y))/µn1 → 0 as n → ∞}.

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ss c ss c Moreover Wloc (x) intersects Wloc transversely. Let y ∈ Wloc (x) ∩ Wloc . Then the orbit of y shadows the orbit of x. We can therefore formulate a more general shadowing property:

Proposition 3.2. — Let x ∈ B such that f n (x) ∈ B for every n ≥ 1. There exists c k > 0 and y ∈ Wloc such that kf n (x) − f n (y)k < kµn1 as n → ∞. An old question posed by Dulac and Fatou is whether there exists orbits converging to an irrationally indiﬀerent fixed point of a holomorphic map. Using the dynamics on the hedgehogs, renormalization theory and Yoccoz estimates for analytic circle diﬀeomorphisms, Pérez-Marco showed that the answer to this question is negative, see Theorem 2.4. It is a natural question to ask whether there exist orbits converging to a semi-Cremer fixed point of a holomorphic germ of (C2 , 0). Also, Pérez-Marco has constructed examples of hedgehogs in which the origin is accumulated by periodic orbits of high periods, see Theorem 2.5. In the two-dimensional setting, the problem of the existence of periodic orbits of f accumulating on the origin or the existence of orbits converging to zero can naturally be reduced to posing the same question for the restriction of the map f to the local center manifold(s). Let us explain this reduction further. If the semi-indiﬀerent fixed point is accumulated by periodic orbits of high period, then these periodic points necessarily live in the intersection of all center manc ifolds Wloc (0), by the weak uniqueness property of local center manifolds. Since we work with dissipative maps, there will always be points that converge to 0 under forward iterations, corresponding to the strong stable manifold of 0. If there exists some other point x, whose forward orbit converges to 0, then the orbit of x must be shadowed by the orbit of a point y that converges to 0 on the center manifold. Therefore, in order to answer the questions about the dynamics of the two-dimensional germ around 0, we should first study the dynamics of the function restricted to the center manifolds. The main obstruction for extending the results of Pérez-Marco directly to our setting is the fact that the center manifolds are not analytic. It is well-known (see e.g., [11], [32]) that there exist C ∞ -smooth germs which do not have any C ∞ -smooth center manifolds. For every finite k one can find a neighborhood Bk of the origin for which there exists a C k -smooth center manifold relative to Bk , however the sets Bk shrink to 0 as k → ∞. A first useful observation in this context is that some analytic structure still exists in some parts of the center manifold. Let (5)

Λ = {z ∈ B : f −n (z) ∈ B, for all n ≥ 0}

be the set of points that never leave B under backward iterations. As a consequence of Theorem 1.1, we know that the set Λ is not trivial, i.e., Λ 6= {0}. By the weak c uniqueness property of center manifolds, Λ is a subset of Wloc . Also, f −1 (Λ) ⊂ Λ, by definition.

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c Proposition 3.3. — Let Wloc be any local center manifold relative to B ′ . The tangent c space Tx Wloc at any point x ∈ Λ is a complex line Exc of Tx C2 . The line field over Λ is df -invariant, in the sense that dfx (Exc ) = Efc (x) for every point x ∈ Λ with f (x) ∈ Λ.

Proof. — Let x ∈ Λ. All iterates f −n (x), n ≥ 0 remain in the domain B, where we have an invariant family of horizontal cones C h preserved by df . The derivative acts on tangent vectors as a vertical contraction by a factor µ1 , close to µ. Let \ Exc = dffn−n (x) Cfh−n (x) . n≥0

This is a decreasing intersection of nontrivial compact subsets in the projective space, hence Exc is a non-trivial complex subspace of Tx C2 . Thus Exc is a complex line included in Cxh . The invariance of the line bundle (Exc )x∈Λ under df follows from the definition. c Any local center manifold Wloc defined relative to B ′ contains the set Λ and the c tangent space Tx Wloc at any point x ∈ Λ is equal to Exc , thus it is a complex line in the tangent bundle T C2 . Proposition 3.3 means in particular that for every point x ∈ Λ, the tangent c space Tx (Wloc ) is J-invariant, where J is the standard almost complex structure obtained from the usual identification of R4 with C2 . Recall that an almost complex structure on a smooth even dimensional manifold M is a complex structure on its tangent bundle T M , or equivalently a smooth R-linear bundle map J : T M → T M with J ◦ J = −Id. c The center manifold Wloc is a real 2-dimensional submanifold of C2 . The standard Hermitian metric of the complex manifold C2 defines a Riemannian metric on the underlying smooth manifold R4 , which restricts to a Riemannian metric on the center c manifold Wloc . Recall that in Cn , the standard Hermitian inner product decomposes into its real and imaginary parts: hu, viH = hu, vi − iw(u, v), where hu, vi is the Euclidean scalar product and w(u, v) is the standard symplectic form of R2n . From now on, whenever we refer to the Riemannian metric we understand the metric defined by the Euclidean scalar product. Every Riemannian metric on an oriented 2-dimensional manifold induces an almost complex structure given by the rotation by 90◦ , i.e., by defining c c Jx′ : Tx Wloc → Tx Wloc , as Jx′ (v) = v ⊥ ,

where v ⊥ is the unique vector orthogonal to v, of norm equal to kvk, such that the choice is orientation preserving. Every almost complex structure on a 2-dimensional manifold is integrable, that is, it arises from an underlying complex structure. Namely, c there exists a (J ′ , i)-holomorphic parametrization function φ : ∆ → Wloc where ∆ is an open subset in C and i is the standard complex structure in C given by multiplication by the complex number i. By (J ′ , i)-holomorphic map, we understand a c C 1 -smooth map with the property that its derivative dφz : Tz ∆ → Tφ(z) Wloc is com′ plex linear, that is dφz ◦iz = Jφ(z) ◦dφz . An introduction to almost complex structures

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and J-holomorphic curves can be found in [37] and [18]. Note that the parametrizing map φ that we have constructed is only (J ′ , i)-holomorphic, but not necessarily c (J, i)-holomorphic, as Wloc is not in general an embedded complex submanifold of C2 . c Note also that the almost complex structure induced by J ′ on Wloc agrees with the 2 standard almost complex structure J from C on the set Λ, so Jx = Jx′ for all x ∈ Λ. c c Let W := B ∩ Wloc and U := φ−1 (W ) ⊂ ∆. The set W ′ = f (W ) belongs to Wloc , ′ by the local invariance of the center manifold. The map f : W → W is an orientationpreserving C 1 diﬀeomorphism. Let g = φ ◦ f ◦ φ−1 : U → U ′ = g(U ) be the orientation-preserving C 1 -diﬀeomorphism induced by f on U : f

W −−−−→ x φ g

W′ x φ

U −−−−→ U ′ . −1 Denote by X := φ (Λ), or equivalently (6)

X = {z ∈ U : g −n (z) ∈ U, for all n ≥ 0},

the set of points that stay in U under all backward iterations by g. The map f is holomorphic on C2 , so it is (J, J)-holomorphic on Λ, which means that ∂¯J f = 0 for ξ ∈ Λ, where 1 (7) ∂¯J f := (dfξ + Jf (ξ) ◦ dfξ ◦ Jξ ). 2 The conjugacy function φ is (J, i)-holomorphic on Λ. In the holomorphic coordinates provided by φ, this means that g is (i, i)-holomorphic on X, or equivalently ∂¯i g = 0, where 1 (8) ∂¯i g := (dgz + ig(z) ◦ dgz ◦ iz ), 2 and z = φ−1 (ξ). It is easy to check that with the standard identifications z = x + iy, g(x, y) = g1 (x, y) + ig2 (x, y), and (9)

iz (∂x ) = ∂y ,

iz (∂y ) = −∂x ,

the relation ∂¯i g = 0 is equivalent to the familiar Cauchy-Riemann equations ∂x g1 − ∂y g2 = 0 and ∂x g2 + ∂y g1 = 0. As usual, consider the linear partial diﬀerential operators of first order 1 1 (10) ∂ = ∂z = (∂x − i∂y ) and ∂¯ = ∂z¯ = (∂x + i∂y ) . 2 2 We have just shown the following proposition: ¯ = 0 on the set X, defined in Equation (6). Proposition 3.4. — ∂g c Let intc (Λ) denote the interior of Λ rel Wloc . Propositions 3.3 and 3.4 immediately imply the following corollary:

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Corollary 3.4.1. — The set intc (Λ) is a complex submanifold of C2 . The conjugacy map φ : int(X) ⊂ C → intc (Λ) ⊂ C2 is holomorphic, and the function g is holomorphic on int(X). c c When x ∈ Wloc \ Λ, the tangent space Tx Wloc is only a real 2-dimensional subspace 2 h of Tx C , included in the horizontal cone Cx . We want to measure how far it is from a complex line. The angle between two real subspaces V1 and V2 of Ty C2 of the same dimension can be defined as Angle(V1 , V2 ) = max min ∠(u1 , u2 ). u1 ∈V1 u2 ∈V2

For each n ≥ 0, let Wn be the set of points from W that stay in W under the first n backward iterates of f . Let Un = φ−1 (Wn ). c Proposition 3.5. — Let x ∈ Wn and v ∈ Tx Wloc . There exists ρ < 1 such that c Angle (Tx Wloc , SpanC {v}) = O (ρn ). c Proof. — Let y = f −n (x). Let w = dfx−n (v) ∈ Ty Wloc . The vector Jy w does not in c general belong to Ty Wloc , but it does belong to the horizontal cone Cyh . The derivative dfyn maps this cone into a smaller cone inside Cxh , with an angle opening O (ρn ) where ρ < 1 as in Remark 3.1. The vectors v and

Jx v = Jf n (y) dfyn (w) = dfyn (Jy w) both belong to the cone dfyn (Cyh ). Note that SpanC (v) = SpanR (v, Jx v). In conclusion the angle between the complex line spanned by v and Jx v and the real tangent c space Tx Wloc is O (ρn ). Define the norm of the ∂¯J ′ -derivative of f on a set W as k∂¯J ′ f kW = sup k(∂¯J ′ f )z k, z∈W

where k(∂¯J ′ f )z k is the operator norm of (∂¯J ′ f )z : Tz W → Tf (z) W . Lemma 3.6. — There exists a constant C such that for every n ≥ 1, k∂¯J ′ f kW < Cρn . n

c c Proof. — Let x ∈ Wn . Then f (x) ∈ Wloc . Let v ∈ Tx Wloc . We will use the fact that ′ the map f is analytic on B , so Jf (x) ◦ dfx = dfx ◦ Jx , to estimate (11) 2k(∂¯J ′ f )x (v)k = kJ ′ ◦ dfx v − dfx ◦ J ′ vk f (x)

x

≤ kJf′ (x) ◦ dfx v − Jf (x) ◦ dfx vk + kdfx ◦ Jx v − dfx ◦ Jx′ vk. Let β = ∠(Jx v, Jx′ v). Since Jx = Jx′ on Λ, we may assume that β < π/2 for x ∈ Wn . c c Note then that β is also equal to the Angle(Jx v, Tx Wloc ), as Tx Wloc = SpanR {v, Jx′ v}, ′ and hv, Jx vi = hv, Jx vi = 0. By a direct computation we get (12)

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kJx′ v − Jx vk = 2kvk sin(β/2) ≤ βkvk.

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Let M = sup kdfx k, where the supremum is taken after x ∈ f (B). Clearly M < ∞ since f is C 1 . In view of Equation (12), we get (13)

kdfx (Jx v − Jx′ v)k ≤ M kJx v − Jx′ vk ≤ βM kvk.

By the same estimate (12), we have kJf′ (x) w − Jf (x) wk ≤ β ′ kwk, where w = dfx v c and β ′ is the angle between the vector Jf (x) w and the tangent space Tf (x) Wloc . The −1 h vector v is in the horizontal cone Cx , so kwk ≤ λ1 kvk, by partial hyperbolicity (4). Putting everything together, Equation (11) becomes k(∂¯J ′ f )x (v)k ≤

1 −1 ′ (λ β + M β)kvk. 2 1

From Proposition 3.5 we know that β = O (ρn ) and β ′ = O (ρn+1 ), for some ρ < 1. Thus there exists a constant C, independent of n, such that k(∂¯J ′ f )x k < Cρn , for every x ∈ Wn . We will now transport the estimates obtained for f in Lemma 3.6 to estimates for ¯ the ∂-derivative of g. Since g = φ−1 ◦ f ◦ φ and φ is (J ′ , i)-holomorphic, we get the ¯ following relation between the ∂-derivatives of f and g computed with respect to the corresponding almost complex structures J ′ and respectively i: (14)

¯ ′ (∂¯i g)z = dφ−1 φ(g(z)) ◦ (∂J f )φ(z) ◦ dφz , for all z ∈ U.

Using Lemma 3.6 and the fact that dφ and dφ−1 are bounded above, we obtain that there exists a constant C ′ such that for every n ≥ 1, k∂¯i gkUn < C ′ ρn .

(15)

Corollary 3.6.1. — There exists a constant C ′ such that for every n ≥ 1, ¯ |∂g(z)| < C ′ ρn , for all z ∈ Un . Proof. — The proof follows directly from (15). For completion, we give the details below. Let z = x + iy and g(x, y) = s(x, y) + it(x, y). We have dgz (∂x ) =

∂s ∂ ∂t ∂ + ∂x ∂s ∂x ∂t

and

dgz (∂y ) =

∂s ∂ ∂t ∂ + . ∂y ∂s ∂y ∂t

The standard complex structure on C satisfies the relations in Equation (9), so we compute ∂s ∂t ∂t ∂ ∂s ∂ dgz ◦ iz − ig(z) ◦ dgz (∂x ) = + + − , ∂y ∂x ∂s ∂y ∂x ∂t ∂s ∂s ∂ ∂t ∂ ∂t − + + . dgz ◦ iz − ig(z) ◦ dgz (∂y ) = − ∂y ∂x ∂s ∂y ∂x ∂t ¯ Using Equation (10), the complex ∂-derivative of g is 1 ∂s ∂t ∂s ∂t ¯ ∂g = . − +i + 2 ∂x ∂y ∂x ∂y

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By inequality (15), k(∂¯i g)z (∂x )k and k(∂¯i g)z (∂y )k are bounded above by C ′ ρn for ¯ z ∈ Un , which implies that |∂g(z)| ≤ C ′ ρn for all z in Un .

4. Quasiconformal conjugacy 4.1. Preliminaries. — In this section we give a brief account of quasiconformal homeomorphisms. We refer to the classical text of Ahlfors [1] for a thorough treatment of quasiconformal maps. Let U, V be two open sets of C and ψ : U → V be a homeomor1,2 phism such that ψ belongs to the Sobolev space Wloc (U ) (that is, ψ has distributional first order derivatives which are locally square-integrable). Let K ≥ 1. The map ψ is called K-quasiconformal if ¯ ≤ K − 1 |∂ψ| |∂ψ| K +1 almost everywhere. Quasiconformal homeomorphisms are almost everywhere diﬀerentiable. If ψ is quasiconformal, then ∂ψ 6= 0 and Jac(ψ) > 0 almost everywhere. The complex dilatation of ψ at z (or the Beltrami coeﬃcient of ψ) is defined as µψ (z) =

¯ ∂ψ(z) . ∂ψ(z)

We have kµψ k∞ < 1, where k · k∞ is the essential supremum. The number K(ψ, z) =

1 + |µψ (z)| 1 − |µψ (z)|

is the conformal distortion of g at z (or the dilatation of g at z). Clearly kµk∞ < 1 is equivalent to K(ψ, z) < ∞ almost everywhere. ¯ =0 By Weyl’s Lemma, if ψ is 1-quasiconformal, then it is conformal (i.e., if ∂ψ a.e. then ψ is conformal). The composition of a K1 -quasiconformal homeomorphism with a K2 -quasiconformal homeomorphism is K1 K2 -quasi-conformal. The inverse of a K-quasiconformal homeomorphism is also K-quasiconformal. Using the chain rule for complex dilatations, we note that if µψ = µϕ almost everywhere, then the composition ψ ◦ ϕ−1 is conformal. Each measurable function µ : V → C with kµk∞ < 1, viewed as a coeﬃcient of a measurable Riemannian metric a(z)|dz+µ(z)d¯ z |2 , is called a Beltrami coeﬃcient in V . The pullback of such a metric by a quasiconformal map ψ is a metric corresponding to a Beltrami coeﬃcient ψ ∗ µ called the pullback of µ. 4.2. Quasiconformal conjugacy to an analytic map. — Let U, U ′ be open sets of C and g : U → U ′ be the orientation-preserving C 1 diﬀeomorphism on a neighborhood of U with g(0) = 0, constructed in Section 3. In this section, we will show how to change the complex structure on U , in order to make the function g analytic.

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For n ≥ 0, consider the sets (16)

Un =

n \

g k (U ) and

U−n =

n \

g −k (U ).

k=0

k=0

The set Un is the of points from U such that the first n backward iterates remain in U . The set U−n consists of the points of U whose first n forward iterates belong to U . Note that U0 is equal to U . Moreover Un+1 ⊆ Un and U−(n+1) ⊆ U−n , for all n ≥ 0. The set X from Equation (6) is equal to U∞ and g −1 (X) ⊆ X. Note that the sets Un and X have already been introduced in Section 3, as the preimages of the c sets Wn and Λ from the center manifold Wloc under the parametrizing map φ. From the definitions given in Equation (16) we have the following invariance properties. Lemma 4.1. — For every n ≥ 0, we have: a) g −n (Un ) = U−n b) g −1 (Un+1 ) ⊂ Un and g(U−(n+1) ) ⊂ U−n c) g j (U−n ) = Uj ∩ U−(n−j) , for 0 ≤ j ≤ n. Proof. — The set equalities g j (U−n ) =

n \

g j−k (U ) =

j \

k=0

k=0

g k (U ) ∩

n−j \

g −k (U ) = Uj ∩ U−(n−j)

k=0

can be used to prove part c). Taking j = n yields part a). The first inclusion in part b) follows from the fact that g(Un ) ∩ U = Un+1 , while the second relation is obtained from part c) by taking j = 1. Let σ0 denote the standard almost complex structure of the plane, represented by the zero Beltrami diﬀerential on U . The following lemma is a restatement of Corollary 3.6.1 in the language of Beltrami coeﬃcients. Lemma 4.2. — There exist ρ < 1 and M independent of n such that sup |µg (z)| < M ρn , for all n ≥ 0. z∈Un

For each positive integer n, let µgn : U−n → C be the restriction of the pullback (g n )∗ σ0 to the set U−n . This is the Beltrami coeﬃcient of the n-th forward iterate g n on the set U−n . Lemma 4.3. — There exists a constant κ < 1, such that for all integers n ≥ 0 we have |µgn (z)| < κ, for all z ∈ U−n . Proof. — Let n > 0 and z ∈ U−n . Let zj = g j (z) for 0 ≤ j ≤ n denote the j-th iterate of z under the map g. By Lemma 4.1 c), zj ∈ Uj for 0 ≤ j ≤ n.

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We want to show that the dilatation K(g n , z) is bounded by a constant independent of n and the choice of z. Recursively using the classical relation (see [1]) K(f ◦ g, z) ≤ K(f, g(z)) K(g, z), we get the following estimate K(g n , z) ≤

(17)

n−1 Y

K(g, zj ).

j=0

By definition and Lemma 4.2 we have K(g, zj ) =

1 + |µ(zj )| 1 + M ρj ≤ . 1 − |µ(zj )| 1 − M ρj

Since ρ < 1, we can choose j0 such that M ρj < 1/3, for j ≥ j0 . Then K(g, zj ) is bounded above by 1 + 3M ρj , for j ≥ j0 . The product of the first j0 terms in Equation (17) is bounded by a constant C. This is an immediate consequence of the fact that g is injective and orientation-preserving on a neighborhood of U , thus kµg k∞ is bounded away from 1. For n > j0 we get K(g n , z) ≤ C

n−1 Y

(1 + 3M ρj ).

j=j0

The infinite product (1+3M ρj ) is convergent. There exists a constant M ′ such that K(g n , z) < M ′ for all points z ∈ U−n and all n ≥ 0. Thus |µgn (z)| < (1−M ′ )/(1+M ′ ) on U−n . Q

For n > 0, let σn : Un → C be the restriction of the pullback (g −n )∗ σ0 to the set Un . This is the Beltrami coeﬃcient of the n-th backward iterate g −n on the set Un . We write σn = µg−n for simplicity. The map g n : U−n → Un is bijective. In fact, the measurable function σn is the push-forward of σ0 under g n , also written as σn = (g n )∗ σ0 . From the standard properties of Beltrami coeﬃcients we have 2 ∂g(z) µg−1 (z) = − µg (g −1 (z)), |∂g(z)| so |µg−1 (z)| = |µg (g −1 (z))|. It follows that |µg−n (z)| = |µgn (g −n (z))|, for all z ∈ Un . By Lemma 4.3 we get kσn k∞ < κ, for all n > 0. By a result of Sullivan [33], a uniformly quasiconformal group is conjugate to a group of conformal transformations. We will give a direct proof of this property in our situation: Theorem 4.4. — The map g −1 : U1 → U−1 is quasiconformally conjugate to a holomorphic map.

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Proof. — Consider the measurable function µ : U → C, given by ( σn on Un \ Un+1 , for n ≥ 0 (18) µ= σ0 on X. Then kµk∞ < 1 by Lemma 4.3 and the observation above. Thus µ is a Beltrami coeﬃcient in U . Moreover, by construction, µ is g −1 invariant, i.e., (g −1 )∗ µ = µ on U1 . The standard almost complex structure is g −1 invariant on X, since µg = 0 on X by Lemma 4.2, which implies that µg−1 = 0 on X, by Definition (18) and the fact that g −1 (X) ⊂ X. The Beltrami coeﬃcient µ is g −1 invariant on U1 \ X by construction. To see this, let n > 0 and pick any point z ∈ Un \ Un+1 . Then µ(z) = σn (z) on Un \ Un+1 , µ(z) = σn−1 (z) on Un−1 \ Un , and g −1 (Un \ Un+1 ) ⊂ Un−1 \ Un by Lemma 4.1 b). We have the following sequence of equalities (g −1 )∗ µ(z) = (g −1 )∗ σn−1 (z) = (g −n )∗ σ0 (z) = σn (z) = µ(z), which shows that (g −1 )∗ µ = µ on U1 \ X as well. By the Measurable Riemann Mapping Theorem, there exists a quasiconformal homeomorphism ψ : U → ψ(U ) ⊂ C with complex dilatation µψ equal to the Beltrami coeﬃcient µ. We choose ψ such that ψ(0) = 0. Let Ω = ψ(U−1 ) and Ω′ = ψ(U1 ) and consider the map h : Ω → Ω′ , h = ψ ◦ g ◦ ψ −1 , as in the diagram below g −1

(U1 , (g −1 )∗ µ) −−−−→ (U−1 , µ) ψ ψy y Ω′

h−1

−−−−→

Ω.

The map ψ ◦ g is a composition of two quasiconformal maps, hence quasiconformal. It has complex dilatation µψ◦g−1 = (g −1 )∗ µ = µ. Since µψ◦g−1 = µψ , the maps h and h−1 = ψ ◦ g −1 ◦ ψ −1 are conformal. The conjugacy map ψ is analytic on the interior of X, since the Beltrami coeﬃcient µ on X is equal to the standard almost complex structure σ0 . −1

4.3. Proof of Theorem A. — Let f be a germ of a holomorphic diﬀeomorphism of (C2 , 0) with a semi-indiﬀerent fixed point at the origin. Let λ = e2πiθ be the neutral eigenvalue of 0. The restriction of the map f to the center manifold, f : W → W ′ is conjugate to the map g : U → U ′ by a C 1 -diﬀeomorphism φ. By Theorem 4.4, and eventually replacing U with U−1 , the map g : U → U ′ is conjugate to a holomorphic map h : (Ω, 0) → (Ω′ , 0) by a quasiconformal homeomorphism ψ. Let Λ be defined as in Equation (5). By Proposition 3.3 , the interior of the set Λ is a (not necessarily connected) complex submanifold of C2 . To show that the conjugacy map ψ ◦ φ−1 is analytic on the interior of Λ, we first recall from Corollary 3.4.1 that φ : int(X) → intc (Λ) is holomorphic, where X = φ−1 (Λ). From the proof of

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Theorem 4.4 it follows that map ψ is analytic on the interior of the set X. So the composition ψ ◦ φ−1 is holomorphic on intc (Λ). The last thing we need to show is that h′ (0) = λ. The map dφ−1 maps the complex eigenspace E0c of the eigenvalue λ to the complex tangent space T0 U . The action of df0 on E0c is multiplication by λ and dg0 = dφ−1 0 ◦ df0 ◦ dφ0 , so dg0 as a real matrix is just a rotation matrix of angle θ. Since the tangent space T0 U is complex, we can view dg0 as a complex function, which means that g ′ (0) = λ. Remark 4.5. — It is well-known that the multiplier of an indiﬀerent fixed point is a quasiconformal invariant (even a topological invariant by a theorem of Naishul [19]). Namely, if g1 and g2 are two quasiconformally conjugate (topologically conjugate) germs of holomorphic maps of (C, 0) with indiﬀerent fixed points at the origin, then g1′ (0) = g2′ (0). We cannot use this fact in our setting to conclude that g ′ (0) = h′ (0), because the map g is not holomorphic. To show that the two rotation numbers coincide, we use a generalization of Naishul’s Theorem in 1D, due to Gambaudo, Le Calvez, and Pécou [10], which says in particular that the multiplier at the origin is a topological invariant for the class of orientationpreserving homeomorphisms of the plane which are diﬀerentiable at the origin and for which the derivative at the origin is a rotation. Alternatively, to show that h′ (0) = λ, we can use the hedgehog constructed in [9]. c compactly contained in W . Let Kf be Let V be a neighborhood of the origin in Wloc a compact, connected, and completely invariant set for f given by Theorem 1.1 such c that 0 ∈ Kf and Kf ∩∂V 6= 0. By construction Kf ⊂ V and Kf is full in Wloc (that is, c Wloc \Kf is connected). As in [9], we can associate to (f, Kf ) an orientation-preserving homeomorphism f˜ of the unit circle, with rotation number θ, by uniformizing the complement of Kf inside the global center manifold W c (0) identified with R2 , by the complement of the unit disk in C. The set Kh = ψ ◦ φ−1 (Kf ) is a nontrivial compact, connected, full, and completely invariant set for h, containing the origin. Moreover, Kh intersects the boundary of V ′ = ψ ◦ φ−1 (V ). By [23, Lemma III.3.4], we can associate ˜ of the unit circle, with rotation to (h, Kh ) an orientation-preserving diﬀeomorphism h ′ ′ number θ , the argument of h (0). The conjugacy map ψ ◦ φ−1 is uniformly continuous on a neighborhood of Kf . Therefore it defines a homeomorphism from the set of prime ˆ \ Kh . We use this to obtain a ends of W c (0) ∪ {∞} \ Kf to the set of prime ends of C ˜ ˜ conjugacy between the homeomorphisms f and h on the unit circle. Hence they have the same rotation number θ = θ′ .

5. Dynamical consequences of Theorem A In this section we give the proofs of the remaining theorems stated in the introduction.

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Theorem B, respectively Corollary B.1 follows from Theorem 2.3, respectively Corollary 2.3.1 by applying Theorem A. We now proceed with the proofs of Theorems C, D, E and Corollary D.1. Proof of Theorem C. — Consider a domain B ′ ⊂ C2 such that B, f (B) are compactly c contained in B ′ and f is partially hyperbolic on B ′ . Let Wloc be a center manifold ′ of 0 constructed with respect to the bigger set B . Let W be the connected compoc nent of Wloc ∩ B ∩ f −1 (B) containing the origin. By Theorem A, the map f | is W quasiconformally conjugate to a holomorphic map h : Ω → Ω′ whose multiplier at the origin is equal to the neutral eigenvalue of f . We can choose B so that Ω ⊂ C is a Jordan domain with C 1 boundary (i.e., an admissible neighborhood in Pérez-Marco’s terminology). Let φ : W → Ω denote the quasiconformal conjugacy. Let H ⊂ B be a hedgehog for f such that H ∩ ∂B 6= ∅. Consider a point x ∈ B, ss which does not belong to Wloc (H ). Suppose that f n (x) ∈ B for n ≥ 1. By the shadc c owing property from Proposition 3.2 there exists y ∈ Wloc \ H such that f n (y) ∈ Wloc for all n ≥ 1 and the orbit of y shadows the orbit of x. Clearly the ω-limit set of x is the same as the ω-limit set of y. The set K = φ(H ) is a hedgehog for h. The point z = φ(y) does not belong to K. However, f n (z) ∈ Ω for all n ≥ 0. By Theorem 2.4, ω(z) ∩ K = ∅. It follows that ω(x) ∩ H = ∅. A similar argument shows that if x ∈ B \ H and f −n (x) ∈ B for all n ≥ 1, then α(x) ∩ H = ∅. Proof of Theorem D. — Suppose that f is conjugate in a neighborhood of the origin to the map f˜(x, y) = (λx, µ(x)y). For some small r, the disk Dr × {0} is invariant under f˜, therefore there exists an embedded holomorphic disk ∆ which is invariant for the dynamics of f . All local center manifolds must contain ∆, therefore 0 ∈ ∆ ⊂ H . Conversely, suppose that 0 ∈ intc (H ), the interior of H relative to a center manifold. Let ∆ be the connected component of intc (H ) which contains 0. By the properties of the hedgehog H from Theorem 1.1, the set ∆ is open, bounded, simply connected, with f (∆) = ∆. By Proposition 3.4, the set ∆ is an analytic submanifold of C2 of dimension 1, hence it is biholomorphic to the unit disk D. Choose a biholomorphism φ : D → ∆ with φ(0) = 0, and set g := φ−1 ◦ f ◦ φ. The map g : D → D is an automorphism of the unit disk, satisfying g(0) = 0 and g ′ (0) = λ, |λ| = 1, hence by the Schwarz Lemma we can conclude that g is a rotation, g(z) = λz for all z ∈ D. Therefore, the restriction of f to ∆ is linearizable. We will show that the map f in conjugate in a neighborhood of the origin in C2 to a linear cocycle (x, y) 7→ (λx, µ(x)y), in a two-step argument. In the first step, we conjugate f to a skew product (x, y) 7→ (λx, ν(x, y)), where ν is a nonlinear cocycle, using ideas from [4, Proposition 6]. In the second step we show how to reduce the nonlinear cocycle to a linear one. Step 1. — We conjugate f to a map of the form (19)

F (x, y) = (λx, ν(x, y)) ,

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where ν(x, y) = µy (B0 (x) + yB1 (x, y)) , for some holomorphic functions B0 (·) and B1 (·, ·) with B0 (x) = 1 + O (x). The parametrizing map φ : D → ∆, φ = (φ1 , φ2 ), is a biholomorphism, hence (φ′1 (x), φ′2 (x)) 6= (0, 0) for all x ∈ D. There exists two holomorphic maps q1 and q2 such that φ′1 (x)q2 (x) − φ′2 (x)q1 (x) = 1 for all x ∈ D. Let s : D × C → C2 be given by s(x, y) = (φ1 (x) + yq1 (x), φ2 (x) + yq2 (x)), and set F := s−1 ◦f ◦s. The maps s and F are local diﬀeomorphisms in a neighborhood of the disk s−1 (∆) = D × {0}, since det ds(x, 0) = 1 for all x ∈ D. Moreover F (D × {0}) = D × {0} and F (x, 0) = (λx, 0) for all x ∈ D. ss Consider the strong stable set Wloc (∆) of ∆ with respect to a small neighbor−1 ss ss hood N of ∆ and let V = s (Wloc (∆)). The strong stable set Wloc (∆) consists of point from N which converge asymptotically exponentially fast to the invariant disk ∆. In Step 1, we show how to straighten the foliation of the local strong stable ss set of ∆ so that the local strong stable manifolds Wloc (x), x ∈ ∆, become vertical. By construction, F (V ) ⊂ V . The sequence of iterates (F n | )n≥0 is a normal family. V Therefore there exists a subsequence of iterates F nj which converges uniformly on compact subsets of V to a holomorphic map ρ : V → D × {0}. One can in fact choose the subsequence nj as in [2] so that the map ρ is a retract of V onto the invariant disk D × {0}, that is, ρ(x, 0) = (x, 0) for all x ∈ D. By construction, ρ commutes with the map F , so we have ρ ◦ F (x, y) = F ◦ ρ(x, y) = λρ(x, y). Consider the map H(x, y) = (ρ(x, y), y) which leaves the disk D × {0} invariant and is invertible in a neighborhood of this disk since ∂x ρ(x, 0) = 1. By replacing F with H ◦ F ◦ H −1 , we may assume that F is linear in the first coordinate and therefore has the form given in Equation (19). Step 2. — Next we show that F is conjugate to f˜(x, y) = (λx, µ(x)y), where µ(x) = µ + O (x) is a holomorphic function. Let F n (x, y) = (λn x, νn (x, y)) denote the n-th iterate of F , where νn is holomorphic and νn = νn−1 ◦F , for all n ≥ 1. By convention, ν0 = ν. Using this recurrence relation, we find (20)

νn (x, y) = µn y

n−1 Y j=0

B0 (λj x) + νj (x, y)B1 (F j (x, y)) ,

for all n ≥ 1. Note that the partial derivative of ν with respect to y has the form ∂y ν(x, y) = µ (B0 (x) + yC1 (x, y)) , for some appropriate holomorphic function C1 . By direct computation, we obtain ∂y νn = (∂y νn−1 ◦ F ) · ∂y ν,

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which yields ∂y νn (x, y) =

n−1 Y

∂y ν(F j (x, y)) = µn

n−1 Y j=0

j=0

B0 (λj x) + νj (x, y)C1 (F j (x, y)) ,

for all n ≥ 1. Note that νj (x, 0) = 0 for all j ≥ 0. Also µ 6= 0 since F is a local diﬀeomorphism. Hence, when y = 0, the formula above simplifies to ∂y νn (x, 0) = Qn−1 µn j=0 B0 (λj x). We will show that the infinite product ∞ Y νn (x, y) νj (x, y)B1 (F j (x, y)) (21) ψ(x, y) = lim 1+ =y n→∞ ∂y νn (x, 0) B0 (λj x) j=0 is uniformly convergent in some neighborhood V ⊂ C2 of 0. Using the local dynamics, we can choose a suﬃciently small neighborhood V of the origin so that F (x, y) ∈ V whenever (x, y) ∈ V . There exists a constant M > 0 such that |B0 (x)| < 1 + M |x| and |B1 (x, y)| < M throughout V . Since |µ| < 1, we can choose V small enough so that the following technical condition holds: |µ|1/2 (1 + |x|M + |y|M ) < 1,

(22) for all (x, y) ∈ V .

Lemma 5.1. — |νn (x, y)| < |µ|n/2 |y| for all n ≥ 0 and for all (x, y) ∈ V . Proof. — We proceed by induction. From the definition of ν and assumption (22), for n = 0 we get |ν0 (x, y)| ≤ |µ||y|(1 + |x|M + |y|M ) < |µ|1/2 |y|. Let n ≥ 1 and suppose that |νj (x, y)| < |µ|j/2 |y| for all 0 ≤ j ≤ n−1. By Equation (20) and the fact that |λ| = 1 and |µ| < 1 we have |νn (x, y)| ≤ |y||µ|n/2

n−1 Y

|µ|1/2 (1 + |x|M + |y|M ) < |y||µ|n/2 ,

j=0

which concludes the proof.

The germ F is a local diﬀeomorphism, so the Jacobian is bounded away from 0. Thus there exists a constant κ > 0 such that |B0 (x)| > κ for all x ∈ ∆. Using Lemma 5.1, we find that the infinite product (21) is bounded above by ∞ ∞ Y Y 1 + |µ|j/2 M κ−1 < ∞. |y| 1 + |νj (x, y)|M κ−1 < |y| j=0

j=0

This shows that the product from Equation (21) is convergent, uniformly on U . From the definition of νn we have νn+1 (x, y) νn (λx, ν(x, y)) = . ∂y νn+1 (x, 0) ∂y νn (λx, 0) · ∂y ν(x, 0)

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By letting n → ∞ we see that the map ψ satisfies the equation ψ(F (x, y)) = µB0 (x)ψ(x, y). Let Ψ(x, y) = (x, ψ(x, y)) and f˜(x, y) = (λx, µB0 (x)y). The map Ψ is a holomorphic function on a neighborhood of the origin with Ψ(0, 0) = (0, 0), which conjugates F to f˜. For simplicity, we denote µB0 (x) by µ(x). This step concludes the proof of Theorem D. Assume as in Theorem D that the germ f is analytically conjugate to f˜(x, y) = (λx, µ(x)y), for some holomorphic function µ(x) = µ + O (x). Since we work in the dissipative setting, we can topologically linearize f˜ in a neighborhood of the origin. Proposition 5.2. — There exists r > 0 and a homeomorphism h of Dr × D which conjugates f˜ to the linear map L : (x, y) 7→ (λx, µy) on Dr × D. Proof. — Since µ(x) = µ + O (x) and |µ| < 1, we can choose r > 0 small enough and ǫ so that |µ(x)| < ǫ < 1 for all x ∈ Dr . Let us define h(x, y) = (x, y), for (x, y) ∈ Dr ×S1 . We then propagate this definition recursively by the dynamics to the inner levels and set h = Ln ◦ h ◦ f˜−n on f˜n (Dr × S1 ), for n ≥ 1. An easy computation gives the formula h (x, Bn (x)y) = (x, µn y), for Qn (x, y) ∈ f˜n (Dr × S1 ), where Bn (x) = j=1 µ(λ−j x), for n ≥ 1. We extend the definition of h on the set Dr × D − f˜(Dr × D) radially in the second coordinate: h x, (t + (1 − t)µ(λ−1 x)y = (x, (t + (1 − t)µ)y) , for t ∈ (0, 1). For each n ≥ 1, we define h on the inner set f˜n (Dr × D) − f˜n+1 (Dr × D) by dynamics as h = Ln ◦ h ◦ f˜−n . This yields the following explicit form: h (x, (tBn (x) + (1 − t)Bn+1 (x)) y) = x, tµn + (1 − t)µn+1 y ,

for t ∈ (0, 1) and (x, y) ∈ f˜n (Dr × D) − f˜n+1 (Dr × D), for all n ≥ 1. Note that kBn k∞ → 0 as n → ∞, which allows us to set h(x, 0) = (x, 0) for x ∈ Dr . By construction, the map h is a homeomorphism and conjugates f˜ to L. Let us now discuss the analytic linearizability of f˜. As in [16], we say that a cocycle h is reducible if it cohomologous to a constant map, that is, if h satisfies the cohomology equation (23)

h(x) − h(0) = φ(λx) − φ(x),

for some function φ. The reducibility of h depends on finer arithmetic properties of the neutral eigenvalue λ. Suppose that there exists a biholomorphic map (x, y) 7→ (x, η(x)y) with η(0) = 1 which conjugates f˜ to the linear map (x, y) 7→ (λx, µy). The maps µ(x) and η(x) are non vanishing in a neighborhood of the origin, so they have holomorphic logarithms

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h(x) = log(µ(x)) and φ(x) = log(η(x)) which must satisfy the cohomology Equation (23). If we compare the Taylor series expansions of h(x) = log(µ)+a1 x+a2 x2 +· · · and φ(x) = x + b1 x + b2 x2 + · · · , we get an = bn (λn − 1) for n ≥ 1. When λ is not a root of unity, we can solve for bn and obtain a formal series defining φ. The problem of convergence of the formal series for φ is strongly related to how fast λn − 1 approaches 0. Let λ = e2πiα , α ∈ / Q and let pn /qn be the convergents of α given by the continued fraction. If λ satisfies the Brjuno condition, X log qn+1 (24) < ∞, qn n≥0

then by [6] we have a convergent power series for φ. On the other hand, for general functions µ(x) and neutral eigenvalues λ which are not roots of unity and do not satisfy the Brjuno condition, small divisor problems could prevent the existence of solutions of the cohomology Equation (23). Non-linearizable germ with a Siegel disk. — Let α be an irrational angle such that lim sup({nα})−1/n = ∞, n→∞

where {nα} denotes the fractional part of nα. Set λ = e2πiα . As in [17], the arithmetic condition imposed on α is equivalent to lim sup |λn − 1|−1/n = ∞. n→∞

Assume that the power series expansion of h(x) = log(µ(x)) has radius of convergence 0 < R < ∞. Since bn = an (λn − 1)−1 for n ≥ 1, it follows that the radius of convergence of the formal series expansion of φ is 0, which shows that there is no holomorphic function φ satisfying (23). As a concrete example, let λ be chosen as above, and let µ(x) = µex/(1−x) , with |µ| < 1. Then f˜(x, y) = (λx, µ(x)y) is an example of a local diﬀeomorphism which has a Siegel disk containing 0 and which cannot be analytically linearized in a neighborhood of the origin in C2 . The next two proofs in this section deal with the case of polynomial automorphisms of C2 . Proof of Corollary D.1. — If f is a polynomial automorphism of (C2 , 0) then it has non-zero constant Jacobian, equal to the product λµ of the two eigenvalues of df0 . By Theorem D, 0 ∈ intc (H ) if and only if f is analytically conjugate to a holomorphic map f˜(x, y) = (λx, µ(x)y). Any conjugacy function constructed in the proof of Theorem D has the property that the determinant of its Jacobian matrix restricted to the invariant disk D × {0} is 1. This is obvious for the conjugacy maps considered at Step 1. For the coordinate transformation Ψ from Step 2, this property follows from the fact that ∂y νn (x, y) det dΨ(x, y) = ∂y ψ(x, y) = lim , n→∞ ∂y νn (x, 0)

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which is equal to 1 when y = 0. Therefore det df˜| is constant and equal to λµ. D×{0} It follows that µ(x) = µ for all x ∈ D, so f˜ is a linear map.

Proof of Theorem E. — Let f be a polynomial diﬀeomorphism of C2 with an irrationally semi-indiﬀerent fixed point at the origin. Since f is assumed non-linearizable in a neighborhood of the origin in C2 , by Corollary D.1, we know that 0 ∈ / intc (H ), c the interior of H relative to a center manifold Wloc . By Theorem A, the restricc tion of f to Wloc is quasiconformally conjugate to a holomorphic diﬀeomorphism ′ c h : (Ω, 0) → (Ω , 0), h(z) = λz + O (z 2 ). Denote by φ : Wloc → Ω the conjugacy map and by K = φ(H ) the corresponding hedgehog for h. It follows that 0 ∈ / int(K), which is equivalent, by Theorem 2.1, to the fact that h is non-linearizable as well. By [24], the interior of a non-linearizable hedgehog is empty, therefore intc (H ) = ∅. Hence H belongs to the Julia set J. We now show that a stronger statement holds true, i.e., H ⊂ J ∗ . The following argument is due to Romain Dujardin. To fix the notations, let B ⋐ B ′ be two nested domains in C2 containing the fixed point 0 such that f is partially hyperbolic in a neighborhood B ′′ of B ′ . We can choose B ′ as in Theorem C. Let H and H ′ be the hedgehogs constructed with respect to B and B ′ . Then H ⊂ H ′ . ss Let x ∈ H . The local strong stable manifold Wloc (x) is a holomorphic disk. The S −n ss ss (f n (x)) is biholomorphic to C global strong stable manifold W (x) = n≥0 f (Wloc (see [15, Proposition 7.3]). Let p be a saddle periodic point of f . By [3], there exists a ss transverse intersection point t between W ss (x) and W u (p). Then t ∈ f −k (Wloc (f k (x)) n ss n for some k ≥ 0. Iterating forward we obtain f (t) ∈ Wloc (f (x)) for every n ≥ k. Using the recurrence of the points of the hedgehog from Corollary B.1 and the strong ss contraction along the leaves of Wloc (H ), we conclude that there exist transverse intersection points z arbitrarily close to x. Therefore, to show that x ∈ J ∗ , it suﬃces to show that z ∈ J ∗ . We claim that z belongs to the relative boundary of K + in W u (p), thus z ∈ J ∗ by [8, Lemma 5.1]. Otherwise, assume by contradiction that there exists a holomorphic disk ∆ ⊂ W u (p) ∩ K + which contains z. The sequence of iterates (f n | )n≥0 is a ∆ normal family. By reducing the size of ∆, we may assume that all forward iterates of ∆ are contained in B ′ . Consider the limit Γ of a convergent subsequence (f nj (∆))j≥0 . For any fixed k ∈ Z, f k (Γ) = lim f nj +k (∆) ⊂ B ′ , so Γ belongs to every center j→∞

c manifold Wloc defined relative to B ′′ . By Theorems A and 2.2 it follows that Γ ⊂ H ′ . s ss Therefore ∆ ⊂ Wloc (H ′ ) = Wloc (H ′ ) by Corollary C.2. On the other hand, notice ss that ∆ \ Wloc (H ′ ) is nonempty, since the holonomy is a local homeomorphism and the hedgehog H ′ has empty relative interior. This is a contradiction. To show the last part of the theorem, suppose there is a wandering component converging to H , and choose any interior point z of the wandering component. Then for n suﬃciently large, we may assume that all points f n (z) ∈ B ′ and ω(z) ⊂ H ′ . This again contradicts Theorem C.

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Suppose f is a polynomial diﬀeomorphism of C2 with a semi-Siegel fixed point at 0 with an eigenvalue λ = e2πiα , α ∈ / Q. Let ϕ : D → C2 be a maximal injective holomorphic mapping such that f (ϕ(ξ)) = ϕ(λξ), for all ξ ∈ D. The image ∆ = ϕ(D) is called a Siegel disk. Proposition 5.3. — If the closure of the Siegel disk ∆ is contained in a hedgehog H , then ∂∆ ⊂ J ∗ . Proof. — The first observation is that since the closure of the Siegel disk is contained in a region where the map f is partially hyperbolic, every point x on the boundary ss of the Siegel disk has a well-defined local strong stable manifold, Wloc (x), which is a holomorphic disk. Let x ∈ ∂∆ and let p be a saddle periodic point of f . Using the same arguments as in the proof of Theorem E, we deduce that there ss exist transverse intersection points z between Wloc (f n (x)) and W u (p), arbitrarily ∗ close to x. To show that x ∈ J , it suﬃces to show that z belongs to the relative boundary of K + in W u (p), thus z ∈ J ∗ . Otherwise, there exists a holomorphic disk D ⊂ W u (p)∩K + which contains z. The sequence of iterates (f n | )n≥0 is a normal family, so by reducing the size of D, we may D assume that all its forward iterates are in B ′ . Let g be the limit of some convergent subsequence (f nj | )j≥0 . Denote by Γ = g(D) and let x′ = g(z) ∈ ∂∆. The rank of g D can be 1 or 0, and in both cases the set Γ belongs to the hedgehog H ′ and so to every local center manifold defined relative to B ′′ . If the rank of g is 1 then Γ is a c holomorphic disk which extends the Siegel disk to a neighborhood of x′ in Wloc ; this is a contradiction to the maximality of the Siegel disk. If the rank of g is 0 then Γ is a ss single point x′ . However, this is not possible, since D ∩ Wloc (∆) 6= ∅ by construction n ss u ss (W (p) intersects Wloc (f (x)) ⊂ Wloc (∂∆) transversely at z, hence it also intersects ss the leaves of the foliation of Wloc (∆) transversely in an open subset of D, which ss (∆), then contains z in its boundary). If we pick any point in the intersection D ∩ Wloc nj its limit under the subsequence f is an interior point of the Siegel disk, and not the boundary point x′ . Hence in both cases we reach a contradiction. We proceed with the proof of Theorem F, which discusses germs with a semiparabolic fixed point at the origin. Proof of Theorem F. Let B ⊂ C2 be a ball containing 0 such that f is partially hyc perbolic on a neighborhood B ′ of B. Let Wloc be any local center manifold of 0, constructed relative to B ′ . By the weak uniqueness property of center manifolds, c ΣB ⊂ Wloc , where ΣB is defined in Equation (1). By Theorem A, the map f rec stricted to Wloc is quasiconformally conjugate to an analytic map h : (Ω, 0) → (Ω′ , 0), ′ where Ω, Ω are domains in C. By Corollary 3.4.1, the quasiconformal map φ−1 is c holomorphic on the interior of Λ rel Wloc , for the set Λ defined in (5). Therefore −1 c φ is holomorphic on the set ΣB , since ΣB belongs to the interior of Λ rel Wloc .

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By Theorem A, the map h has a parabolic fixed point at 0, with multiplier λ, so it is conjugate to a normal form h(z) = λz + z νq+1 + az 2νq+1 + O (z 2νq+2 ). By the Leau-Fatou theory, h has ν cycles of q attracting and q repelling petals, containing 0 in their boundary. On each repelling petal Prep , there exists an outgoing Fatou coordinate ϕo : Prep → C, which satisfies the Abel equation ϕo (f q ) = ϕo + 1, that is, it conjugates f q to the translation z 7→ z + 1 on a left half plane. The repelling petals for f are just the pullback of the repelling petals for h under the holomorphic map φ−1 | , and on each such repelling petal we have a holomorphic ΣB Fatou coordinate ϕo ◦ φ−1 . We define the parabolic basin of 0 with respect to the neighborhood B as Bpar (0)

= {x ∈ B : f n (x) ∈ B ∀n ∈ N, f n (x) → 0 locally uniformly}.

Note that this set has complex dimension two. Therefore we cannot use the same strategy as in the proof of Theorem F to construct incoming Fatou coordinates, since the conjugacy map φ−1 is only quasiconformal on the set one-dimensional slice c Wloc ∩ Bpar (0). We conclude this section with a generalization of Naishul’s theorem to higher dimensions, which is of independent interest. Theorem 5.4. — For i = 1, 2, let fi be a germ of a holomorphic diﬀeomorphism of (C2 , 0) with a fixed point with eigenvalues λi = e2πiθi and |µi | < 1. If f1 and f2 are topologically conjugate by a homeomorphism ϕ : C2 → C2 with ϕ(0) = 0, then θ1 = ±θ2 . Proof. — Let ϕ be a homeomorphism which fixes the origin and conjugates f1 and f2 , that is ϕ ◦ f1 = f2 ◦ ϕ. Let B1′ ⊂ C2 be a small enough ball containing the origin such that f1 is partially hyperbolic on a neighborhood of B1′ and f2 is partially hyperbolic on a neighborhood of the closure of B2′ = ϕ(B1′ ). Consider a ball B1 ⋐ B1′ and let H1 be a hedgehog for f1 and B1 . Let H2 = ϕ(H1 ) be the corresponding hedgehog for f2 and B2 = ϕ(B1 ). Let i = 1 or 2. Fix r ≥ 1 and let Wic be a C r -smooth local center manifold of 0 for fi , constructed relative to Bi′ . The hedgehog Hi is compactly contained in the center manifold Wic . By Theorem 1.1, there exists a dynamically defined strong ss stable lamination Wloc (Hi ) invariant under fi , whose leaves are holomorphic disks. Note that the conjugating homeomorphism ϕ preserves the strong stable lamination of the hedgehogs. By the general theory of partially hyperbolic systems (see e.g., [29] pp. 278-282), ss there exists a C r−1 -smooth invariant foliation Floc with C r -smooth leaves transverse ss c (Hi ). to the center manifold Wi , which extends the strong stable lamination Wloc c Consider the map π : B2 → W2 obtained by projecting along the leaves of the stable foliation to W2c . The projection π is continuous. There exist neighborhoods Ui of Hi in Wic such that the homeomorphism ϕ induces a continuous surjection ϕc : U1 → U2

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given by ϕc = π ◦ ϕ. The map ϕc is a homeomorphism from H1 to H2 which conju. to f2 | gates f1 | to f2 | , but it might not conjugate f1 | U2 \H2 U 1 \ H1 H2 H1 c By Theorem A there exists a quasiconformal map φi : Wi → Ωi ⊂ C which conjugates fi | c to an analytic map hi : Ωi → Ω′i , with multiplier h′i (0) = e2πiθi . Let Wi Ki = φi (Hi ) be the corresponding hedgehog for the map hi and the Jordan domain Di = φi (Bi ∩ Wic ). By the Uniformization Theorem, there exist a Riemann map ˆ \ Ki which fixes ∞. When Ki is not locally connected, ψi does not ˆ \D → C ψi : C extend as a continuous map from S1 to Ki , but has radial limits almost everywhere on S1 . Let gi = ψi−1 ◦ hi ◦ ψi be the corresponding holomorphic map induced by fi in ˆ \ D. By [23, Theorem 2], the map gi extends to an annular neighborhood of S1 in C an analytic circle diﬀeomorphism with rotation number θi . The map η = φ2 ◦ ϕc ◦ φ−1 : Ω1 → Ω2 is continuous. It is a homeomorphism 1 from K1 to K2 conjugating h1 | to h2 | and it maps the complement of K1 to the K2 K1 complement of K2 . The following result follows from the theory of prime ends: Lemma 5.5. — Let η : K1 → K2 be a homeomorphism between full compact sets Ki ⊂ C which extends continuously to a neighborhood Ω1 ⊃ K1 such that η(Ω1 \K1 ) ⊂ (C \ K2 ). Then η induces a continuous map ̺ : S1 → S1 between the corresponding spaces of prime ends. By Lemma 5.5, the map ̺ = ψ2−1 ◦ η ◦ ψ1 extends continuously to S1 . We will show that ̺ is a homeomorphism which conjugates g1 and g2 on S1 , i.e., ̺ ◦ g1 = g2 ◦ ̺, which will prove that θ1 = ±θ2 . It is easy to check that θ1 is rational iﬀ θ2 is rational. The rational case is easy to deal with and is left to the reader. Suppose θ1 is irrational. A point z ∈ ∂K1 is accessible if there is a path γ in C \ K1 landing at z. By a theorem of Lindelöf, z is accessible if and only if it is the landing point of some ray Rt = {ψ1 (re2πit ), r > 1}. Let z1 ∈ K1 ∩ ∂D1 . There is a path γ1 lying outside D1 landing at z1 , so z1 is accessible. The image η(γ1 ) is a curve in C \ K2 , so the point z2 = η(z1 ) of K2 ∩ ∂D2 is also accessible. Let t1 , t2 ∈ S1 be the angles corresponding to z1 , respectively z2 . Then ̺(t1 ) = t2 and ̺ conjugates g1 to g2 on the orbit of t1 under g1 . By Denjoy’s theorem, g1 is topologically conjugate to an irrational rotation, so the orbit of t1 is dense in the circle. Similarly, the orbit of t2 is dense in S1 , so ̺ : (S1 , t1 ) → (S1 , t2 ) semi-conjugates g1 and g2 on S1 . By replacing f1 with f2 and reversing the entire construction, one obtains a semi-conjugacy ̺′ : (S1 , t2 ) → (S1 , t1 ) between the two analytic circle diﬀeomorphisms g1 and g2 . It follows that ̺ and ̺′ must be mutually inverse conjugacies. We remark that Theorem 5.4 generalizes in a straightforward way to the case of germs of holomorphic diﬀeomorphisms of (Cn , 0), for n > 2, that have a fixed point at the origin with exactly one eigenvalue on the unit circle and n − 1 eigenvalues inside the unit disk.

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6. A generalization to germs of (Cn , 0) In this section we consider germs f of holomorphic diﬀeomorphisms of (Cn , 0) such that the linear part of f at 0 has eigenvalues λi , 1 ≤ i ≤ n, with |λk | = 1 and (25)

0 < |λ1 | ≤ · · · ≤ |λk−1 | < 1 < |λk+1 | ≤ · · · ≤ |λn |,

for some k between 1 and n. The presence of the neutral eigenvalue permits the existence of a rich type of local invariant sets and induces more complicated local dynamics. The tangent space at 0 has an invariant splitting into three subspaces T0 Cn = E0s ⊕ E0c ⊕ E0u , of dimensions k − 1, 1, and respectively n − k. E0s is strongly contracted and E0u is strongly expanded by df , and the center direction E0c is the eigenspace corresponding to the neutral eigenvalue λk . When k 6= 1 and k 6= n, f is partially hyperbolic (in the narrow sense) (see [13], [7]). Partial hyperbolicity is an open condition which can be extended to a suitable neighborhood of the origin. Let B ′ a small ball containing the origin. As in Section 3 we explain in terms of invariant cone fields what it means for f to be partially hyperbolic on B ′ . Let E be a subspace of Tx Cn and denote by Cx (E, α) = {v ∈ Tx Cn , ∠(v, E) ≤ α} the cone at x of angle α centered around E. There exist (not necessarily invariant) continuous distributions E s , E c and E u , extending E0s , E0c , E0u , such that Tx Cn = Exs ⊕ Exc ⊕ Exu for any x in B ′ . Let Excs = Exs ⊕ Exc and Excu = Exc ⊕ Exu . There exist invariant cone families of stable and unstable cones s s u u Cx = Cx (Ex , α), Cx = Cx (Ex , α) and center-stable and center-unstable cones cs

Cx

= Cx (Excs , α),

cu

Cx

= Cx (Excu , α)

such that dx f −1 (Cxs ) ⊂ Int Cfs−1 (x) ∪ {0},

dx f (Cxu ) ⊂ Int Cfu(x) ∪ {0}

dx f −1 (Cxcs ) ⊂ Int Cfcs−1 (x) ∪ {0},

dx f (Cxcu ) ⊂ Int Cfcu(x) ∪ {0}

and there are constants 0 < µs < µcu ≤ 1 ≤ µcs < µu such that µcu < µcs , |λk−1 | < µs , |λk+1 | > µu , and the following inequalities hold: kdfx (v)k ≤ µs kvk,

for v ∈ Cxs

kdfx (v)k ≤ µcs kvk,

for v ∈ Cxcs

kdfx (v)k ≥ µu kvk,

for v ∈ Cxu

kdfx (v)k ≥ µcu kvk,

for v ∈ Cxcu .

The fact that f is partially hyperbolic on the set B ′ implies that there exists local cs cu center-stable manifolds Wloc and center-unstable manifolds Wloc of class C 1 , tangent

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at 0 to the subspaces E0cs and respectively to E0cu . By intersecting the local centerstable and the local center-unstable manifolds, one shows the existence of center manc ifolds Wloc of class C 1 , tangent at 0 to the eigenspace E0c of the neutral eigenvalue λk . A local center manifold is the graph of a C 1 function ϕf : E0c ∩ B ′ → E0s ⊕ E0u , and c c is locally invariant, in the sense that f (Wloc ) ∩ B ′ ⊂ Wloc . The center manifold is not unique in general, but all center manifolds defined with respect to the ball B ′ must contain the set of points which never escape from B ′ under forward and backward iterations. A weak uniqueness property can therefore be formulated as follows: c if f n (x) ∈ B ′ for all n ∈ Z, then x ∈ Wloc . c For the rest of the section, fix a local center manifold Wloc defined with respect to c the ball B ′ . We show that the map f restricted to Wloc is quasiconformally conjugate to an analytic map, in a two-step argument. Most of the analysis will be similar to Sections 3 and 4.2, so we refer the reader to these sections for most proofs, and we will only outline the diﬀerences, whenever they occur. We prove the following: Theorem G. — Let f be a germ of a holomorphic diﬀeomorphism of (Cn , 0). Suppose df0 has eigenvalues λj , 1 ≤ j ≤ n, with |λk | = 1 for some k and |λj | 6= 1 when c j 6= k. Let Wloc (0) be a C 1 -smooth local center manifold of the fixed point 0. There c exist neighborhoods W, W ′ of the origin inside Wloc (0) such that f : W → W ′ is quasiconformally conjugate to a holomorphic diﬀeomorphism h : (Ω, 0) → (Ω′ , 0), h(z) = λk z + O (z 2 ), where Ω, Ω′ ⊂ C. c Moreover, the conjugacy map is holomorphic on the interior of Z rel Wloc (0), where Z is the set of points that stay in W under all forward and backward iterations of f . Remark 6.1. — Note that if |λk | = 1 and |λj | < 1 for all j 6= k or |λj | > 1 for all j 6= k, then the proof is identical to the proof of Theorem A. It is worth mentioning that the set Z from Theorem G belongs to the intersection of all center manifolds defined relative to the ball B ′ . Denote by J the standard almost complex structure of Cn . Consider a ball B c containing 0, such that B ⊂ B ′ . We first endow the two-dimensional real manifold Wloc 1 ′ with a C -smooth almost complex structure J , induced by the restriction of the c Riemannian metric of Cn to Wloc . By integrating the almost complex structure J ′ , c we show that the map f on W = Wloc ∩ B is conjugate to a map g : U ⊂ C → C of 1 ′ class C , via a (J , i)-holomorphic conjugacy map φ, as in the diagram below: f

W −−−−→ x φ g

W′ x φ

U −−−−→ U ′ .

We will estimate how far g is from being an analytic map by measuring how far c the tangent space Tx Wloc is from being a complex subspace of Tx Cn , when x ∈ W . To carry on the analysis, we fix some notations for the dynamically relevant sets for f and g. For each n ≥ 0, let Wn (respectively W−n ) be the set of points whose first n

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backward (respectively forward) iterates remain in W . Similarly, we define the sets Un = φ−1 (Wn ) and U−n = φ−1 (W−n ) for the map g. With these notations, W∞ and W−∞ represent the set of points from W that do not escape from B under backward, and respectively forward iterations. Let Z := W∞ ∩ W−∞ . For simplicity, let X = U∞ and Y = U−∞ . c Proposition 6.2. — a) The tangent space Tx Wloc at any point x ∈ Z is a complex c n line Ex of Tx C . The line field over Z is df -invariant.

b) There exists ρ < 1 such that for all integers m, n ≥ 0 and for all x ∈ Wn ∩ W−m c and v ∈ Tx Wloc the following estimate holds c Angle (Tx Wloc , SpanC {v}) = O ρmin(m,n) .

Proof. — Part a) follows from the fact that for x ∈ Z \ \ −n c Tx Wloc = dffn−n (x) Cfcu−n (x) ∩ dff n (x) Cfcsn (x) , n≥0

n≥0

and the counterpart of Proposition 3.3, which is straightforward. For part b) we observe that since x ∈ Wn , the tangent vector v belongs to dffn−n (x) Cfcu−n (x) which, by Remark 3.1, is a cone of angle opening α1 = O (ρn ) inside Cxcu , centered around Excu . cs Similarly, since x ∈ W−m , the tangent vector v belongs to dff−m m (x) Cf m (x) , which is a m cs cs cone of angle opening α2 = O (ρ ) inside Cx , centered around Ex . Hence v belongs to the complex cone centered around E c , of angle less than the maximum of the c angles α1 and α2 . As in the proof of Proposition 3.5, it follows that both Tx Wloc and SpanC {v} are included in this cone. c Corollary 6.2.1. — Let intc (Z) denote the interior of Z relative to Wloc . The set c n int (Z) is a complex submanifold of C of complex dimension 1. The conjugacy map φ : int(X ∩ Y ) ⊂ C → intc (Z) ⊂ Cn is holomorphic.

Lemma 6.3. — There exists a constant C such that for every m, n ≥ 1, k∂¯J ′ f kWn ∩W−m < Cρmin(m,n) , where ∂¯J ′ f is the derivative of f with respect to the almost complex structure J ′ c on Wloc . The proof of this lemma uses Proposition 6.2. The argument is the same as in the proof of Lemma 3.6, so we omit it here. Proposition 6.4. — There exists a constant C ′ such that for every m, n ≥ 1, ¯ |∂g(z)| < C ′ ρmin(m,n) , for all z ∈ Un ∩ U−m . Proof. — The proof follows from Lemma 6.3 and Corollary 3.6.1.

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In terms of Beltrami coeﬃcients the proposition above implies that there exist ρ < 1 and M independent of m, n such that (26)

sup

|µg (z)| < M ρmin(m,n) , for all m, n ≥ 0.

z∈Un ∩U−m

¯ = 0 on X ∩ Y . Corollary 6.4.1. — ∂g ¯ Note that in Section 3 we obtained that the ∂-derivative of g is 0 on the entire set X, whereas when we have stable, neutral and unstable eigenvalues we can only ¯ show that the ∂-derivative of g is 0 on X ∩ Y . Let σ0 denote the standard almost complex structure of the plane, represented by the zero Beltrami diﬀerential. For n ≥ 0, consider the Beltrami diﬀerential σn on Un , given by σn = (g −n )∗ σ0 . Similarly, we define the Beltrami diﬀerentials σ−n on U−n by σ−n = (g n )∗ σ0 . Lemma 6.5. — There exists a constant κ < 1, such that for all integers n ≥ 0 kσn k∞ = kσ−n k∞ < κ. Proof. — Let z ∈ U−n . Let zj = g j (z) for 0 ≤ j ≤ n denote the j-th iterate of z under the map g. By Lemma 4.1 part c), we know that zj ∈ Uj ∩ U−(n−j) for all 0 ≤ j ≤ n. Hence, by Equation (26), |µg (z)| < M ρmin(j,n−j) , 1 ≤ j ≤ n.

sup z∈Uj ∩U−(n−j)

As in the proof of Lemma 4.3, we show that the dilatation K(g n , z) is bounded by a constant independent of n and the choice of z. We have (27)

K(g n , z) ≤

n−1 Y

K(g, zj ),

j=0

where K(g, zj ) = 1 + O ρmin(j,n−j) and the conclusion follows.

¯ to prove the following theorem. We now use the estimates obtained on ∂g

Theorem 6.6. — The map g −1 : U1 → U−1 is quasiconformally conjugate to a holomorphic map. Proof. — Consider the measurable function µ : U → C, given by on Un \ Un+1 , for n ≥ 0 σn µ= σ−n on X ∩ U−n \ U−(n+1) , for n ≥ 0 σ0 on X ∩ Y.

Then kµk∞ < 1 by Lemma 6.5. Thus µ is a Beltrami coeﬃcient, which is g −1 invariant by construction, i.e., (g −1 )∗ µ = µ on U1 . The set X ∩ Y is forward and backward invariant so (g −1 )∗ σ0 = σ0 on X ∩Y , by Corollary 6.4.1. The invariance of µ on U \X is

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discussed in the proof ofTheorem 4.4. The only new case to check is when n > 0 and z ∈ X ∩ U−n \ U−(n+1) . By Lemma 4.1, g X ∩ U−n \ U−(n+1) ⊂ X ∩ U−(n−1) \ U−n .

We have the following sequence of equalities ∗ ∗ ∗ g −1 σ−n (z) = g −1 (g n ) σ0 (z) = σ−(n−1) (z), ∗ which shows that g −1 µ = µ on X \ Y as well. The Measurable Riemann Mapping Theorem concludes the proof.

We proved that f : W → W ′ is quasiconformally conjugate to a holomorphic diﬀeomorphism h : (Ω, 0) → (Ω′ , 0), where Ω, Ω′ ⊂ C. By Corollary 6.2.1 and Theorem 6.6 it follows that the quasiconformal conjugacy map is holomorphic on the interior of Z, the set of points that remain in W under all forward and backward iterates of f . The fact that h(z) = λk z + O (z 2 ), where λk is the neutral eigenvalue of df0 , follows from the generalization of Naishul’s theorem due to Gambaudo, Le Calvez, and Pécou [10]. This concludes the proof of Theorem G. Types of hedgehogs. — Let H denote the connected component containing 0 of the set Z. Then H is the maximal hedgehog associated to the neighborhood B of the origin. Using Theorem G and the local dynamics of the holomorphic germ h of (C, 0) with an indiﬀerent fixed point at 0, we can further describe the dynamical nature of the hedgehog. If λk is a root of unity, λk = e2πip/q , and the parabolic multiplicity of h at 0 is ν, then the hedgehog H of f is the closure of 2νq holomorphic petals Pinv , which are invariant under f q and f −q , and where points converge both forward and backward to 0. In addition, when k = n in Equation (25), we can prove as in Theorem F the existence of holomorphic one-dimensional repelling petals with holomorphic outgoing c Fatou coordinates. When k 6= 1, n, we can fix any center manifold Wloc ⊃ H and use the quasiconformal conjugacy to construct νq one-dimensional attracting and c repelling petals in Wloc , with the same regularity as the center manifold, consisting of c points whose forward, respectively backward, orbit is contained in Wloc and converges to 0. However, the attracting/repelling petals will change as we change the center manifold. To visualize the phenomenon better, one may think of slicing the parabolicattracting basin of 0 (of complex dimension k), and the parabolic-repelling basin of 0 (of dimension n − k + 1) with diﬀerent center manifolds. Theorem G also guarantees the existence of quasiconformal incoming/outgoing Fatou coordinates ϕi /ϕo , with holomorphic transition maps ϕi ◦ (ϕo )−1 : Pinv → C. If λk = e2πiα , α ∈ / Q, and h is linearizable at 0 (that is, analytically conjugate to the rigid rotation z → λk z in a neighborhood of 0 in C), then H contains a holomorphic disk, called a Siegel hedgehog in our context. Lastly, if the angle α is irrational and h is not linearizable at the origin, then H is a Cremer hedgehog, with a complicated topology: H has no interior, and is non-locally connected at any point diﬀerent from the origin.

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7. Semi-parabolic germs of (Cn , 0) Let f be a germ of a holomorphic diﬀeomorphism of (Cn , 0), with an isolated fixed point at the origin, with a neutral eigenvalue λ = e2πip/q a root of unity, k − 1 eigenvalues λ2 , . . . , λk inside the unit disk, and n − k eigenvalues λk+1 , . . . , λn outside of the unit disk. Let νq +1 be the order of 0 as a fixed point of the equation f q (x) = x. The germ f (x, y, z) = (x1 , y1 , z1 ) may be written near the origin in the form x1 = λx + f1 (x, y, z) y1 = A2 y + f2 (x, y, z) z1 = A3 z + f3 (x, y, z), where x ∈ C, y ∈ Ck−1 , z ∈ Cn−k , A2 and A3 are Jordan blocks corresponding to the attracting/repelling eigenvalues, and fi , 1 ≤ i ≤ 3 are holomorphic functions which vanish at the origin along with their first order derivatives. Let r > 2νq+1 be a positive integer. Consider as before a ball B ′ containing the origin on which f is partially hyperbolic. By choosing a suﬃciently small neighborhood B of the origin, compactly contained in B ′ , we may apply the Center Manifold Theorem ss uu to obtain unique analytic local strong stable/strong unstable manifolds Wloc , Wloc , cu cs , and (non unique) center-stable, center-unstable and center manifolds, Wloc , Wloc c of class C r , embedded in B ′ and tangent to the linear subspaces E0s , E0u , E0cs , Wloc cu cs E0 and E0c . The center-stable manifold Wloc = {z = ϕ(x, y)} has real dimension 2k r and is the graph of a C -smooth function ϕ : E0cs ∩ B → E0u with ϕ(0, 0) = 0. The cu center-unstable manifold Wloc = {y = ψ(x, z)} has real dimension 2(n − k + 1) and is r the graph of a C -smooth function ψ : E0cu ∩ B → E0s , with ψ(0, 0) = 0. The centerstable and center-unstable manifolds intersect transversely, and their intersection is c . the center manifold Wloc − Let Λ (respectively Λ+ ) be the set of points that do not escape from B under cs cu forward (respectively backward) iterations. Clearly Λ− ⊂ Wloc and Λ+ ⊂ Wloc . We define the local parabolic-attracting (respectively parabolic-repelling) basins of 0 to be the set of points from Λ+ (respectively Λ− ) which converge to 0 under cs forward (respectively backward) iterations, locally uniformly in Wloc (respectively cu in Wloc ). We will construct holomorphic Fatou coordinates on the parabolic basins. In the case when we have one neutral eigenvalue and all the other eigenvalues are inside the unit disk (k = n), this is done in [34] and [12]. When we allow eigenvalues both inside and outside the unit disk we cannot apply the results in [34], [12] directly to the germ f , but we will be able to apply them to the germ f restricted to the centercs cu stable, and respectively to the center-unstable manifold of 0. Even if Wloc and Wloc r are only C -smooth, the parabolic basins are holomorphic, since they are contained in the analytic parts Λ− /Λ+ of the center-stable/center-unstable manifolds. cs cu Proposition 7.1. — Let Wloc and Wloc be any local center-stable and center-unstable ′ manifolds of 0 relative to B .

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cs a) The tangent space Tx Wloc at any point x ∈ Λ+ is a complex subspace Excs n of Tx C . The distribution over Λ+ is df −1 -invariant, i.e., dfx−1 (Excs ) = Efcs−1 (x) for every point x ∈ Λ+ . cu b) The tangent space Tx Wloc at any point x ∈ Λ− is a complex subspace Excu n of Tx C . The distribution over Λ− is df -invariant, i.e., dfx (Excu ) = Efcu(x) for every point x ∈ Λ− .

Proof. — We have: \ \ cs − Excu = dffn−n (x) Cfcu−n (x) ∀x ∈ Λ+ and Excs = dff−n n (x) Cf n (x) ∀x ∈ Λ . n≥0

n≥0

The proof follows as in Proposition 3.3. We outline here the construction of the Fatou coordinate for the parabolicattracting basin, as the case of the parabolic-repelling basin is obtained analogously, by replacing f by f −1 . First we do a holomorphic change of coordinates in a neighborhood of the oriss gin to straighten the analytic strong stable manifold Wloc of the fixed point 0 to uu the axis {x = 0, z = 0} and the analytic strong unstable manifold Wloc to the axis {x = 0, y = 0}. So we may assume without loss of generality that in the new coordics cu nates we have Wloc = {z = ϕ(x, y)}, with ϕ(0, y) = 0, and Wloc = {y = ψ(x, z)}, cs with ψ(0, z) = 0. Next we straighten the center-stable manifold Wloc to the plane cu z = 0, and the center-unstable manifold Wloc to the plane y = 0 via a local change of coordinates φ(x, y, z) = (x1 , y1 , z1 ), x1 = x, y1 = y − ψ(x, z), and z1 = z − ϕ(x, y), ss uu of class C r , which leaves Wloc and Wloc invariant. In the new coordinates, the center c manifold Wloc is just the x-axis. The map φ is in fact analytic where ψ and ϕ are analytic. It is easy to see that φ(0, y, 0) = (0, y, 0) and φ(0, 0, z) = (0, 0, z), hence in the new ss coordinate system we can think of π2 : Wloc → Ck−1 , π2 (0, y, 0) = y as a natural ss uu complex coordinate on Wloc and of π3 : Wloc → Cn−k , π3 (0, 0, z) = z, as a natural uu complex coordinate on Wloc . The center manifold is only of class C r , but we can c put a reference complex structure on the center manifold to make π1 : Wloc → C, c π1 (x, 0, 0) = x a complex coordinate on Wloc . Consider a new complex structure on Cn by taking the product of the complex manifolds C × Ck−1 × Cn−k , each considered with the complex structures described above. The new almost complex structure on Cn agrees with the usual almost complex structure of Cn up to order r. Consider now the restriction of the map f to the center-stable manifold. cs Let intcs (Λ− ) denote the interior of the set Λ− rel Wloc . Notice that f | cs is holomorphic on intcs (Λ− ). In the new coordinates, f |

cs Wloc

with (28)

ASTÉRISQUE 416

(

x1 = f˜1 (x, y) y1 = g(y) + f˜2 (x, y),

Wloc

has the form f (x, y) = (x1 , y1 ),

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247

where f˜1 , f˜2 are germs of C r -smooth functions of (Ck , 0), and g is a germ of a holomorphic transformation of (Ck−1 , 0). The restriction f | ss is analytically conjugate Wloc

to g. The map g is contracting with dg0 = A2 , the Jordan block corresponding to the attracting eigenvalues {λj }2≤j≤k . The germs f˜1 and f˜2 are holomorphic on intcs (Λ− ). Moreover, f˜1 (0, y) = 0, f˜2 (0, y) = 0, and ∂x f˜1 at (0, 0) is λ. ss We consider the Taylor series expansion of f˜1 around a point (0, y) ∈ Wloc . The map r ˜ f1 is not holomorphic, so we think of it as a C -smooth function in the variables x, x ¯ and y and use the Wirtinger derivatives. Consider now an integer 1 < r′ < r. We obtain: f˜1 (x, y) =

(29)

X

1≤i+j≤r ′

′ ∂xi ∂x¯j f˜1 (0, y) i j xx ¯ + Oy (|x|r +1 ). i!j!

′

′

The tail Oy (|x|r +1 ) is a C r -smooth function of x and x ¯. It is a C r−r -smooth function of y. ss The map f˜1 is holomorphic on intcs (Λ− ). Since Wloc = {(0, y)} lies in the closure cs − of int (Λ ), it follows by continuity that all partial derivatives of ∂¯f˜1 vanish up to order r′ at (0, y). Therefore the Taylor series expansion (29) reduces to ′

(30)

f˜1 (x, y) =

r X

′

ai (y)xi + Oy (|x|r +1 ),

i=1

where the coeﬃcients ai (·), 1 ≤ i ≤ r′ , are C r−i -smooth functions from (Ck−1 , 0) to C, with a1 (0) = λ. Note that these coeﬃcients are in fact holomorphic on intcs (Λ− ). As in [34], [12], [27], we will do a series of changes of variables in the variable x to make the coeﬃcients ai (·) constants, for 1 ≤ i ≤ r′ . The variable y will remain unchanged. We first make a1 (y) = λ by considering the coordinate change X = u(y)x, Y = y, where u is a germ of C r−1 functions on (Ck−1 , 0). We need to find u so that X1 = u(y1 )x1 = u g(y) + f˜2 (x, y) a1 (y)x + a2 (y)x2 + · · · = u (g(Y ) + · · · ) (a1 (Y )X/u(Y ) + · · · ) =

u(g(Y ))a1 (Y ) X + ··· u(Y )

))a1 (Y ) We seek a function u such that u(g(Yu(Y = λ. We successively substitute Y with ) g(Y ) in this equation and obtain

(31)

u(Y ) =

∞ Y a1 (g n (Y )) , λ n=0

which converges in a neighborhood of 0 since kgk < 1 and a1 (0) = λ. Suppose now that the first i − 1 coeﬃcients are constants and proceed by induction on i. The base case was treated above.

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Consider the coordinate transformation X = x + v(y)xi , Y = y, where v is a germ of C r−i functions on (Ck−1 , 0). We have X1

= x1 + v(y1 )xi1 =

i−1 X j=1

=

i−1 X j=1

aj xj + ai (y) + λi v(g(y)) xi + · · ·

aj X j + ai (Y ) + λi v(g(Y )) − λv(Y ) X i + · · ·

We choose v such that the coeﬃcient of X i is constant. This yields the functional equation λv(Y ) − λi v(g(Y )) = ai (Y ) − ai (0), with solution (32)

v(Y ) =

∞ 1X ai g j (Y ) − ai (0) λj(i−1) . λ j=0

The series converges in a neighborhood of 0 since kgk < 1. We proved the following result. ′

Proposition 7.2. — There exists a C r−r local change of coordinates in which f | cs has the form f (x, y) = (x1 , y1 ), where Wloc r′ ′ x = λx + P ai xi + Oy (|x|r +1 ) 1 (33) i=2 y = g(y) + f˜ (x, y) 1 2 and ai are constants, 2 ≤ i ≤ r′ . The change of coordinates is analytic on intcs (Λ− ).

Proposition 7.3. — There exists a C r−2νq−1 local change of coordinates in which f | cs has the form f (x, y) = (x1 , y1 ), where Wloc ( x1 = λx + xνq+1 + Oy (|x|2νq+1 ) (34) y1 = g(y) + f˜2 (x, y). The change of coordinates is analytic on intcs (Λ− ). Proof. — We start from the normal form in Equation (33) and perform the same coordinate changes as in [34], [12], [27]. If k is not congruent to 1 modulo q (i.e., λk 6= λ), then we can set ak b= λ − λk and consider the coordinate transformation X = x + bxk , Y = y to eliminate the term ak xk . The first term which we cannot eliminate in this way is aνq+1 xνq+1 . We can further make the coeﬃcient aνq+1 = λ by considering a transformation of the form X = Ax, Y = y, where A is a constant such that Aνq = aνq+1 . One can also eliminate all the intermediate terms of degree jq + 1, for ν < j < 2ν, by doing changes ajq+1 . of variables of the form X = x + bx(j−ν)q+1 , Y = y, where b = (2ν−j)q

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The changes of coordinates are holomorphic on intcs (Λ− ), so in particular they are holomorphic on the parabolic-attracting basin of 0. Holomorphic Fatou coordinates. — The choice of r in the beginning of this section guarantees that the local change of coordinates in Proposition 7.3 is at least C 1 smooth. We can use the C 1 -normal form (34) to show the existence of νq disjoint domains attracted to 0 and invariant under f q , as in [12], [34]. We call these doj mains parabolic-attracting petals and denote them by Ppar -att (0), 1 ≤ j ≤ νq. The key observation is that the parabolic-attracting petals belong to the analytic part j of the center-stable manifold and the restriction of the map f to any Ppar -att (0) is j holomorphic. Therefore, by Theorem 7.3, f restricted to Ppar-att (0) is analytically conjugate to the normal form (34). Assuming (34), we can then use the same anaj lytic changes of variables as in [12], [34] on each Ppar -att (0) to show the existence of a j i holomorphic incoming Fatou coordinate ϕ : Ppar-att (0) → C, verifying the functional equation ϕi (f q (p)) = ϕi (p)+1. The parabolic-attracting basin of 0 has νq components j Bpar-att (0), 1 ≤ j ≤ νq, which are defined as j

Bpar-att (0)

=

∞ [

n=1

j f −nq Ppar -att (0) .

j i By putting ϕi (p) = ϕi (f n (p)) − n on f −nq Ppar -att (0) we can extend ϕ to the

entire set Bjpar-att (0). Analogously, we can construct holomorphic outgoing Fatou coordinates on each component of the parabolic-repelling basin of 0. Having holomorphic Fatou coordinates has important consequence for the global dynamics. Hakim [12] and Ueda [34] have shown that in the case of polynomial automorphisms of C2 with a semi-parabolic fixed point at the origin, each component of the parabolic basin of 0 is a Fatou-Bieberbach domain, i.e., a proper subset of C2 biholomorphic to C2 . The main strategy is to show that each component of the parabolic basin is a trivial fiber bundle over C with fibers biholomorphic to C, given by the level sets of the incoming Fatou coordinate. The same technique, together with the holomorphic incoming/outgoing Fatou coordinates discussed in this section yield the following more general result: Theorem H. — Let n ≥ 3 and consider f a dissipative polynomial diﬀeomorphism of Cn with a fixed point at 0. Suppose df0 has a neutral eigenvalue λ = e2πip/q , a stable eigenvalue 0 < |λ2 | < 1, while the rest of the eigenvalues λ3 , . . . , λn are outside of the closed unit disk. Suppose the equation f q (x) = x has multiplicity νq + 1 at the origin. Then the parabolic-attracting basin of 0 has νq components and each component is biholomorphic to C2 .

Corollary H.1. — If n = 3, then the parabolic-repelling basin of 0 has νq components and each component is biholomorphic to C2 .

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[29]

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M. Lyubich, Stony Brook University, Stony Brook, United States E-mail : [email protected] R. Radu, Uppsala University, Uppsala, Sweden E-mail : [email protected] R. Tanase, Uppsala University, Uppsala, Sweden E-mail : [email protected]

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Astérisque 416, 2020, p. 253–299 doi:10.24033/ast.1116

TIME-CHANGES OF HEISENBERG NILFLOWS by Giovanni Forni & Adam Kanigowski

In memory of Jean-Christophe Yoccoz, il miglior fabbro.

Abstract. — We consider the three dimensional Heisenberg nilflows. Under a full measure set Diophantine condition on the generator of the flow we construct Bufetov functionals which are asymptotic to ergodic integrals for sufficiently smooth functions, have a modular property and scale exactly under the renormalization dynamics. By the asymptotic property we derive results on limit distributions, which generalize earlier work of Griffin and Marklof [17] and Cellarosi and Marklof [8]. We then prove analyticity of the functionals in the transverse directions to the flow. As a consequence of this analyticity property we derive that there exists a full measure set of nilflows such that generic (non-trivial) time-changes are mixing and moreover have a “stretched polynomial” decay of correlations for sufficiently smooth functions (this strengthens a result of Avila, Forni, and Ulcigrai [2]). Moreover we also prove that there exists a full Hausdorff dimension set of nilflows such that generic non-trivial time-changes have polynomial decay of correlations. Résumé (Changements de temps des flots nilpotents d’Heisenberg). — Nous étudions les flots nilpotents de Heisenberg en dimension trois. Sous une condition Diophantienne de mesure pleine sur le générateur du flot, nous montrons l’existence de fonctionnelles de Bufetov, qui sont asymptotiques aux intégrales ergodiques pour toutes les fonctions suffisamment différentiables, qui ont une propriété modulaire, et satisfont une identité de changement d’échelle sous la dynamique de renormalisation. De la propriété asymptotique, nous dérivons des résultats sur les distributions limites des moyennes ergodiques, qui généralisent les travaux de Griffin et Marklof [17], et Cellarosi et Marklof [8]. Ensuite nous montrons une propriété d’analyticité des fonctionnelles dans les directions transverses au flot. Comme conséquence de cette propriété d’analyticité, nous dérivons l’existence d’un ensemble de mesure pleine de flots nilpotents dont les changements de temps génériques (non-triviaux) sont mélangeant, et de plus ont une vitesse de mélange « polynomiale étirée » pour toutes les fonctions suffisamment différentiables (cela améliore un résultat de Avila, Forni, 2010 Mathematics Subject Classification. — 37A25, 37A17, 37A50, 60F05. Key words and phrases. — Heisenberg nilpotent flows, renormalization, limit distributions, timechanges, mixing, decay of correlations.

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et Ulcigrai [2]). De plus, nous montrons qu’il existe un ensemble de dimension de Hausdorff maximale de flots nilpotents tels que les changements de temps génériques non-triviaux ont une vitesse de mélange polynomiale.

1. Introduction This paper concerns the smooth ergodic theory of parabolic ﬂows, that is, ﬂows characterized by polynomial (sub-exponential) divergence of nearby orbits. In particular we prove results on limit distributions of Heisenberg nilﬂows and on the decay of correlations of their non-trivial reparametrizations (time-changes). Our approach is based on the construction of finitely additive Hölder measures and Hölder cocycles for Heisenberg nilﬂows, asymptotic to ergodic integrals, following the work of A. Bufetov [6] on translation ﬂows and of Bufetov and G. Forni [5], [9] on horocycle ﬂows. Hölder cocycles for translation ﬂows are closely related to “limit shapes” of ergodic sums for Interval Exchange Transformations, studied in the work of S. Marmi, P. Moussa and J.-C. Yoccoz [22] on wandering intervals for aﬃne Interval Exchange Transformations. In fact, roughly speaking, “limit shapes” are related to graphs of Hölder cocycles as functions of time. We recall that the mixing property for generic, non-trivial time-changes of Heisenberg nilﬂows was proved by A. Avila, G. Forni and C. Ulcigrai [2]. The main result of that paper was that for uniquely ergodic Heisenberg nilﬂow all non-trivial timechanges, within a dense subspace of time-changes, are mixing. Under a Diophantine condition the set of trivial time-changes has countable codimension and can be explicitly described in terms of invariant distributions for the nilﬂow. Results on limit theorems for skew-translations, which appear as return maps (with constant return time) of Heisenberg nilﬂows, limited however to a single character function, have more recently been proved by J. Griﬃn and J. Marklof [17] and refined by F. Cellarosi and Marklof [8] by an approach based on theta functions. Their worked raised the question of possible relations between theta functions and Bufetov’s Hölder cocycles, developed for other analogous dynamical systems in [6] (translation ﬂows), [5] (horocycle ﬂows), [7] (tilings), as a formalism to derive asymptotic theorem for ergodic averages and prove limit theorems. In this paper we generalize the results of Griﬃn and Marklof [17] on limit distributions, proving in particular that almost all limits of ergodic averages of arbitrary suﬃciently smooth functions are distributions of Hölder continuous functions on the Heisenberg nilmanifold, hence in particular they have compact support. Our main results, however, are on the decay of correlations of smooth functions for time-changes: we prove that it has polynomial (power law) speed for all non-trivial smooth timechanges of Heisenberg nilﬂows of bounded type, within a generic subspace of timechanges. As mentioned above, the study of limit distributions for parabolic ﬂows has been developed only in recent years after Bufetov’s work [6] on translations ﬂows (and

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Interval Exchange Transformations). A comprehensive study of spatial and temporal limit theorems for dynamical systems of diﬀerent type has been more recently carried out in the work of D. Dolgopyat and O. Sarig [10]. The study of mixing properties of elliptic and parabolic ﬂows and their timechanges has a longer history. For instance, mixing properties of suspension ﬂows over rotations and Interval Exchange Transformations have been investigated in depth (see for instance [18], [19], [27], [26],[28], [29] and reference therein), mixing for reparametrizations of linear toral ﬂows were investigated by B. Fayad (see for instance [12]), finally mixing for time-changes of classical horocycle ﬂows was proved in a classical paper of B. Marcus [21] after a partial result of Kushnirenko [20]. As mentioned above mixing for time-changes of Heisenberg nilﬂows was investigated in [2]. Work of Avila, Forni, Ravotti and Ulcigrai [1] extends the methods developed there to prove mixing for a dense set of nontrivial time-change for any uniquely ergodic nilﬂows. Ravotti’s paper [25] was a first step in that direction. It should be remarked that there is an important diﬀerence between time-changes of linear toral ﬂows and parabolic ﬂows. In the parabolic case there are often countably many obstructions to triviality of time-changes for Diophantine ﬂows, while in the elliptic case of linear toral ﬂows non-trivial time-changes can exist only in the Liouvillean case. Estimates on the decay of correlations of smooth functions for non-homogenous elliptic or parabolic ﬂows are harder to come by and there are much fewer results in the literature. A classical paper of M. Ratner [23] established the decay rate for classical horocycle ﬂows (as well as geodesic ﬂows) on surfaces of constant negative curvature. This result was generalized to suﬃciently smooth time-changes of horocycle ﬂows by Forni and Ulcigrai [15], who also proved that the spectrum remains Lebesgue. Fayad [11] proved polynomial decay for a class of Kochergin-type ﬂows on the 2-torus and only recently, in [13], it was shown that there exists a class of Kochergin ﬂows on the 2-torus with countable Lebesgue spectrum. For locally Hamiltonian ﬂows with a saddle loop on surfaces (or, more generally, for suspension ﬂows over Interval Exchange Transformations with asymmetric logarithmic singularities of the roof function), Ravotti [24] was able to prove (logarithmic) estimates on decay of correlations. For these ﬂows mixing was proved by Khanin and Sinai [27] in the toral case, and by C. Ulcigrai [28] for suspension ﬂows over Interval Echange Transformations in the significant special case of roof functions with a single asymmetric logarithmic singularity. We expect non-trivial time-changes of nilﬂows to have polynomial decay of correlations. However, we are able to prove this result only for Heisenberg nilﬂows of bounded type. Our methods do not generalize to higher step nilﬂows, since they are based on the renormalization dynamics introduced by L. Flaminio and G. Forni in [14], which has no known generalization to the higher step case. We are also unable to decide whether the spectral measures of time-changes of Heisenberg nilﬂows are absolutely continuous with respect to Lebesgue. Indeed, the approach of [15], considerably refined in [13] , fails since the “stretching of Birkhoﬀ sums” is at best borderline square

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integrable (it grows at most as the square root of the time, up to logarithmic terms). In fact, our bounds on the decay of correlations are significantly worse than that, and we have no control on the size of the exponent. This follows from the general principle that proving “lower bounds” on Birkhoﬀ sums or ergodic integrals is much harder than proving “upper bounds”. In our case we are able to prove polynomial (power-law) lower bounds outside appropriate sublevel sets of Bufetov’s Hölder cocycles, which are asymptotic to ergodic integrals up to a well-controlled error. Polynomial estimates on the measure of such sublevel sets (for small parameter values) are derived from general results (see [3], [4]) on the measure of the sublevel sets of analytic functions. In fact, at the core of our argument we establish the real analyticity of the Bufetov cocycles along the leaves of a foliation transverse to the ﬂow. This outline is diﬀerent from the proof of mixing in [2]. In that paper the stretching of Birkhoﬀ sums for Heisenberg nilﬂows was derived from a more general result on the growth of Birkhoﬀ sums of functions which are not coboundaries with measurable transfer function, essentially based on a measurable Gottschalk-Hedlund theorem, and on the parabolic divergence of orbits. However, it is completely unclear whether it is possible to prove an eﬀective version of this argument. For this reason we have followed here a diﬀerent approach. Outline of the paper. — In Section 2 we give basic definitions on Heisenberg nilﬂows, the Heisenberg moduli space, renormalization ﬂow and Sobolev spaces. Finally we state two main theorems. In Section 3 we recall some basic results in representation theory of Heisenberg group. In Section 4 we compute the stretching of arcs (in the central direction) under the reparametrized ﬂow. Sections 5 and 6 are crucial since Bufetov functionals are constructed and their main properties are studied. In particular we prove the expected asymptotic formula according to which Bufetov fuctionals control orbital integrals. In Section 7 we derive from the asymptotic formula results on limit distributions of ergodic integrals for Diophantine Heisenberg niﬂows, following the method developed in [6], [5]. We also give an alternative proof, based on representation theory, of a substantial part of the work of Griﬃn and Marklof [17] on limit theorems for skew-shifts of the 2-torus, and generalize most of their conclusions to arbitrary smooth functions. Our approach also naturally gives results on the regularity of limit distributions, in particular their Hölder property (with exponent 1/2−) first derived for quadratic Weyl sums in the work of Cellarosi and Marklof [8]. In Section 8 we prove sharp square mean lower bounds for Bufetov functionals along the leaves of a one-dimensional foliation transverse to the ﬂow. Our aim is to prove measure estimates for the sub-level sets of Bufetov functionals, a key result in establishing the stretching of ergodic integrals outside sets of small measure. For that we prove in Section 9 that Bufetov functionals are real analytic on the leaves of a 2-dimensional foliation (the weak-stable foliation of the renormalization dynamics on the Heisenberg nilmanifold). We then recall in Section 10 a result of A. Brudnyi [3] on the measure of the sub-level sets of real analytic functions. These estimates depend on the so-called Chebyshev degree and valency of the function. We prove that under

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certain conditions the valency is uniformly bounded on every normal family of analytic functions. In Sections 11 and 12 we apply results of the previous section to finally prove measure estimates on the sets where Bufetov functionals are small (Lemmas 11.2 and 12.1). We conclude in Section 13 with an analysis of correlations and derive from results of Sections 11 and 12 (Corollary 11.3 and 12.2) the proof of our main Theorems 2.2 and 2.3. 2. Definitions In this section we will recall the definitions of Heisenberg nilﬂows, of moduli space of Heisenberg frames M and we state our main results. We then recall the definitions of the renormalization ﬂow gR on M and of the renormalization cocycle ρR on the Hilbert bundle of Sobolev distributions. For more details see [14] or [16]. We also introduce an extended renormalization ﬂow gˆR on an extended moduli space Mˆ , which is a tautogical bundle over M with fibers isomorphic to the Heisenberg nilmanifold. 2.1. Heisenberg Nilflows. — The (three-dimensional) Heisenberg group H is equal (up to isomorphisms) to the group of matrices 1 x z H := 0 1 y : x, y, z ∈ R . 0 0 1 Let Γ be a lattice in H. A Heisenberg manifold M is a quotient Γ\H. It is known that up to an automorphism of H any lattice has the form p 1 m K Γ = ΓK = 0 1 n : m, n, p ∈ Z , 0 0 1

where K is a positive integer. Notice that M has a probability measure vol, given by the projection of the Haar (bi-invariant) volume on H. Let W be any element of the Lie algebra h of H. The Heisenberg nilﬂow of generator W is the ﬂow given by φW t (x) = x exp(tW ) for x ∈ M. Notice that the ﬂow φW t on M preserves the (Haar) volume element vol. 2.2. Heisenberg moduli space. — A Heisenberg frame is any triple (X, Y, Z) of elements, generating the Heisenberg Lie algebra h, such that Z is a fixed generator of the center of the Lie algebra Z(h) and [X, Y ] = Z (of course we have [X, Z] = [Y, Z] = 0). One can for instance take 0 0 1 Z = Z0 = 0 0 0 . 0 0 0

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The set of all Heisenberg frames can be identified with the subgroup A of all automorphisms of H which are identity on the center. Notice that up to identification, 2 2 A is equal to the group SL(2, R) ⋉ R∗ , where SL(2, R) acts on R∗ via the standard linear contragredient action of SL(2, R) on the space R∗2 of linear forms on R2 . Let AΓ be the subgroup of A which stabilizes Γ, i.e., a ∈ AΓ if and only if a ∈ A and a(Γ) = Γ. We have the following definition Definition 2.1 ([14]). — The moduli space of the Heisenberg manifold M is the quotient space M = AΓ \A. It follows that AΓ is isomorphic to ΛΓ ⋉ (K −1 Z∗ )2 where ΛΓ is a finite index subgroup of SL(2, Z). Therefore the space M is a finite volume orbifold which fibers over the homogeneous space ΛΓ \SL(2, R) with fiber T2 (see Proposition 3.4. in [14]). The homogeneous space ΛΓ \SL(2, R) can be identified with the unit tangent bundle of the hyperbolic surface SΛ , quotient of the hyperbolic plane with respect to the action of ΛΓ by Möbius transformations. 2.3. Main results. — Let W s (M ) denote the Sobolev space of square-integrable functions on the compact manifold M , with square integrable weak derivatives up to order s > 0, and let W+s (M ) denote the subspace of strictly positive functions. For any function α ∈ W+s (M ), let φαX be the time-change of the Heisenberg R αX nilﬂow φX R with generator αX and let ωαX denote the measure preserved by φR . In this paper we prove the following results. We recall that a subset of a Baire space (for instance a complete metric space) is called generic, or residual, if it contains the intersection of countably many dense open subsets. Theorem 2.2. — There exists a set F ⊂ M of full Hausdorﬀ dimension and, for all s > 7/2, a generic set Ω ⊂ W s (M ) such that, for a = (X, Y, Z) ∈ F and α ∈ W+s (M ) with Z(1/α) ∈ Ω the following holds. Either 1/α is X-cohomologous to a constant, or there exist constants Ca,α > 0 and δa,α > 0 such that, for every pair of functions h ∈ W s (M ), g ∈ L2 (M, ωαX ) of zero average, such that Zg ∈ L2 (M, ωαX ), and for all t ∈ R, we have |hh ◦ φαX t , giL2 (M,ωαX ) |

7/2, a generic set Ω ⊂ W s (M ) such that for a = (X, Y, Z) ∈ F ′ and α ∈ W+s (M ) with Z(1/α) ∈ Ω the following holds. Either 1/α is X-cohomologous to a constant, or for every δ > 1/2 there exists a constant Ca,α,δ > 0 such that, for every pair of functions h ∈ W s (M ), g ∈ L2 (M, ωαX ) of zero average, such that Zg ∈ L2 (M, ωαX ), and for all t ∈ R, we have −

|hh ◦ φαX t , giL2 (M,ωαX ) | < Ca,α,δ (1 + |t|)

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1 1+logδ (1+|t|)

khks (kgk0 + kZgk0 ).

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2.4. The renormalization flow. — Following the notation from [14], for an element a ∈ A we denote a ¯ := AΓ a ∈ M . Let (X0 , Y0 , Z0 ) be a fixed Heisenberg triple. Let aR be the following one-parameter subgroup of A: at (X0 , Y0 , Z0 ) = (et X0 , e−t Y0 , Z0 ). Definition 2.4. — The renormalization ﬂow gR on M is defined by gt (¯ a) = a ¯at = AΓ aat . The renormalization ﬂow projects onto the ﬂow g¯R given by the diagonal subgroup of SL(2, R) acting on ΛΓ \SL(2, R) by right multiplication. Let SΓ denote the finitearea hyperbolic surface defined as a quotient of the hyperbolic plane with respect to the action of the finite index subgroup ΛΓ < SL(2, Z). Under the identification of the space ΛΓ \SL(2, R) with the unit tangent bundle of SΓ , the homogenous ﬂow g¯R is isomorphic to the geodesic ﬂow of the hyperbolic metric on SΓ . In what follows we will also consider the extended renormalization ﬂow gˆR on extended moduli space Mˆ , defined as follows. The extended moduli space is the quotient ˆ := AΓ \(A × M ), M with respect to the action of AΓ on A × M by multiplication on the left on A and by the embedding AΓ < Diﬀ(M ) on M . The extended renormalization ﬂow is the projection to the extended moduli space of the ﬂow (t, a, x) → (aat , x),

for all (t, a, x) ∈ R × A × M.

Note that Mˆ is a fiber bundle over M with fiber diﬀeomorphic to M and the extended renormalization ﬂow gˆR projects onto the renormalization ﬂow gR . 2.5. Sobolev spaces. — For (a, x) ∈ A × M denote Xa = a∗ (X0 ), Ya = a∗ (Y0 ). Let ∆a := −Xa2 − Ya2 − Z02 be the Laplace operator. For every s ∈ R and any C ∞ function f ∈ L2 (M, vol) we define kf ka,s = hf, (1 + ∆a )s f i1/2 . Let Was (M ) be the completion of C ∞ (M ) with the above norm. Let Wa−s (M ) denote the dual space. Following [14] again, we can define Ws := AΓ \(A × W s (M )) and W−s := AΓ \(A × W −s (M )), where A × W s (M ) (A × W −s (M )) denotes the Hilbert bundle over A, where k(a, f )ks = kf ka,s (k(a, D)k−s = kDka,−s ). We denote elements of Ws (respectively W−s ) by (a, f ) (respectively (a, D)). We now define the renormalization cocycle. Definition 2.5. — [14] The renormalization cocycle ρR is a ﬂow on Ws and W−s given by ρt (a, f ) = (aat , f ) and ρt (a, D) = (aat , D).

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3. Representation theory Recall that the right quasi regular representation U of the Heisenberg group H on L2 (M, vol) is given by U (g)F = F (R(g)), here R(g)(x) = xg. Notice that L2 (M ) := L2 (M, vol) =

M

Hn ,

n∈Z

where Hn = {f ∈ L2 (M ) : exp(tZ)f = exp(2πınKt)f } are closed and U -invariant. Moreover each Hn further splits into irreducible subrepresentations spaces. A complete classification of irreducible representations (with non-zero central parameter) is given by the Stone-Von Neumann theorem. Theorem 3.1 (Stone-Von Neumann). — For any a Heisenberg triple a = (X, Y, Z) and for the derived representation Drn of any irreducible unitary representation rn of the group H, of non-zero central parameter n ∈ Z \ {0}, on a Hilbert space H ⊂ Hn there exists a unique unitary operator UaH : H → L2 (R, λ) such that (UaH ◦ Drn (X) ◦ (UaH )−1 )(f )(u) = f ′ (u), (UaH ◦ Drn (Y ) ◦ (UaH )−1 )(f )(u) = 2πınKuf (u), (UaH ◦ Drn (Z) ◦ (UaH )−1 )(f )(u) = 2πınKf (u). Moreover, by Proposition 4.4 in [14] it follows that, for every irreducible component H ⊂ Hn with non-zero central parameter, the space of X-invariant distributions has dimension 1. Definition 3.2. — For all a = (X, Y, Z) ∈ A, let DaH be the unique distribution such that DaH corresponds by the unitary equivalence UaH (given by the Stone-Von Neumann theorem) to the Lebesgue measure on R. It follows from the Stone-Von Neumann theorem, as proved in [14], that the invariant distribution DaH belongs the dual Sobolev space Wa−s (M ), for all s > 1/2. Lemma 3.3. — For all a ∈ A and for any irreducible representation on a Hilbert space H of non-zero central parameter, we have the following scaling formula: DgHt (a) = e−t/2 DaH ,

for all t ∈ R.

Proof. — The argument was essentially already given in [14], Prop. 4.8 (but only for the scaling of Sobolev norms of invariant distributions). We reproduce it below for the convenience of the reader. Let Dr denote the derived representation (defined on the Heisenberg Lie algebra) of an irreducible unitary representation r on a Hilbert space H of non-zero central parameter. The unitary operator Ut : L2 (R, λ) → L2 (R, λ) given by (1)

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Ut (f ) = et/2 f (et u)

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intertwines a := (X, Y, Z) and gt (a) = gt (X, Y, Z) := (et X, e−t Y, Z), in the sense that Ut (UaH ◦ Dr(et X) ◦ (UaH )−1 )Ut−1 = UaH ◦ Dr(X) ◦ (UaH )−1 , (2)

Ut (UaH ◦ Dr(e−t Y ) ◦ (UaH )−1 )Ut−1 = UaH ◦ Dr(Y ) ◦ (UaH )−1 , Ut (UaH ◦ Dr(Z) ◦ (UaH )−1 )Ut−1 = UaH ◦ Dr(Z) ◦ (UaH )−1 .

It follows by the above definitions and by the uniqueness part of the Stone-Von Neumann theorem that UgHt (a) = Ut ◦ UaH , hence by the definition of DaH , we have DgHt (a) = LebR ◦ UgHt (a) = (LebR ◦ Ut ) ◦ UaH = e−t/2 (LebR ◦ UaH ) = e−t/2 DaH . This finishes the proof.

4. Stretching of curves Fix a Heisenberg triple (X, Y, Z). Let α > 0 denote a smooth time-change function (of the ﬂow φX R generated by X) and V = αX. We have the commutations [V, Y ] = [αX, Y ] = −(Y α)X + αZ = − [V, Z] = [αX, Z] = −

Yα V + αZ, α

Zα V. α

Let φVR denote the ﬂow generated by the vector field V on the nilmanifold M . We will compute the tangent vector of the push forwards of curves under the ﬂow φVR . Let W be any vector in the Lie algebra. We write (φVt )∗ (W ) = at V + bt Y + ct Z. By diﬀerentiation we derive dbt dct dat V + Y + Z = −V at V − V bt Y − bt [V, Y ] − V ct Z − ct [V, Z] dt dt dt Yα Zα = −(V at − bt − ct )V − V bt Y − (bt α + V ct )Z, α α or in other terms dat Yα Zα = −V at + bt + ct , dt α α dbt = −V bt , dt dct = −V ct − bt α. dt

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It follows that Yα Zα d (at ◦ φVt ) = (bt ◦ φVt ) ◦ φVt + (ct ◦ φVt ) ◦ φVt , dt α α d (bt ◦ φVt ) = 0, dt d (ct ◦ φVt ) = −(bt ◦ φVt )(α ◦ φVt ). dt At this point analogously to [2] we will look at the case W = Z (curves tangent to the central direction), hence (a0 , b0 , c0 ) = (0, 0, 1). We have Z t Zα V at ◦ φt = ◦ φVτ dτ, α 0 bt ◦ φVt = 0,

ct ◦ φVt = 1. In other terms Z (φVt )∗ (Z) = (

0

−t

Zα ◦ φVτ dτ )V + Z. α

To understand the above orbital integrals we write them in terms of the nilﬂow φX R. We have relations Z t Z t −1 X τV (x, t) = α ◦ φr (x)dr and τX (x, t) = α ◦ φVr (x)dr. 0

0

By these formulas and by change of variables, we have Z τX (x,t) Z t f V ( ) ◦ φX f ◦ φτ (x)dτ = (3) r (x)dr. α 0 0 We will therefore investigate time averages Z t (4) f ◦ φX r (x)dr, 0

for functions f of zero average with respect to the Haar volume on M .

5. Construction of the functionals Let γ be any rectifiable curve. The curve γ defines a current, that is, a continuous functional on 1-forms. We recall that the renormalization cocycle ρR acts on currents (see Definition 2.5). Fix an irreducible representation H ⊂ L2 (M ) contained in the eigenspace of eigenvalue 2πıKn ∈ 2πıKZ \ {0} of the action of the center of the Heisenberg group on M and fix a Heisenberg triple a = (X, Y, Z).

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There exists a unique basic current BaH (of degree 2 and dimension 1) associated to DaH . Let ω denote the invariant volume form (with unit total volume) and let ηX = ıX ω. The basic current BaH is defined as BaH = DaH ηX . The above formula means that for every 1-form α we have BaH (α) = DaH (

ηX ∧ α ). ω

The current BaH is basic in the sense that ıX BaH = LX BaH = 0. The basic current BaH belongs to a dual Sobolev space of currents, defined as folˆ Yˆ , Z} ˆ denote the frame of the cotangent bundle T ∗ M dual to the lows. Let {X, frame {X, Y, Z} of T M . We can write any smooth 1-form α as follows: ˆ + αY Yˆ + αZ Z. ˆ α = αX X It follows that the space of smooth 1-forms is identified to the product (C ∞ (M ))3 by the isomorphism α → (αX , αY , αZ ). By the above isomorphism, it is also possible to define Sobolev spaces of currents Ωsa (M ) ≡ Was (M )3

for s ≥ 0,

s ∗ and their dual spaces Ω−s a (M ) := (Ωa (M )) of currents. By the Sobolev embedding theorem, for every rectifiable arc γ, the current H −s γ ∈ Ω−s a (M ) for all s > 3/2. Since Da ∈ Wa (M ) for all s > 1/2 (see Section 3 or H −s [14], Prop. 4.4), all basic currents Ba ∈ Ωa (M ) for all s > 1/2. Notice that the Hilbert structure of Ωsa (M ) and Ω−s a (M ) depends on a = (X, Y, Z).

Let H ⊂ L2 (M ) denote irreducible component H of non-zero central parameter. −s Let Ω−s a (H) denote the subspace of Ωa (M ) of forms with functional coeﬃcients in −s −s −s −s the subspace Wa (H) := Wa (M ) ∩ H and let Π−s H : Ωa (M ) → Ωa (H) denote the −s orthogonal projection. Let BH,a : Ω−s (M ) → C denote the orthogonal component map in the direction of the 1-dimensional space of basic currents CBaH ⊂ Ω−s a (H). We recall that for any automorphism a ∈ A we denote a ¯ the equivalence class AΓ a of a in the quotient AΓ \A (see Section 2). We introduce below a crucial Diophantine condition. Let δM : M → R+ ∪ {0} be the distance function (which projects to distance function of the hyperbolic metric of curvature −1 on the hyperbolic surface SΓ , defined in Section 2.4) from the base point id ∈ M . For any L > 0, let DC(L) denote the set of a ∈ M such that Z +∞ t 1 exp[ δM (g−t (a)) − ]dt ≤ L. (5) 4 2 0 SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2020

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Let DC denote the union of the sets DC(L) over all L > 0. By Kinchine’s theorem, or the logarithmic law of geodesics, it follows that, for almost all a ∈ M , we have lim sup t→+∞

δM (g−t (a)) = 1. log t

It follows immediately that the set DC ⊂ M has full Haar volume. The Bufetov functionals are defined for all Diophantine a ∈ DC as follows : Lemma 5.1. — Let a ∈ DC(L). For s > 7/2 and every rectifiable arc γ on M , the limit t βˆH (a, γ) = lim e− 2 B−s H,g−t (a) (γ). t→+∞

exists, is finite and defines a finitely-additive measure on the space of rectifiable arcs. There exists a constant Cs′′ > 0 such that the following estimate holds: Z Z −s H ′′ ˆ ˆ ˆ |ΠH (γ) − βH (a, γ)Ba |a,−s ≤ Cs (1 + L) 1 + |Y | + |Z| . γ

γ

For every L > 0, the function βˆH (·, γ) is continuous on DC(L) ⊂ M . Proof. — For simplicity of notation (since H is fixed) we suppress the dependence on H ⊂ L2 (M ). We will use subscript a, t to denote any dependence on g−t (a) = (Xt , Yt , Z), for example Π−s (γ) := Π−s H (γ),

−s

Ba,t

:= B−s H,g−t (a) ,

Ba,t := BgH−t (a) .

Since the inner product of the Sobolev space Wa−s (M ) depends smoothly on a ∈ A, the projection map B−s a,t , which can be written in terms of the scalar product with a (M ), depends smoothly on t ∈ R. normalized invariant distribution in the space Wa−s t For every t ∈ R we have the following splitting: Π−s (γ) = B−s a,t (γ)Ba,t + Ra,t . Moreover this splitting is orthogonal in Ω−s g−t (a) (M ). By construction, for any h ∈ R we have −s

Ba,t+h (γ)Ba,t+h

+ Ra,t+h = B−s a,t (γ)Ba,t + Ra,t .

Since by Lemma 3.3 we have Bt+h = e−h/2 Bt it follows that −s

Ba,t+h (γ)

−s = eh/2 B−s a,t (γ) + Ba,t+h (Ra,t ).

By diﬀerentiating the expression at h = 0, we get d −s 1 −s d −s B (γ) = Ba,t (γ) + [ B (Ra,t )]h=0 . dt a,t 2 dh a,t+h The derivative on the right hand side of the above equation can be computed in representation. Let h·, ·it denote the inner product in the Hilbert space Ω−s g−t (a) . From

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the intertwining Formulas (2) it follows that −s

Ba,t+h (Ra,t )

Ba,t+h ia,t+h |Ba,t+h |2t+h Ba,t+h ◦ U−h = hRa,t ◦ U−h , ia,t |Ba,t+h |2a,t+h = hRa,t ,

= hRa,t ◦ U−h ,

Ba,t ia,t = B−s a,t (Ra,t ◦ U−h ). |Ba,t |2a,t

Now by the definition of the intertwining operators Uh in Formula (1) it follows that, in the sense of distributions, d 1 1 (Ra,t ◦ U−h ) = −Ra,t ◦ (Xt + ) ◦ U−h = [(Xt − )Ra,t ] ◦ U−h . dh 2 2 We conclude that d 1 −s [ B−s a,t+h (Ra,t )]h=0 = −Ba,t ((Xt − )Ra,t ). dh 2 We finally claim that the following estimate holds: for all rectifiable curve γ on M and all t ∈ R, we have 1 |B−s a,t ((Xt − )Ra,t (γ))| ≤ |Ra,t (γ)|g−t (a),−(s+1) 2 Z Z 1 ˆ ˆ ≤ Cs exp[ δM (g−t (a))] 1 + |Y | + |Z| . 4 γ γ

The above remainder estimate will be proved in the lemma below. We get therefore a scalar diﬀerential equation d −s 1 −s B (γ) = Ba,t (γ) + Ra,t (γ) dt a,t 2 with a bounded non-negative function Ra,t (γ) satisfying the estimate Z Z 1 ˆ ˆ |Y | + |Z| . (6) Ra,t (γ) ≤ Cs exp[ δM (g−t (a))] 1 + 4 γ γ

The solution of the above diﬀerential equation is Z t t τ −s −s Ba,t (γ) = e 2 Ba,0 (γ) + e− 2 Ra,τ (γ)dτ . 0

It follows that, under the Diophantine assumption that a ¯ ∈ DC(L), t ˆ lim e− 2 B−s a,t (γ) = βH (a, γ)

t→+∞

exists. Since by definition the distributions B−s a,t (γ) and Ra,t (γ) depend continuously on (a, t) ∈ A × R, by the Diophantine bound (6), which implies the convergence of the integral Z +∞ τ e− 2 Ra,τ (γ)dτ, 0

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it follows that the complex number βˆH (a, γ) = B−s a,0 (γ) +

Z

+∞

τ

e− 2 Ra,τ (γ)dτ

0

depends continuously on a ∈ DC(L). Moreover, we have Z +∞ −s H − τ2 ˆ ΠH (γ) − βH (a, γ)Ba = R0 − e Ra,τ dτ BaH 0

and by the above bound on the remainder terms Ra,t and by the Diophantine condition on a ∈ A, it follows that Z Z ˆH (a, γ)B H |a,−s ≤ C ′′ (1 + L) 1 + |Yˆ | + |Z| ˆ . |Π−s (γ) − β a s H,a γ

γ

The argument is thus concluded, up to the above claim on the remainder bounds.

We then prove the claim on the remainder bounds. Lemma 5.2. — There exists Cs > 0 such that, for all t ≥ 0 and all rectifiable arcs γ we have Z Z 1 ˆ ˆ |Rt (γ)|a,−s ≤ Cs exp[ δM (gt (a))] 1 + |Y | + |Z| . 4 γ γ Proof. — Let α be any 1-form. For simplicity, for all t ∈ R, we let gt (a) = (Xt , Yt , Z). We can write ˆ ˆ t + αY Yˆt + αZ Z. α = αX X t

t

Let us assume now that α is supported on a single irreducible component H. Since ˆ ˆ t ∧ Yˆt ∧ Z, ˆ hence ηX = Yˆt ∧ Z, ω=X t

we have the identity ˆ = DtH (αX ). BtH (α) = DtH (αXt ηXt ∧ Xˆt + αYt ηXt ∧ Yˆt + αZ ηXt ∧ Z) t Let us then assume that BtH (α) = DtH (αXt ) = 0. It follows that αXt is a coboundary for the cohomological equation, that is, there exists a smooth function u on M (with a loss of Sobolev regularity of 1+) such that αXt = Xt u. By the Sobolev embedding theorem, for any s > r + 1 > 7/2, there exists a constant Br (gt (a)) we have |u|C 0 (M ) + |Yt u|C 0 (M ) + |Zu|C 0 (M ) ≤ Br (gt (a))|u|gt (a),r ≤ Br (gt (a))|αXt |gt (a),s . By [14], Corollary 3.11, there exists a universal constant Cr > 0 such that the best Sobolev constant Br (a) is bounded above as follows: 1 (7) Br (a) ≤ Cr exp[ δM (a)]. 4

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We remark that we can write ˆ t + (Yt u) Yˆt + (Zu) Z, ˆ du = (Xt u) X hence, by the Sobolev embedding theorem and the fact that Yˆt = e−t Yˆ , for all s > 7/2, we have Z Z ˆ ˆ | α| = du + (αYt − Yt u)Yt + (αZ − Zu)Z γ γ Z Z 1 ˆ . ≤ Cs |α|gt (a),s exp[ δM (gt (a))] 1 + |Yˆ | + |Z| 4 γ γ Let us now consider an arbitrary smooth 1-form α on M supported on the irreducible component H. There exists an orthogonal decomposition s α = κα + κ⊥ α ∈ Ωgt (a) (M )

such that κα ∈ Ker(BtH ). Since Rt (γ) ∈ {BtH }⊥ ∈ Ω−s gt (a) (M ), and s H ⊥ H ⊥ κ⊥ α ∈ [Ker(Bt )] = Ker({Bt } ) ∈ Ωgt (a) (M ),

it follows that Rt (γ)(α) = Rt (γ)(κα ) =

Z

κα ,

γ

hence the above estimate leads to the bound Z Z 1 ˆ ˆ |Rt (γ)(α)| ≤ Cs |κα |gt (a),s exp[ δM (gt (a))] 1 + |Y | + |Z| . 4 γ γ

The conclusion immediately follows by the orthogonality of the decomposition. 6. Main properties of the functionals By definition, the Bufetov functional has the additive property, that is, for all rectifiable arcs γ1 and γ2 on M , by linearity of projections and limits, we have (8) βˆH (a, γ1 + γ2 ) = βˆH (a, γ1 ) + βˆH (a, γ2 ) ; it has the scaling property, that is, for every rectifiable arc γ and t > 0, we have (9) βˆH (gt (a), γ) = e−t/2 βˆH (a, γ), The Bufetov functional also has the following invariance property: for all rectifiable arc γ and for all τ > 0, (10) βˆH (a, (φY )∗ (γ)) = βˆH (a, γ). τ

The above invariance property follows from the fact that, by the Sobolev embedding theorem, we have δM (g−t (a)) |φYτ (γ)) − γ|g−t (a),−s ≤ Cτ Bs (g−t (a)) ≤ Cτ exp[ ]. 4 In fact, the current φYτ (γ) − γ is equal, up to two bounded orbit arcs of the ﬂow φYR , to the boundary of a 2-dimensional current ∆, which has uniformly bounded

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2-dimensional area with respect to the frame g−t (a) := (Xt , Yt , Z). This follows from the fact that ∆ can be taken to be a surface tangent to the ﬂow φYR , that is, ∆ = {φt (p)|(t, p) ∈ [0, τ ] × γ}, ˆ t ∧ Yˆt = X ˆ ∧ Yˆ , but also hence, not only X Z Z τ Z −t ˆ ˆ ˆ |Z|]dσ [ |Yt ∧ Zt | = e

and

φY σ (γ)

0

∆

Z

ˆ t ∧ Zˆt | = 0. |X

∆

The above invariance property then follows immediately from the Diophantine condition and from the existence of the Bufetov functional. Finally, the Bufetov functional has the following vanishing property: for every rectifiable arc γ tangent to the central-stable foliation of the extended renormalization ˆ , that is, the foliation generated ﬂow gˆR on the fibers of the extended moduli space M by the integrable distribution {Y, Z} above each point a = (X, Y, Z), we have βˆH (a, γ) = 0.

(11)

The vanishing property is a direct consequence of the definition, as the length of any arc γ tangent to the central-stable foliation is uniformly bounded along the backward orbit of the renormalization ﬂow: Z Z Z Z Z ˆ t | = 0, ˆ |X |Yˆt | = e−t |Yˆ |, |Zˆt | = |Z|. γ

γ

γ

γ

γ

Let now γTX (x) denote the arc of orbit of the ﬂow φX R , that is, γTX (x) = {φX t (x)|t ∈ [0, T ]}, and, for all (x, T ) ∈ M × R+ , let βH (a, x, T ) := βˆH a, γTX (x) .

From the additive property we derive the following cocycle property: for all (x, T1 , T2 ) ∈ M × R × R we have βH (a, x, T1 + T2 ) = βH (a, x, T1 ) + βH (a, φX T1 (x), T2 ). Moreover, for α ∈ AΓ , we have (12)

βH (αa, α(x), T ) = βH (a, x, T ),

which means that the function βH (·, ·, T ) is a well defined function on the extended moduli space Mˆ . By Lemma 5.1 for any smooth function f which belongs to a single irreducible component H, we have Z T X H (13) f ◦ φt (x)dt − βH (a, x, T )Da (f ) 0 ˆ − βˆH a, γTX (x) Ba (f X)| ˆ ≤ C ′′ (1 + L)|f |a,s . = |hγTX (x), f Xi s ASTÉRISQUE 416

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By the scaling property of the Bufetov functional and (12) we derive the following scaling identities: for all (x, T ) ∈ M × R+ and t ∈ R, we have (14)

βH (a, x, T t) = T 1/2 βH (glog T (a), x, t) = T 1/2 βH (ˆ glog T ([a, x]AΓ ), t).

We have therefore derived the following asymptotic formula for ergodic averages. For every x ∈ M and t, T > 0 we have Z tT H X 1/2 (15) f ◦ φτ (x)dτ − T βH (ˆ glog T [a, x]AΓ , t) Da (f ) ≤ Cs′′ (1 + L)|f |a,s . 0

As an immediate consequence of the above asymptotic property we can derive the following orthogonality property : for all a ∈ DC and for all t ∈ R+ , for any smooth function f ∈ H we have βH (a, ·, t) ∈ H ⊂ L2 (M ).

(16)

In fact, by Equations (13) and (14) we have 1

DaH (f )βH (a, x, t) = lim

βH (g− log T (a), x, T t) Z Tt Xg 1 (a) f ◦ φτ − log T (x)dτ. 1/2

T →+∞ T 1/2

(17)

= lim

T →+∞

T

0

It follows that βH (a, ·, t) ∈ H as a pointwise uniform limit of (normalized) ergodic integrals functions of any given function f ∈ H. It can also be proved (as in the work of Bufetov [6], or [5]) that, for almost all a ∈ A, the Bufetov functionals are Hölder for exponent 1/2− along the orbits of the ﬂow φX R . In fact, the Hölder property for the Bufetov functionals on rectifiable arcs takes the following form. For every (a, T ) ∈ A × R+ , we introduce the excursion function Z log T δM (glog T −t (¯ a)) t dt − exp EM (a, T ) := 4 2 0 (18) Z log T δM (gt (¯ a)) t = T −1/2 exp dt. + 4 2 0 Lemma 6.1. — For all Diophantine a ∈ DC(L) the Bufetov functional βˆH has the following Hölder property. There exists a constant C > 0 such that whenever a ∈ DC(L) we have, for any rectifiable arc γ, Z Z Z Z Z ˆ ˆ ˆ ˆ ˆ ˆ 1/2 . |βH (a, γ)| ≤ C 1 + L + |Z| + ( |X|)( |Y |) + EM (a, |X|) ( |X|) γ

γ

γ

γ

γ

Proof. — The Hölder property is an immediate consequence of the scaling property and of uniform bounds for the Bufetov functionals on arcs of bounded length. By the scaling property, for all t ∈ R, βˆH (a, γ) = et/2 βˆH (gt a, γ).

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Let γRws be a rectifiable arc tangent to the weak-stable distribution {Xa , Z}. Let ˆ a | so that by definition we have T = γ |X Z ˆg |X | = 1. log T a γws

By the definition of the set DC(L) in Formula (5) we have that whenever a ∈ DC(L) then glog T (a) ∈ DC(LT ) with (19)

LT ≤ LT −1/2 + EM (a, T ),

hence, by Lemma 5.1 and by the scaling property, we derive Z −1/2 ˆ ˆ |βH (a, γws )| ≤ C 1 + |Z| + LT + EM (a, T ) T 1/2 . γws

The above estimate immediately implies the result for rectifiable arcs tangent to the distribution {Xa , Z}. For a general rectifiable arc γ, let γws denote any projection of the arc γ on a leaf of the distribution {Xa , Z} by holonomy along the orbits of the ﬂow φYRa . By the commutation relations we have Z Z Z Z Z Z Z Z ˆ ˆ + ( |X|)( ˆ ≤ ˆ ˆ = |Yˆ |), |Z| |Z| |Yˆ |, |Yˆ | = |X|, |X| γws

γ

γws

γ

γws

γ

γ

γ

and by the invariance property

βˆH (a, γ) = βˆH (a, γws ). The estimate in the general case then follows. From the above property we can easily derive the Hölder property for the Hölder cocycles β(a, x, T ) with respect to x ∈ M along the orbits of the ﬂow φX R or with respect to the time T ∈ R. We conclude this section by constructing Bufetov functionals of smooth functions. By the theory of unitary representations we can write L2 (M ) =

M n∈Z

Hn :=

M M µ(n)

Hi,n ,

n∈Z i=1

where Hn ⊂ L2 (M ) denotes the eigenspace of the action of the center with eigenvalue Lµ(n) (central parameter) n ∈ Z. For n ∈ Z\{0}, we have a decomposition Hn = i=1 Hi,n into irreducible representation spaces Hi,n , for i = 1, . . . , µ(n). It follows from the Howe-Richardson multiplicity formula (or by a direct calculation of irreducible representations) that µ(n) = |n|, for all n 6= 0. The space H0 , with central parameter n = 0, coincides with the pull-back of the space L2 (T2 ) under the canonical projection of M onto the torus T2 , hence by the Fourier decomposition on the torus, it is a direct sum of countably many irreducible components, each isomorphic to a complex line. For functions in H0 , the ergodic theory of Heisenberg nilﬂows reduces to that of linear ﬂows on T2 . In particular, for

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Diophantine linear ﬂows on T2 , all suﬃciently smooth functions of zero average are coboundaries with bounded transfer function, so that their ergodic integrals are uniformly bounded. For this reason, we will restrict to consider functions in the space H0⊥ of functions orthogonal to all toral functions, and its Sobolev subspaces. Let L20 (M ) := H0⊥ ⊂ L2 (M ) and W0s (M ) := W s (M ) ∩ L20 (M ) for all s ≥ 0. The space W0s (M ) has an orthogonal decomposition µ(n)

M

M

W0s (M ) =

W s (Hi,n )

n∈Z\{0} i=1

into the Sobolev subspaces W s (Hi,n ) ⊂ Hi,n of the irreducible components. For every n 6= 0 and every i ∈ {1, . . . , µ(n)}, let Dai,n denote the unique normalized Xa -invariant distribution supported on W −s (Hi,n ) and β i,n = βHi,n the associated Bufetov functional. Since any function f ∈ W0s (M ) has a decomposition µ(n)

f=

X

X

fi,n ,

n∈Z\{0} i=1

where each component fi,n ∈ W s (Hi,n ), we can define the Bufetov cocycle associated to the function f ∈ W0s (M ) as the sum µ(n)

(20)

f

β (a, x, T ) :=

X

X

Dai,n (f )β i,n (a, x, T ).

n∈Z\{0} i=1

For general functions f ∈ W s (M ), the Bufetov cocycle can be defined as the sum of the Bufetov cocycle of the orthogonal projection of f onto W0s (M ) and of the average of the function. The following result is a version for Bufetov functionals of the bound on ergodic integrals proved in [16], Lemma 1.4.9. Lemma 6.2. — For all Diophantine a ∈ DC(L) and for all function f ∈ W0s (M ) for s > 2, the Bufetov functional β f is defined by a uniformly convergent series, hence the function βaf is a Hölder function on M × R. In addition there exists a constant Cs > 0 such that whenever a ∈ DC(L) we have, for all (x, t, T ) ∈ M × (R+ )2 , |β f (a, x, tT )| ≤ Cs L + T 1/2 (1 + t + EM (a, T )) |f |a,s .

Proof. — It follows from Lemma 5.1 that there exists a constant C > 0 such that whenever a ∈ DC(L) then (21)

|β i,n (a, x, t)| ≤ C(1 + L + t),

for all (x, t) ∈ M × R+ .

By the exact scaling property in Formula (14), we have β i,n (a, x, tT ) = T 1/2 β i,n (glog T (a), x, t) = T 1/2 β i,n (ˆ glog T ([a, x]AΓ ), t).

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By Formula 19 and by the bound in Formula (21) we have that for all (x, t, T ) ∈ M × (R+ )2 .

|β i,n (glog T (a), x, t)| ≤ C(1 + LT + t), It follows that for all r > 1/2 we have µ(n)

|β f (a, x, tT )| ≤ Cr T 1/2 (1 + LT + t)

X

X

|fi,n |a,r

n∈Z\{0} i=1

µ(n)

≤ Cr T 1/2 (1 + LT + t)(

X

′

(1 + n2 )−r )−1/2 (

X

[

X

′

|(1 − Z 2 )r /2 fi,n |a,r ]2 )1/2 .

n∈Z\{0} i=1

n∈Z\{0}

We therefore conclude that for all r > 3/2 there exists a constant Cr,r′ > 0 such that ′

|β f (a, x, tT )| ≤ Cr,r′ T 1/2 (1 + LT + t)|f |a,r+r′ , hence, in view of Formula (19), the statement is proved. It follows from the convergence result given in Lemma 6.2 that all properties of the Bufetov functionals βH , each associated to a single irreducible component, extend to the Bufetov functionals β f for any f ∈ W0s (M ) (s > 2). In particular, for every Diophantine a = (X, Y, Z) ∈ DC the function βaf on M × R is a Hölder cocycle for the ﬂow φX R , which satisfies the scaling property (14), that is, + 2 for all (x, t, T ) ∈ M × (R ) , we have (22)

β f (a, x, T t) = T 1/2 β f (glog T (a), x, t) = T 1/2 β f (ˆ glog T ([a, x]AΓ ), t).

Finally from the asymptotic Formula (15) on each irreducible component we derive the following asymptotic result: Theorem 6.3. — For all s > 7/2 there exists a constant Cs > 0 such that for all a = (X, Y, Z) ∈ DC(L), for all f ∈ W0s (M ) and for all (x, T ) ∈ M × R+ , we have Z T X f (23) f ◦ φt (x)dt − β (a, x, T ) ≤ Cs (1 + L)|f |a,s . 0

All the results of this paper, about limit distributions and about decay of correlations for time-changes, are derived from the above asymptotic result. 7. Limit distributions

In this section we derive some corollaries on limit distributions of ergodic integrals, which generalize results of J. Griﬃn and J. Marklof [17] to arbitrary smooth functions and recover the Hölder property of limit distributions, proved by F. Cellarosi and J. Marklof [8]. Let H ⊂ L2 (M ) denote an irreducible component contained in the eigenspace of eigenvalue 2πıKn ∈ 2πıKZ \ {0} of the action of the center of the Heisenberg group on M , hence orthogonal to the subspace H0 ⊂ L2 (M ) of toral functions.

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Lemma 7.1. — There exists a continuous AΓ -invariant, that is, modular, function θH : A → H ⊂ L2 (M ) such that for any f ∈ Was (H) with s > 1/2, we have

1 Z T

H lim 1/2 f ◦ φX = 0. t (·)dt − θH (glog T (a)) Da (f ) T →+∞ T

2 0 L (M )

The family {θH (a)|a ∈ A} has (positive) constant norm in L2 (M ): there exists a constant C > 0 such that for all a ∈ A we have kθH (a)kL2 (M ) = C. Proof. — We refer the reader to [14] or [16] for background on the application of this theory to the cohomological equation of Heisenberg nilﬂows. By the Stone-Von Neumann theorem, the space H := Hz is unitarily equivalent to the space L2 (R, du) and such a unitary equivalence can be chosen so that the group φX R is represented as the group of translations on the real line and the group φYR is a group of the form {eızt Id}. In other terms we have the infinitesimal representation X→

d , du

Y → ızu.

The space of smooth vectors (for an irreducible unitary representation of central parameter z 6= 0) is the Schwartz space S (R) and the space of translation invariant tempered distribution on the real line is given by all scalar multiples of the Lebesgue measure. It follows that invariant distributions supported on H for the ﬂow φX R are represented as scalar multiples of the Lebesgue measure. Finally the Sobolev space Was (H) is represented as the space S s (R) of functions f ∈ L2 (R, du) such that Z d2 |(1 + 2 + z 2 u2 )s/2 fˆ(u)|2 du < +∞. du R

The statement is equivalent to the claim that there exists θ(a) ∈ L2 (R, du) such that for all f ∈ S s (R) we have

1 Z T

f (u + t)dt − θ (glog T (a)) Leb(f ) lim 1/2 = 0. T →+∞ T

2 0 L (R,du)

An equivalent formulation, by the standard Fourier transform on R:

1 Z T

eıtˆu fˆ(ˆ u)dt − θˆ (glog T (a)) fˆ(0) = 0. lim 1/2 T →+∞ T

2 0 L (R,dˆ u)

Let χ ∈ L2 (R, dˆ u) denote the function defined as χ(ˆ u) =

eıˆu − 1 , ıˆ u

for all u ˆ ∈ R.

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ˆ Let θ(a)(ˆ u) := χ(ˆ u), for all u ˆ ∈ R. Let us compute θ (glog T (a)). By definition glog T (a) = (T X, T −1 Y, Z), hence the induced representation d dˆ u is intertwined to the normalized representation by the unitary equivalence UT : L2 (R) → L2 (R) defined as T −1 Y → T −1 z

T X → ıT u ˆ,

UT (f )(ˆ u) = T 1/2 f (T u ˆ),

for all u ∈ R.

In fact, we have d d = UT−1 ◦ (T −1 z ) ◦ UT . dˆ u dˆ u It follows that, for all a ∈ A and all T > 0, ˆ) ◦ UT , ıˆ u = UT−1 ◦ (ıT u

z

θˆ (glog T (a)) (ˆ u) = UT (χ)(ˆ u) = T 1/2 χ(T u ˆ). The function θH (a) ∈ H is uniquely defined in representation as the unique Fourier ˆ anti-transform θ(a) ∈ L2 (R) of the function θ(a) ∈ L2 (R). By its definition the function θH is modular, that is, it is invariant under the action of the lattice AΓ on A. As a consequence, it induces a well-defined function on the moduli space M = AΓ \A. By unitary equivalence

ıˆu

e − 1 ˆ

:= C > 0. kθH (a)kH = kθ(a)kL2 (R) = kθ(a)kL2 (R) = ıˆ u L2 (R,dˆu) By integration we have Z T eıˆut fˆ(ˆ u)dt = T χ(T u ˆ)fˆ(u) 0 u) − fˆ(0) + T 1/2 θˆ (glog T (a)) (ˆ u)fˆ(0). = T χ(T u ˆ) fˆ(ˆ

The claim is therefore reduced to the following statement u) − fˆ(0) kL2 (R,dˆu) = 0. lim sup kT 1/2 χ(T u ˆ) fˆ(ˆ T →+∞

Since by hypothesis f ∈ S s (R) with s > 1/2, the function fˆ ∈ C 0 (R) and it is bounded, hence v u) − fˆ(0) kL2 (R,dˆu) = kχ(v) fˆ( ) − fˆ(0) kL2 (R,dv) → 0, kT 1/2 χ(T u ˆ) fˆ(ˆ T by the dominated convergence theorem. The continuous dependence of the the function θH (a) ∈ H on a ∈ A follows from the continuous dependence of the intertwining operator Ua : L2 (R) → L2 (R), which conjugates a representation of the form Xa → α

ASTÉRISQUE 416

d + ıγzu + v, du

Ya → β

d + ıδzu + w du

TIME-CHANGES OF HEISENBERG NILFLOWS

275

to the standard representation, on the parameters ! α β ∈ SL(2, R) and (v, w) ∈ R2 , γ δ of the automorphism a ∈ A. The intertwining operator Ua can be computed explicitly. The details are left to the reader. Corollary 7.2. — There exists a constant C > 0 such that, for any s > 1/2, for any a = (X, Y, Z) ∈ A and for any f ∈ Was (H), we have

Z

T 1

X = C|DaH (f )|. f ◦ φ dt lim

t T →+∞ T 1/2 0

2 L (M )

The above statement strengthens Lemma 15 of [2]. In fact, there the authors proved a slightly weaker statement for linear skew-shifts of the torus T2 , that is for maps of the form Tρ,σ (y, z) = (y + ρ, z + y + σ) , for all (y, z) ∈ T2 .

As explained in [2], the minimal ﬂow φX R has constant return time to a transverse torus with return map a linear skew-shift, that is, a map of the form Tρ,σ for constants ρ ∈ R \ Q and σ ∈ R. From Corollary 7.2 we derive the following limit result for the L2 norm of Bufetov functionals. Corollary 7.3. — There exists a constant C > 0 such that for all irreducible components H ⊂ L2 (M ) and a ∈ DC, we have lim

1

T →+∞ T 1/2

kβH (a, ·, T )kL2 (M ) = C.

Proof. — By the normalization of the invariant distributions in the Sobolev space W s (M ) for any given s > 1/2 the following holds. For all irreducible components H and all a ∈ A , there exists a (non-unique) function faH ∈ W s (H) such that DaH (faH ) = kfaH ks = 1. For all a ∈ DC(L), and for s > 7/2, we derive from the asymptotic formula in Theorem 6.3 that Z T H X fa ◦ φt (x)dt − βH (a, x, T ) ≤ Cs (1 + L). 0 The L2 estimates in the statement then follow from Corollary 7.2.

A relation between the Bufetov functionals and the above theta functionals is established below.

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Corollary 7.4. — For any irreducible component H ⊂ L2 (M ) of non-zero central parameter the following holds. For any L > 0 and for any gR -invariant probability measure µ supported on DC(L) ⊂ M , we have βH (a, ·, 1) = θH (a) ∈ L2 (M ),

for µ-almost all a ¯ ∈ M.

Proof. — By Theorem 6.3 and Lemma 7.1 we immediately derive that there exists a constant Cµ > 0 such that for all a ∈ supp(µ) ⊂ DC(L), for all T > 0 we have (24)

kβH (glog T (a), ·, 1) − θH (glog T (a))kL2 (M ) ≤

Cµ . T 1/2

By Luzin’s theorem, for any δ > 0 there exists a compact subset E (δ) ⊂ M such that we have the measure bound µ(M \ E (δ)) < δ and such that the function βH (a, ·, 1) ∈ L2 (M ) depends continuously on a ¯ ∈ E (δ). By Poincaré recurrence there is a full measure subset E ′ (δ) ⊂ E (δ) of gR -recurrent points. In particular, for every a0 )} ⊂ E (δ) with a ¯0 ∈ E ′ (δ) there is a diverging sequence (tn ) such that {gtn (¯ a0 ) = a ¯0 . lim gtn (¯

n→∞

By the continuity of the function θH : M → L2 (M ), by the continuity at a ¯0 of the function βH (¯ a, ·, 1) ∈ L2 (M ) on the set E (δ), and by the above L2 estimate, we have a0 ))kL2 (M ) = 0. a0 ), ·, 1) − θH (gtn (¯ kβH (¯ a0 , ·, 1) − θH (¯ a0 )kL2 (M ) = lim kβH (gtn (¯ n→∞

We have thus proved that βH (¯ a, ·, 1) = θH (a) ∈ L2 (M ) for all a ∈ E ′ (δ). It follows that the set where the latter identity fails has µ-measure less than any δ > 0, hence the identity holds for µ-almost all a ¯ ∈ A, as stated. Corollary 7.5. — There exists a constant C > 0 such that for all irreducible components H ⊂ L2 (M ) the following holds. For any L > 0 and for any gR -invariant probability measure µ supported on DC(L) ⊂ M , and for all T > 0 we have kβH (a, ·, T )kL2 (M ) = CT 1/2

for µ-almost all a ¯ ∈ M.

Remark 7.6. — Since by Lemma 7.1 the function θH : A → L2 (M ) is continuous and approximates ergodic integrals, it is possible to write it (at least for the first irreducible component) in terms of the theta function Θχ introduced in [17], as both functions are continuous, modular, and provide a similar asymptotic formula for ergodic averages (sums) (see formula (13) in [17]). It follows that Bufetov functional βH (for the first irreducible component) essentially coincide with the function Θχ almost everywhere on the moduli space. The main advantage of the approach of Cellarosi and Marklof [8] is that it provides an explicit Diophantine condition which describes the set where the function Θχ is absolutely convergent and 1/2-Hölder (see [8], Theorem 3.10).

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For general smooth functions we proceed as in the previous section. Since any function f ∈ W0s (M ), orthogonal to the subspace H0 of toral functions, has a decomposition µ(n)

f=

X

X

fi,n

n∈Z\{0} i=1

where each component fi,n ∈ W s (Hi,n ), we can define the functional θf : A → L2 (M ) associated to the function f ∈ W s (M ) as the weighted sum over all functionals θi,n := θHi,n associated to the irreducible representations Hi,n : µ(n)

(25)

θf (a) :=

X

X

Dai,n (f )θi,n (a).

n∈Z\{0} i=1

Lemma 7.7. — For all a ∈ A and for all function f ∈ W0s (M ) for s > 2, the functional θf is defined by a convergent series, hence the function θf (a) is an L2 function on M and θf : A → L2 (M ) is a continuous function. From Lemma 7.1 we derive a general asymptotic theorem: Theorem 7.8. — For all a ∈ A and for all f ∈ W0s (M ) with s > 2 we have

1 Z T

X f lim 1/2 = 0. f ◦ φt dt − θ (glog T (a)) T →+∞ T

2 0 L (M )

We leave to the reader the derivation of Lemma 7.7 and Theorem 7.8.

From Theorem 7.8 we can derive most of a result of Griﬃn and Marklof [17] on limit distribution of theta sums in the related context of Heisenberg nilﬂows. We also prove the result established later by Cellarosi and Marklof [8] (see in particular [8], § 3) that limit distributions are the distributions of Hölder function of exponent equal to 1/2−. Our results have the advantage of holding for all suﬃciently smooth functions, while the work of Griﬃn and Marklof [17], and Cellarosi and Marklof [8] holds only for a single (toral) character. However, they are much less explicit and less detailed, especially as far as the the behavior at infinity in the moduli space is concerned, and in particular we have no results on limit distributions for time sequences corresponding to escape in the cusp of the moduli space. The following result summarizes our results on limit distributions of ergodic averages of suﬃciently smooth functions for Heisenberg nilﬂows: Theorem 7.9. — Let a = (X, Y, Z) ∈ A and let (Tn ) be any sequence such that a) = a∞ ∈ M . lim glog Tn (¯

n→+∞

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For every function f ∈ W0s (M ) with s > 7/2 which is not a coboundary, the limit distribution of the family of random variables Z Tn 1 f ◦ φX ETn (f ) := 1/2 t (·)dt 0 Tn exists and is equal to the distribution of the function θf (a∞ ) ∈ L2 (M ). In particular, if a∞ ∈ DC belongs to the ω-limit set of any gR -orbit on the set DC of Diophantine points and is a continuity point of the Bufetov functional β f on DC ⊂ A, then θf (a∞ ) is almost everywhere equal to a bounded 12 -Hölder function on M , hence in particular the limit distribution has compact support. Proof. — Since a∞ ∈ M , the existence and characterization of the limits follows from Lemma 7.7 and Theorem 7.8. The regularity statement follows from Corollary 7.4. With the exception of the Hölder regularity statement, equivalent results were proved in [17] for linear toral skew-shift and for function cohomologous to the principal toral character. For such functions, the authors also investigated the case when the limit point a∞ = +∞ and proved that in that case the limit distribution is the Dirac delta δ0 at 0 ∈ R. The Hölder regularity property was proved in [8]. 8. Square mean lower bounds In this section we prove transverse square mean lower bound for ergodic integrals. Let Γ = ΓK < H be a lattice and M = Γ\H a Heisenberg nilmanifold. Let {X0 , Y0 , Z0 } be a fixed Heisenberg frame, integral with respect to the lattice Γ, in the sense that p Γ = {exp(mX0 + nY0 + Z0 )|(m, n, p) ∈ Z3 }. K 0 Let T2Γ denote the 2-dimensional torus transverse to the ﬂow φX R , defined as follows: T2Γ := {Γ exp(yY0 + zZ0 )|(y, z) ∈ R2 }. 0 Since the torus T2Γ is transverse to the nilﬂow φX on M , it is transverse to all R X nilﬂows φR such that the component hX, X0 i of the vector field X in the direction of the vector field X0 , with respect to the orthonormal basis {X0 , Y0 , Z0 } ⊂ h, does not vanish. For all a = (X, Y, Z), let 1 ta := |hX, X0 i|

2 denote the return time of the ﬂow φX R to the transverse tori TΓ . We will prove bounds for the square mean of ergodic integrals along the leaves of the foliation of the torus T2Γ into circles transverse to the central direction:

{ξ exp(yY0 )|y ∈ T}ξ∈T2Γ . (Since by definition exp(Y0 ) ∈ Γ, for any ξ ∈ T2Γ the map y → ξ exp(yY0 ) is welldefined over T := R/Z and thus defined a closed curve embedded in T2Γ .)

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Lemma 8.1. — There exists a constant C > 0 such that for all a = (X, Y, Z) and for every irreducible component H of central parameter n 6= 0, there exist a function fH ∈ C ∞ (M ) ∩ H such that H |fH |L∞ (M ) ≤ Ct−1 a |Da (fH )|,

s H −1 |fH |a,s ≤ Ct−1 (1 + n2 )s/2 , a |Da (fH )| 1 + ta kY k

and such that, for all x ∈ T2Γ and T ∈ Zta , we have Z Z ! T X Y0 fH (φs (φy (x)))ds dy ≤ 1, T 0

Z

T T

Y0 = |DaH (fH )|( )1/2 . fH (φX

s (φy (x)))ds

2

0 ta L (T,dy)

2

′

In addition, whenever H ⊥ H ⊂ L (M ) the functions Z T Z T Y0 Y0 X fH ′ (φX fH (φs (φy (x)))ds and s (φy (x)))ds 0

0

2

are orthogonal in L (T, dy). Proof. — We recall that whenever hX, X0 i = 6 0 the return map of the ﬂow φX R to the 2 transverse torus TΓ is a linear skew-shift, that is, a map of the form Tρ,σ (y, z) = (y + ρ, z + y + σ)

on R/Z × R/K −1 Z,

for constants ρ, σ ∈ R. The operator Ia : L2 (M ) → L2 (T2Γ ) defined as Z ta f → Ia (f ) := f ◦ φX for all f ∈ L2 (M ) t (·)dt, 0

2

is a surjective linear map of L (M ) onto L2 (T2Γ ) with right inverse Raχ defined as follows: let χ ∈ C0∞ (0, 1) denote any function of integral equal to one and, for any F ∈ L2 (T2Γ ), let Raχ (F ) ∈ L2 (M ) be the function uniquely defined by the identity −1 Raχ (F )(φX t (x)) = ta χ(t/ta )F (x),

for all (x, t) ∈ T2Γ × [0, ta ].

It is immediate from the definition that there exists a constant Cχ > 0 such that −1 s |Raχ (F )|a,s ≤ Cχ t−1 a (1 + ta kY k) kF kW s (T2Γ ) .

As explained in [2], § 5, the space L2 (T2Γ ) can be decomposed as a direct sum of irreducible subspaces invariant under the action of the map Tρ,σ on L2 (T2Γ ) by composition. The subspace of functions with non-zero central character can be split as a direct sum of components H(m,n) with (m, n) ∈ ZK|n| × Z \ {0}. These spaces are defined as follows. Let {em,n |(m, n) ∈ Z2 } denote the basis of characters of T2Γ , that is the basis given by the functions em,n (y, z) := exp[2πı(my + nKz)],

for all (y, z) ∈ T2Γ .

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As described in [2], § 5, the functions F ∈ Hm,n are characterized by a Fourier expansion of the form X F = Fj em+jn,n . j∈Z

A generator of the space of invariant distributions is the distribution D(m,n) such that j

D(m,n) (em+jn,n ) = e−2πı[(ρm+σKn)j+ρKn(2)] . It follows that |D(m,n) (em+jn,n )| = 1. For any irreducible component H of central parameter n 6= 0, there exists m ∈ Z|n| such that the operator Ia : L2 (M ) → L2 (T2Γ ) maps the space H onto the space Hm,n , hence the function fH := Raχ (em,n ) ∈ C ∞ (M ) ∩ H has the property that |DaH (fH )| = |D(m,n) (em,n )| = 1, and Z ta fH ◦ φX for all x ∈ T2Γ . t (x)dt = em,n (x), 0

In addition the following estimates hold: |fH |L∞ (M ) ≤ Cχ t−1 a

s 2 2 s/2 −1 |fH |a,s ≤ Cχ t−1 . a (1 + ta kY k) (1 + K n )

and

Since for every n ∈ Z \ {0} and m ∈ ZK|n| , the system j ]}j∈Z ⊂ L2 (T, dy) {exp[2πı(m(y + jρ) + Kn(z + j(σ + y) + ρ 2 is orthonormal, we have the identity

J−1 2

X j

)] exp[2πı(m(y + jρ) + Kn(z + j(σ + y) + ρ

2

2

j=0

= J.

L (T,dy)

In addition, by an immediate computation Z J−1 X j )] dy ∈ {0, 1}. exp[2πı(m(y + jρ) + Kn(z + j(σ + y) + ρ 2 T j=0 By the above formula it follows that, whenever T /ta ∈ Z we have Z T

]−1 Z [ tTX a X Y0 k fH ◦ φt (φy (x))dt dy = | ( em,n ◦ Tρ,σ (y, z))dy| ≤ 1, T k=0 0

T

[ ta ]−1

1/2

X

T

j X Y0

= fH ◦ φt (φy (x))dt . em,n ◦ Tρ,σ =

2 t a

j=0

2 L (T,dy)

Z

Z

T

0

T

!

The argument is therefore complete.

ASTÉRISQUE 416

L (T,dy)

TIME-CHANGES OF HEISENBERG NILFLOWS

281

For p ≥ 1, let ℓp denote the (standard) Banach space of complex sequences c := (ci,n ) such that 1/p X X µ(n) |ci,n |p < +∞. |c|ℓp := n∈Z\{0} i=1

For any infinite dimensional vector c := (ci,n ) ∈ ℓ2 , let βc denote the Bufetov functional defined as follows µ(n)

(26)

X

βc =

X

ci,n β i,n .

n∈Z\{0} i=1

It follows from the orthogonality property and from Corollary 7.3 that the function βc (a, ·, T ) ∈ L2 (M ) for all (a, T ) ∈ A × R+ . In fact, µ(n)

X

kβc (a, ·, T )k2L2 (M ) =

X

|ci,n |2 kβ i,n (a, ·, T )k2L2 (M ) ≤ C 2 |c|2ℓ2 T.

n∈Z\{0} i=1

For any vector c := (ci,n ) ∈ ℓ2 , let |c|s denote the norm defined as µ(n)

|c|2s

=

X

X

(1 + K 2 n2 )s |ci,n |2 .

n∈Z\{0} i=1

We remark that since the multiplicity µ(n) = n, for all n ∈ Z \ {0}, it follows from the Hölder inequality that for all s > 1 there exists a constant CK,s > 0 such that |c|ℓ1 ≤ CK,s |c|s . For any a = (X, Y, Z) ∈ A such that hX, X0 i = 6 0 or, equivalently, such that the return time ta > 0 is finite, and for any x ∈ M , let Sa,x denote the transverse cylinder defined as follows: Sa,x

= {x exp (y ′ Y + z ′ Z)|(y ′ , z ′ ) ∈ [0, t−1 a ) × T} ⊂ M.

Let Φa,x : T2Γ → Sa,x denote the maps defined as follows. For any ξ ∈ T2Γ , let ξ ′ ∈ Sa,x denote the first intersection of the orbit {φX t (ξ)|t ≥ 0} with the transverse cylinder Sa,x . By definition there exists t(ξ) ≥ 0 such that (27)

ξ ′ = Φa,x (ξ) = φX t(ξ) (ξ),

for all ξ ∈ T2Γ .

Let (y, z) and (y ′ , z ′ ) denote the coordinates, respectively on T2Γ and Sa,x , given by the exponential map, as follows (y, z) → ξy,z := Γ exp(yY0 + zZ) ∈ T2Γ and (y ′ , z ′ ) → ξy′ ′ ,z′ := x exp(y ′ Y + z ′ Z). Let X = αX0 + βY0 + vZ and Y = γX0 + δY0 + wZ with α 6= 0 and αδ − βγ = 1. Let x = Γ exp(yx Y0 + zx Z) exp(tx X0 ) with (yx , zx ) ∈ T × R/KZ and tx ∈ [0, 1). By the Baker-Campbell-Hausdorﬀ formula, we derive that (28)

|t(ξ)| = |δtx + γ(y − yx )| ≤ kY k

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and that the map Φa,x : T2Γ → Sa,x is given by formulas of the following form: there exists a polynomial P (a, x, y) of total degree 4, quadratic with respect to each of the variables (a, x, y) ∈ M × M × R, such that ( y ′ = α(y − yx ) + βtx , (29) Φa,x (y, z) =: z ′ = z + P (a, x, y). In particular, the map Φa,x is invertible and such that Φ∗a,x (dy ′ ∧ dz ′ ) = α dy ∧ dz = t−1 a dy ∧ dz. The cylinders Sa,x are foliated by images of the circles {ξ exp(yY0 )|y ∈ T} ⊂ T2Γ under the map Φa,x . Lemma 8.2. — For any s > 7/2 there exists a constant Cs > 0 such that, for all Diophantine a ∈ DC(L), for all c ∈ ℓ2 , for all z ∈ T and all T > 0, we have Z −1 s βc (a, Φa,x (ξy,z ), T )dy ≤ Cs (ta + t−1 a )(1 + L)(1 + ta kY k) |c|s , T 1/2 T −1 s |c|0 ≤ Cs (ta + t−1 kβc (a, Φa,x (ξy,z ), T )kL2 (T,dy) − a )(1 + L)(1 + ta kY k) |c|s . ta

Proof. — By Lemma 8.1, for every n 6= 0 and every i ∈ {1, . . . , µ(n)}, there exists a function fi,n ∈ C ∞ (Hi,n ) with Di,n (fi,n ) = 1 such that, for all T ∈ Zta we have, for all ξ ∈ T2Γ the identities Z Z ! T Y0 fi,n (φX s (φy (ξ)))ds dy ∈ {0, 1}, T 0

Z

T T

Y0 = ( )1/2 . fi,n (φX (φ (ξ)))ds

s y

2

0 ta L (T,dy)

In addition, the integrals

Z

T Y0 fi,n (φX s (φy (ξ)))ds

0

form an orthogonal system of functions in L2 (T, dy). Let then µ(n)

(30)

fc :=

X

X

ci,n f i,n .

n∈Z\{0} i=1

By construction we have βc = β fc . It is immediate that Z Z ! T X Y0 fc (φs (φy (ξ)))ds dy ≤ |c|ℓ1 . T 0 ASTÉRISQUE 416

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By orthogonality we also have

Z

T

X Y0 fc (φs (φy (ξ)))ds

0

=(

L2 (T,dy)

283

T 1/2 ) |c|0 . ta

From the estimates on the functions fi,n stated in Lemma 8.1 we derive the bounds s |fc |L∞ (M ) ≤ C|c|ℓ1 and |fc |a,s ≤ Ct−1 1 + t−1 |c|s . a a kY k

From this estimate it follows that, for every z ∈ T and for all T > 0, we have

Z

Z T

T

X X ≤ 2|fc |L∞ (M ) kY k. fc (φs (ξy,z ))ds fc (φs ◦ Φa,x (ξy,z ))ds −

2

0 0 L (T,dy)

Finally let Ta := ta ([T /ta ] + 1) ∈ Zta . We have

Z Z Ta

T

X X fc (φs (ξy,z ))ds − fc (φs (ξy,z ))ds

0

0

≤ ta |fc |L∞ (M ) .

L2 (T,dy)

We have therefore derived that, for some constant C ′ > 0 and for all T > 0, the following bounds hold: Z Z ! T ′ −1 fc (φX s (Φa,x (ξy,z ))ds dy ≤ C ta 1 + ta kY k |c|ℓ1 , T 0 Z 1/2 T T X fc (φs (Φa,x (ξy,z ))dskL2 (T,dy) − |c|0 ≤ C ′ ta 1 + t−1 k a kY k |c|ℓ1 . 0 ta

By the asymptotic property of Theorem 6.3, for all s > 7/2 there exists a constant Cs > 0 such that we have the uniform estimate Z T X f f ◦ φt (x)dt − β (a, x, T ) ≤ Cs (1 + L)|f |a,s . 0 Since, by the above bounds on the function fc , there exists constant Cs′ > 0 such that −1 −1 s −1 ′ C ′ ta (1 + t−1 a kY k)|c|ℓ1 + Cs ta (1 + L)|fc |a,s ≤ Cs (ta + ta )(1 + L)(1 + ta kY k) |c|s ,

we arrive at the estimates claimed in the statement.

9. Analyticity of the functionals In this section we will prove that, for all a = (X, Y, Z) ∈ DC and for all T ∈ R, the Bufetov functionals βH (a, ·, T ) are real analytic along the foliation tangent to the integrable distribution {Y, Z}. This result is crucial in deriving measure estimates for the level sets of the Bufetov functionals and for our results on decay of correlations of time changes.

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By the orthogonality property, for every T > 0, the Bufetov cocycle belongs to a single irreducible component H (with central parameter n ∈ Z \ {0}), hence in particular (or from its definition), for all (x, T ) ∈ M × R and for all z ∈ R, 2πıKnz βH (a, φZ βH (a, x, T ). z (x), T ) = e

(31)

Let γ : [0, T ] → M a C 1 (or piece-wise C 1 parametrized path). For every y ∈ R \ {0} we define γyZ (s) = φZ for all s ∈ [0, T ]. ys (γ(s)), Lemma 9.1. — The following formula holds: βˆH (a, γyZ ) = e2πınKT y βˆH (a, γ) − 2πınKy

Z

T

e2πınKys βˆH (a, γ|[0,s] )ds.

0

Proof. — Let a = (X, Y, Z) and let α be a 1-form supported on a single irreducible component H. As above we have the decomposition ˆ ˆ + αY Yˆ + αZ Z. α = αX X Let us compute the pairing of the current γtZ with the 1-form α on M . By definition the tangent vector of the path γyZ is given by the formula dγyZ dγ = (φZ ) ◦ γyZ + yZ ◦ γyZ . ys )∗ ( ds ds It follows that the pairing is given by the formula Z T dγ Z Z Z Z α (φys )∗ ( ) ◦ γy (s) + y(ıZ α ◦ γy )(s) ds. hγy , αi = ds 0

Since Z belongs to the center of the Heisenberg Lie algebra h and the coeﬃcients αX , αY and αZ of α are eigenfunctions of the subgroup generated by Z of eigenvalue 2πınK ∈ 2πıKZ \ {0}, it follows that Z T dγ 2πınKys Z Z e α( (s)) + y(ıZ α ◦ γy )(s) ds. hγy , αi = ds 0

Integration by parts gives Z T Z T dγ dγ e2πınKys α( (s))ds = e2πınKT y α( (s))ds ds ds 0 0 Z Z T e2πınKys − 2πınKy 0

s

α(

0

hence we have the formula

hγyZ , αi = e2πınKT y hγ, αi − 2πınKy

Z

T

e2πınKys hγ|[0,s] , αids + y

0

Since the ﬂow gR is identity on the center Z, it follows that Z T − 2t ıZ (ρ−t )∗ α ◦ γyZ (s)ds = 0, lim e t→+∞

ASTÉRISQUE 416

0

dγ (r))dr ds, dr Z

0

T

(ıZ α ◦ γyZ )(s)ds.

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hence the stated formula follows by the definition of the Bufetov functional and by the linearity of the projection operators. Lemma 9.2. — The following formula holds: βH (a, φYy (x), T ) = e−2πınKT y βH (a, x, T ) + 2πınKy

T

Z

e−2πınKsy βH (a, x, s)ds.

0

Proof. — We have the following commutation identities: x exp(sX) exp(yY ) = x exp(yY ) exp(sX) exp(ysZ). X Z Let then := φX s (x) for all s ∈ [0, T ]. By definition the symbol [γT ]y denotes the path given by the formula

γTX (s)

Z X [γTX ]Z y (s) := φys (γT (s)),

for all s ∈ [0, T ].

It then follows by the definitions that φYy (γTX (x)) = [γTX (φYy (x))]Z y. By the invariance property of the Bufetov functional and by Lemma 9.1 we have βH (a, x, T ) = βˆH a, φYy (γTX (x)) = e2πınKT y βˆH a, γTX (φYy (x)) Z T e2πınKsy βˆH (a, γTX (φYy (x))|[0,s] )ds − 2πınKy 0

= e2πınKT y βH (a, φYy (x), T ) − 2πınKy

Z

T

e2πınKsy βH (a, φYy (x), s)ds.

0

The statement is an immediate consequence of the above formula. It follows from Lemma 9.2 and Formula (31) that the Bufetov functional is real analytic (real and complex part are real analytic) on every leaf of the foliation tangent to the integrable distribution {Y, Z}. For any R > 0 let us introduce the analytic norm defined for all c ∈ ℓ2 as kckω,R :=

X µ(n) X

enR |ci,n |.

n6=0 i=1 2

Let ΩR denote the subspace of c ∈ ℓ such that kckω,R is finite. Lemma 9.3. — For any c ∈ ΩR , any a ∈ DC(L) and T > 0, the functions defined as βc (a, φYy φZ z (x), T ),

for all (y, z) ∈ R × T,

extends to a holomorphic function in the domain R }. 2πK The following bounds hold: for any R′ < R there exists a constant CR,R′ > 0 such that, for all (y, z) ∈ DR′ ,T we have 1/2 |βc (a, φYy φZ (1 + EM (a, T )) (1 + K|Im(y)|T ). z (x), T )| ≤ CR,R′ kckω,R L + T

(32)

DR,T := {(y, z) ∈ C × C/Z||Im(y)|T + |Im(z)|

0 ΩR defined as (η) follows. A sequence c ∈ Ω∞ if there exists Cη > 0 such that, for all R > 0 we have βH (a, φYy

X µ(n) X

(33)

φZ z (x), T )

n|ci,n |enR ≤ Cη eR

2−η

.

n6=0 i=1 (η)

The set Ω∞ is dense in ℓ2 , since it contains all finitely supported sequences. (η)

Lemma 9.4. — For any c ∈ Ω∞ , any a ∈ DC(L) and T > 0, the functions defined as βc (a, φYy φZ for all (y, z) ∈ R × T, z (x), T ), extends to a holomorphic function in the domain C × C/Z. In addition, there exists a constant Cη > 0 such that, for all T > 0 and for all (y, z) ∈ C × C/Z, we have 1/2 (1 + EM (a, T )) |βc (a, φYy φZ z (x), T )| ≤ Cη L + T × (1 + 2πK|Im(y)|T ) exp[(|Im(y)|T + |Im(z)|)2−η ].

(η)

Proof. — Since Ω∞ ⊂ ΩR , for all R > 0, it follows already from Lemma 9.3 that the function βc (a, φYy φZ z (x), T ) extends to a holomorphic function on C × C/Z. As in the proof of Lemma 9.3, by Lemma 6.2 for (y, z) ∈ C × C/Z we have ′ 1/2 |βc (a, φYy φZ (x), T )| ≤ C L + T (1 + E (a, T )) M z × (1 + 2πK|Im(y)|T )

X µ(n) X

n6=0 i=1

ASTÉRISQUE 416

n|ci,n |e2πK(|Im(y)|T +|Im(z)|)n .

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(η)

Since by assumption c ∈ Ω∞ , the stated estimates is proved. In the Sections 11 and 12 we will use Lemmas 9.3 and 9.4 to get uniform measure estimates on sets where the Bufetov functional is small. This is possible thanks to results on the measure of small sets for analytic functions (see [4], [3]).

10. Bounds on the valency For convenience of the reader we recall a result of A. Brudnyi on the measure of level sets of analytic functions. For any r > 1, let Or denote the space of holomorphic functions on the ball BC (0, r) ⊂ Cn . Let BR (0, 1) := BC (0, 1) ∩ Rn denote the real euclidean unit ball. Theorem 10.1 ([3], Thm. 1.9). — For any f ∈ Or there is a constant d := d(f, r) > 0 such that for any convex set D ⊂ BR (0, 1), for any measurable subset ω ⊂ D d 4nLeb(D) sup |f |. sup |f | ≤ Leb(ω) ω D The best constant d in the above theorem is called the Chebyshev degree, denoted by df (r), of the function f ∈ Or in BC (0, 1). The Chebyshev degree can be estimated by the valency of the function. We recall the definition of the valency. A holomorphic function f defined in a disk is called p-valent if it assumes no value more than p-times there (counting multiplicities). We also say that f is 0-valent if it is a constant. For any t ∈ [1, r), let Lt denote the set of one-dimensional complex aﬃne spaces L ⊂ Cn such that L ∩ BC (0, t) 6= ∅. Definition 10.2 ([3], Def. 1.6). — Let f ∈ Or . The number vf (t) := sup {valency of f |L ∩ BC (0, t)} L∈Lt

is called the valency of f in BC (0, t). By [3], Prop. 1.7, for any f ∈ Or and any t ∈ [1, r) the valency vf (t) is finite and there is a constant c := c(r) > 0 such that 1+r (34) df (r) ≤ cvf . 2 In this section we prove the following result. Lemma 10.3. — Let R > r > 1. For any normal family F ⊂ OR such that no function in F is constant along a one-dimensional complex line, we have sup vf (r) < +∞. f ∈F

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Proof. — We argue by contradiction. If the statement does not hold, then there exists a sequence of functions (fk ) ⊂ F , a sequence of aﬃne one-dimensional subspaces (Lk ) and a bounded sequence of complex numbers such that #fk−1 {wk } ∩ Lk ∩ BC (0, r) → +∞. By compactness, up to passing to subsequence we can assume that fk → f uniformly on all compact subset of the ball BC (0, R), that Lk → L, a one-dimensional aﬃne complex line such that L ∩ BC (0, r) 6= ∅, in the Hausdorﬀ topology, and that wk → w ∈ C. By hypothesis f |L is non-constant, hence we can assume that fk |Lk is also non-constant for all k ∈ N. Since for any r′ > r the valency vf (r′ ) of the function f on BC (0, r′ ) is finite we have that #f −1 {w} ∩ L ∩ BC (0, r′ ) < +∞. Let f −1 {w} = {p1 , . . . , pv } ⊂ L ∩ BC (0, r′ ). Let ǫ > 0 be chosen so that BC (pi , ǫ) ∩ BC (pj , ǫ) = ∅ and f |∂BC (pi , ǫ) 6= 0 for all i, j ∈ {1, . . . , v}. Since Lk → L there exists a sequence of aﬃne holomorphic maps Ak : Cn → Cn such that Ak → Id uniformly on compact sets and Ak (L) = Lk for all k ∈ N. By uniform convergence we have that for n ∈ N suﬃciently large all the solutions z ∈ L ∩ BC (0, r′ ) of the equation fk ◦ Ak (z) − wk = 0 are contained in the union of the balls BC (pi , ǫ) ∩ L. For all k ∈ N, let φk := (fk ◦ Ak )|L, and let φ = f |L. The sequence of holomorphic functions (φk ) converges to φ uniformly on compact sets of L ∩ BC (0, R). Since L ≈ C by the residue theorem we have that Z φ′ (z)k 1 dz. #(fk ◦ Ak )−1 (wk ) ∩ BC (pi , ǫ) ∩ L = 2πı ∂BC (pi ,ǫ) φk (z) − wk By uniform convergence on compact sets it follows that Z Z 1 φ′k (z) φ′ (z) 1 dz → dz, 2πı ∂BC (pi ,ǫ) φk (z) − wk 2πı ∂BC (pi ,ǫ) φ(z) − w hence we have #(fk ◦ Ak )−1 (wk ) ∩ BC (pi , ǫ) ∩ L → #f −1 (w) ∩ BC (pi , ǫ) ∩ L. We conclude that for k suﬃciently large #fk−1 (wk ) ∩ Lk ∩ BC (0, r) ≤ #f −1 (w) ∩ L ∩ BC (0, r′ ) < +∞. Since by assumption the LHS in the above inequality diverges, we have reached a contradiction. The argument is concluded. In one complex dimension we prove a quantitative bound on the valency. Lemma 10.4. — For any R > r > 3t > 3, there exists a constant Cr,t > 0 such that the following holds. For any non-constant holomorphic function of one complex variable f ∈ OR , let Mf (r) denote the maximum modulus of f on the closed ball BC (0, r) ⊂ C

ASTÉRISQUE 416

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and let Of (t) its oscillation in the ball BC (0, t). The valency vf (t) of the function f in the ball BC (0, t) satisfies the following estimate Mf (r) vf (t) ≤ Cr,t log 4 . Of (t) Proof. — Since there exists a single complex one-dimensional aﬃne space L ⊂ C, it suﬃces to estimate the valency of the function f on BC (0, t), that is, the number of solutions z ∈ BC (0, t) of equations of the form f (z) = w. By definition, the above equation has solutions only if |w| ≤ Mf (r). Let fw ∈ OR denote the holomorphic function fw (z) = f (z) − w. By definition, the maximum modulus of fw on the closed ball BC (0, r) ⊂ C is at most 2Mf (r). Let w ∈ f (BC (0, t) and let zw ∈ BC (0, t) be any point such that |fw (zw )| = |fw (z) − w| ≥ Of (t)/2. Let {z1 , . . . , zν } ⊂ BC (zw , 2t) \ {zw } denote the (non-empty) set of zeros of the function fw in BC (zw , 2t) listed with their multiplicities. Since BC (0, t) ⊂ BC (zw , 2t) it follows that the number of solution of the equation fw (z) = 0 in BC (0, t) is at most ν ∈ N. Let us define gw (z) = fw (zw + z)

ν Y

k=1

(1 −

zw + z −1 ) , zk

z ∈ BC (0, R − t).

By definition the function gw in holomorphic in BC (0, R − t). By the maximum modulus principle r−t 1 Of (t) ≤ |fw (zw )| = |gw (0)| ≤ max |gw (z)| ≤ 2Mf (r)( − 1)−ν , 2 3t |z|=r−t which immediately implies, by taking logarithms, r−t 4Mf (r) ν ≤ log( . − 1) log 3t Of (t) The statement is therefore proved.

11. Measure estimates: the bounded-type case Finally, we derive a bound on the valency, hence on the Chebyshev degree of the holomorphic extensions of Diophantine Bufetov functionals, uniform over compact invariant subset of the moduli space. Lemma 11.1. — Let L > 0 and let B ⊂ DC(L) be a bounded subset. Given R > 0, for all c ∈ ΩR and T > 0, let F (c, T ) denote the family of real analytic functions of the variable y ∈ [0, 1) defined as follows: F

(c, T ) := {βc (a, Φa,x (ξy,z ), T ) |(a, x, z) ∈ B × M × T}.

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There exist TB > 0 and ρB > 0, such that for every (R, T ) such that R/T ≥ ρB and T ≥ TB , and for all c ∈ ΩR \ {0}, we have sup

vf < +∞.

f ∈F (c,T )

Proof. — Since B ⊂ M is bounded, we get 0 < tmin = inf ta ≤ sup ta = tmax < +∞. B B a∈B

a∈B

For any a ∈ B and x ∈ M , the map Φa,x : [0, 1) × T → [0, t−1 a ) × T introduced in ˆ Formula (29) extends to a complex analytic diﬀeomorphism Φa,x : C × C/Z → C × Z. By Lemma 9.3, it follows that the real analytic function βc (a, Φa,x (ξy,z ), T ), for (y, z) ∈ [0, 1) × T, ˆ −1 extends to a holomorphic function on the domain Φ a,x (DR,T ) ⊂ C × C/Z. In particular, for every z ∈ T the function βc (a, Φa,x (ξy,z ), T ), for y ∈ [0, 1), extends to a holomorphic function defined on a strip Ha,x,R,T := {y ∈ C | | Im(y)| < ha,x,R,T }. Moreover by the boundedness of the set B ⊂ M it follows that inf

(a,x)∈B×M

ha,x,R,T := hR,T > 0.

In fact, the function ha,x,R,T and its lower bound hR,T can be computed explicitly from the Formula (29) for the polynomial map Φa,x and from definition of the domain DR,T in Formula (32). In particular, it follows that for every r > 1 there exists ρB ≫ 1 such that, for every R, T such that R/T ≥ ρB , then for every (a, x, z) ∈ B × M × T we have that, as a function of y ∈ [0, 1], βc (a, Φa,x (ξy,z ), T ) ∈ Or . It then follows from Lemma 9.3 that the family F (c, T ) is uniformly bounded and hence normal. Moreover, from Lemma 8.2 it follows that for suﬃciently large T > 0 no sequence from F (c, T ) can converge to a constant function. The statement finally follows from Lemma 10.3. We can finally derive crucial measure estimates on Bufetov functionals. Lemma 11.2. — Let a ∈ DC be such that the forward orbit {gt (¯ a)}t∈R+ is contained in a compact subset of M . There exist R > 0, T0 > 0 and C > 0, δ > 0 such that, for every c ∈ ΩR \ {0}, T ≥ T0 and for every ǫ > 0, we have (35)

ASTÉRISQUE 416

vol({x ∈ M ||βc (a, x, T )| ≤ ǫT 1/2 }) ≤ Cǫδ .

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Proof. — Let R > 0 and T0 > 0 be chosen so that the conclusions of Lemma 11.1 hold and let c ∈ ΩR . By the scaling property of Bufetov functionals βc (a, x, T ) = (T /T0 )1/2 βc (glog(T /T0 ) (a), x, T0 ). Since a ∈ DC and the gR -forward orbit {gt (¯ a)}t∈R+ is contained in a compact set, there exists L > 0 such that gt (a) ∈ DC(L) for all t > 0. Since the volume on M is invariant under the action AΓ , it is enough to estimate (uniformly over (a, x) ∈ B ×M for any given bounded subset B ⊂ DC(L) such that {gt (a)}t∈R+ ⊂ AΓ \B), for any ǫ > 0, the volume vol({x ∈ M ||βc (a, x, T0 )| ≤ ǫ}). By Fubini’s theorem it is enough to estimate, uniformly over (a, x, z) ∈ B × M × T, Leb ({y ∈ [0, 1]||βc (a, Φa,x (ξy,z ), T0 )| ≤ ǫ}) . Let δ −1 := c(r) supf ∈F (c,T0 ) vf ( 1+r 2 ) < +∞. Since by Lemma 8.2 we have sup |βc (a, Φa,x (ξy,z ), T0 )| > 0.

inf

(a,x,z)∈B×M ×T y∈[0,1]

It follows from Theorem 10.1 for D = BR (0, 1) and ω := {y ∈ [0, 1]||βc (a, Φa,x (ξy,z ), T0 )| ≤ ǫ}, and by the bound in Formula (34) for the Chebychev degree, that the following estimate holds: there exists a constant C > 0 such that, for all (a, x, z) ∈ B × M × T and for all ǫ > 0, we have Leb ({y ∈ [0, 1]||βc (a, Φa,x (ξy,z ), T0 )| ≤ ǫ) ≤ Cǫδ . The statement then follows by the Fubini theorem. Corollary 11.3. — Let a = (X, Y, Z0 ) be as in Lemma 11.2. There exist R > 0, T0 > 0 and C > 0, δ > 0 such that, for every c ∈ ΩR \ {0}, T ≥ T0 and for every ǫ > 0, we have ! Z T X 1/2 fc (φt x)dt| ≤ ǫT } ≤ Cǫδ . vol {x ∈ M || 0

12. Measure estimates: the general case Bufetov functionals were constructed for a = (X, Y, Z) ∈ A under a (full measure) Diophantine condition (DC) on a ¯ ∈ M , which depends on the backward orbit under the renormalization ﬂow gR in the moduli space M . Throughout this section we assume that a ∈ DC satisfies an additional (full measure) Diophantine condition DClog (which depends on the forward orbit): a ∈ DClog if a ¯ ∈ M satisfies the logarithmic law of geodesics, that is, if (36)

lim sup t→+∞

δM (gt (¯ a)) = 1. log t

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(η)

Lemma 12.1. — Let a ∈ DC ∩ DClog . Let η ∈ (0, 1) and let c ∈ Ω∞ . For every δ ∈ (1 − η/2, 1) and for every ζ > 0, there exist constants Cδ,ζ > 0 and Cζ > 0 such that, for every ǫ > 0, we have ! 1 T 1/2 Cδ,ζ logδ T } ≤ 4ǫ . (37) vol {x ∈ M ||βc (a, x, T )| ≤ ǫ Cζ log1/4+ζ T Proof. — Let us assume a ∈ DClog ∩ DC(L). By the Diophantine condition DClog , there exists a bounded set B ⊂ M such that the following holds. For any ζ > 0 there exist a constant Cζ > 0 and a sequence (tn ) with gtn (¯ a) ∈ B such that etn+1 −tn ≤ Cζ t1+ζ n .

(38)

Let T ≫ 1 and for all n ∈ N let Tn = e−tn T . By the Diophantine condition DClog , for any ζ > 0 there exists a constant Cζ′ > 0 such that gtn (a) ∈ DC(Ln ) with 1

(39)

Ln = e−tn /2 L + EM (a, etn ) ≤ e−tn /2 L + Cζ′ tn4

+ζ

1

= (T /Tn )−1/2 L + Cζ′ log 4 +ζ (T /Tn ).

By the scaling property of the Bufetov functionals βc (a, x, T ) = (T /Tn )1/2 βc (gtn (a), x, Tn ).

(40)

Since B is a bounded, hence relatively compact set, we have 0 < tmin = inf ta ≤ sup ta = tmax < +∞. B B a∈B

a∈B

By Lemma 9.4 the real analytic function βc (gtn (a), φYy φZ z (x), Tn ),

for all (y, z) ∈ R × R/Z (n)

extends to a complex analytic function on the strip DR ⊂ C2 defined as RTn (n) DR = (y, z) ∈ C × C/Z | |Im(y)|Tn + |Im(z)| ≤ , 2πK (η)

(n)

and, for any c ∈ Ω∞ there exists Cη > 0 such that for any (y, z) ∈ DR we have the uniform upper bound 1/2 (RTn )2−η βc (gtn (a), φYy φZ . z (x), Tn ) ≤ Cη Ln + Tn (1 + EM (gtn (a), Tn )) (1+RTn )e

In particular it follows that, for any z ∈ T, the function βc (gtn (a), Φgtn (a),x (ξy,z ), Tn ), defined for y ∈ [0, 1] extends to a holomorphic function on the strip ( ) Rtmin B , HR = y ∈ C| |Im(y)| ≤ 4πK

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and there exists a constant Cη′ > 0 such that the following uniform upper bound holds: for any y ∈ HR and any z ∈ T, we have (41) βc (gtn (a), Φgtn (a),x (ξy,z ), Tn ) 2−η ≤ Cη′ Ln + Tn1/2 (1 + EM (gtn (a), Tn )) (1 + RTn )e(RTn ) . By a calculation, for all tn ≥ 0 and for Tn = e−tn T ∈ [0, T ] we have that EM (gtn (a), Tn ) ≤ EM (a, T ) ≤ Cζ log1/4+ζ T. By Lemma 8.2 it follows that, for any s > 7/2, whenever we have !1/2 Tn −1 s+1 (42) |c|0 ≥ 10Cs (1 + Ln )|c|s sup t−1 , a (1 + ta kY k) max tB a∈B

then there exists a constant CB > 0 such that

βc (gtn (a), Φg (a),x (ξy,z ), Tn ) 2 ≥ CB |c|0 Tn1/2 ; tn L (T,dy) Z (43) βc (gt (a), Φg (a),x (ξy,z ), Tn )dy ≤ CB |c|0 Tn1/2 . n tn 4 T

In particular, we derive a uniform lower bound for the oscillation On (c, T ) of the function βc (gtn (a), Φgtn (a),x (ξy,z ), Tn ) for y ∈ [0, 1]:

CB |c|0 Tn1/2 . 2 It remains to optimize the choice of tn > 0, hence of Tn ∈ [0, T ), given T > 0. It follows from Formulas (39) and (42) that for any ζ > 0, there exists a constants Lζ > 0 such that we want to choose Tn to be the smallest solution of the inequality 2 1 Tn ≥ L2ζ 1 + log 4 +ζ (T /Tn ) . (44)

On (c, T ) ≥

By this definition and by the condition in Formula (38) we then have 1

etn ≤ Lζ etn (1 + tn4

1

+ζ 2

+ζ

1

4 )2 ≤ Lζ Cζ′′ etn (1 + tn4 ) ≤ T < Lζ etn+1 (1 + tn+1

+ζ 2

) ,

which in turn implies T T = etn ≥ (LCζ′′ )−1 . 1 Tn (1 + log 4 +ζ T )2 It follows in particular that (45)

1

Tn ≤ (LCζ′′ )(1 + log 4 +ζ T )2 ,

1/2 Ln ≤ L + Cζ′ L−1 ζ Tn .

With this choice there exists a constant Cη,ζ > 0 such that (46) βc (gtn (a), Φgtn (a),x (ξy,z ), Tn ) ≤ Cη,ζ 1 + Tn1/2 log1/4+ζ T

× (1 + R log1/2+2ζ T ) exp[(LCζ′′ R(1 + log1/4+ζ T )]4−2η .

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For any R > r > 10πK(tmin )−1 , by Formulas (44) and (46), from Lemma 10.4 we B derive that there exists a constant C > 0 such that, for all t ∈ (1, 3/2), for all fixed z ∈ T, and for all suﬃciently large T > 0, the valency νβ (t) ∈ N of the function of one-complex variable βc (gtn (a), Φgtn (a),x (ξy,z ), Tn ) is bounded above as follows: ′ νβ (t) ≤ C log Cη,ζ R(1 + log3/4+3ζ T ) exp[(LCζ′′ R(1 + log1/4+ζ T )2 ]2−η η ′ ′′ log(Cη,ζ R) + log log T + log1− 2 +2ζ(2−η) T . ≤ Cη,ζ

By Theorem 10.1, it follows that, for all δ ∈ (1−η/2, 1), there exists a constant Cδ,ζ > 0 such that, for all (x, z, n) ∈ M × Z × N, and for all ǫ > 0 we have (47)

1 δ Leb {y ∈ [0, 1]||βc (gtn (a), Φgtn (a),x (ξy,z ), Tn )| < ǫ} < 4ǫ Cδ,ζ log T .

Finally, from the scaling identity (40), and from the measure estimates (47) and Fubini’s theorem, it follows that for all ζ > 0 and for all δ ∈ (1 − η/2, 1) there exists a constant Cζ > 0 such that ! 1 T 1/2 δ vol {x ∈ M ||βc (a, x, T )| ≤ ǫ } ≤ 4ǫ Cδ,ζ log T , 1/4+ζ 1 + Cζ log T as claimed in the statement. Corollary 12.2. — Let a = (X, Y, Z0 ) be as in Lemma 12.1. Let η ∈ (0, 1) and let (η) c ∈ Ω∞ . For every δ ∈ (1 − η/2, 1) and for every ζ > 0, there exist constants Cδ,ζ > 0 and Cζ > 0 such that, for every ǫ > 0, we have ! 1 T 1/2 Cδ,ζ logδ T . (48) vol {x ∈ M ||βc (a, x, T )| ≤ ǫ } ≤ 4ǫ Cζ log1/4+ζ T

13. Correlations. Proof of Theorems 2.2 and 2.3 We will first analyze correlations and then derive Theorems 2.2 and 2.3 from respectively Corollaries 11.3 and 12.2. We analyze correlations as follows. Let ωV the V -invariant volume form. It follows from the definition of V = αX that ωV = α−1 dvol. We have g hh ◦ φVt , giL2 (M,ωV ) = hh ◦ φVt , iL2 (M,dvol) , α hence it is equivalent to analyze correlations with respect to Haar volume form dvol. As the Haar volume form is Z-invariant we have Z 1 S V Z hh ◦ φt , giL2 (M,dvol) = hh ◦ φVt ◦ φZ s , g ◦ φs iL2 (M,dvol) ds. S 0 ASTÉRISQUE 416

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Integrating by parts we finally derive the formula (see the paper [15]): 1 S

Z

S

hh ◦

φVt

◦

φZ s ,g

◦

φZ s iL2 (M,dvol) ds

0

Z S 1 Z h ◦ φVt ◦ φZ = h s ds, g ◦ φS iL2 (M,dvol) S 0 Z Z 1 S s Z h ◦ φVt ◦ φZ − h r dr, Zg ◦ φs iL2 (M,dvol) ds. S 0 0

We have thus written correlations in terms of integrals Z

S

h ◦ φVt ◦ φZ s ds.

0

Let Dt denote the function on M defined as Z t Zα (49) Dt (x) := ◦ φVτ dτ. α 0 We then have the formula Z S S 1 + Dt2 ◦ φZ s V Z h ◦ φVt ◦ φZ ds = s 2 ◦ φZ h ◦ φt ◦ φs ds 1 + D 0 0 t s Z S Z S Dt2 ◦ φZ 1 s V Z ds + = h ◦ φ ◦ φ h ◦ φVt ◦ φZ t s s ds. 2 2 Z Z 0 1 + Dt ◦ φs 0 1 + Dt ◦ φs

Z

We also have Z S Z S Dt ◦ φZ Dt2 ◦ φZ s s V Z Z h ◦ φ ◦ φ ds = (h ◦ φVt ◦ φZ t s s )(Dt ◦ φs ) ds 2 2 Z Z 0 1 + Dt ◦ φs 0 1 + Dt ◦ φs Z S Z Dt ◦ φS Z (h ◦ φVt ◦ φZ = s )(Dt ◦ φs )ds 1 + Dt2 ◦ φZ S 0 Z s Z S d Dt ◦ φZ s V Z Z − (h ◦ φ ◦ φ )(D ◦ φ )dr ds . ( ) t t r r 2 Z 0 ds 1 + Dt ◦ φs 0 σ For every σ > 0, let γx,t be the path defined as σ γx,t (s) = (φVt ◦ φZ s )(x),

for all s ∈ [0, σ].

We have computed above that σ dγx,t σ σ (s) = [Dt ◦ φZ s (x)]V (γx,t (s)) + Z(γx,t (s)). ds

It follows that Z

σ γx,t

hVˆ =

Z

σ Z (h ◦ φVt ◦ φZ s )(x) (Dt ◦ φs )(x)ds.

0

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In other terms, we have the following identity: "Z # Z S Dt2 ◦ φZ Dt ◦ φZ s (x) V Z S (x) ˆ h ◦ φt ◦ φs (x)ds = hV 2 Z S 1 + Dt2 ◦ φZ 0 1 + Dt ◦ φs (x) γx,t S (x)

−

Z

0

S

# "Z Dt ◦ φZ d s (x) hVˆ ds. s ds 1 + Dt2 ◦ φZ γx,t s (x)

It remains to estimate the term d Dt ◦ φZ d 1 − Dt2 ◦ φZ s (x) s (x) Z . = Dt ◦ φs (x) 2 ds 1 + Dt2 ◦ φZ ds (1 + Dt2 ◦ φZ s (x) s (x)) Our estimate is thus reduced to a bound on the term Z t d Zα Zα Z V V Dτ [V ( Dt ◦ φs (x) = ) ◦ φτ ] + Z( ) ◦ φτ ◦ φZ s (x)dτ ds α α 0 Z t Zα Zα 2 Zα Z( (x) + ) ◦ φVt ◦ φZ ) − ( ) ◦ φVτ ◦ φZ = Dt ( s s (x)dτ α α α 0 Z t Zα Zα = Dt ( αZ( 2 ) ◦ φVτ ◦ φZ (x) + ) ◦ φVt ◦ φZ s s (x)dτ. α α 0 In particular by Lemma 6.2 and Theorem 6.3 we conclude that for every L > 0 there exists a constant Cα (L) > 0 such that, for all a ∈ DC(L) we have d 1/2 Dt ◦ φZ (1 + EM (a, t)) . s (x) ≤ Cα (L) 1 + t ds

σ Since the arc γx,t is smooth and contained in a weak-stable leaf, it follows from Lemma 5.1 and from the Hölder property of Lemma 6.1 that, for all s > 7/2 there exists a constant Cs (L) > 0 such that, for all t > 0, we have Z !1/2 ! Z Z ˆ ˆ |X| |X|) . hVˆ ≤ Cs (L)|h|a,s 1 + EM (a, γx,t σ σ σ γx,t γx,t

Finally by Lemma 6.2 and Theorem 6.3 we have Z ˆ ≤ Cα σ(1 + t1/2 ) (1 + EM (a, t)) . |X| σ γx,t

For a given t > 1 and ǫ > 0, let now consider the set n o Mα (t, ǫ) := x ∈ M ||Dt (x)| ≥ ǫ (1 + t1/2 ) (1 + EM (a, t)) .

There exists a constant Cα′ (L) > 0 such that φZ s (Mα (t, 2ǫ)) ⊂ Mα (t, ǫ),

ASTÉRISQUE 416

for all s ∈ (−

ǫ ǫ , ). Cα′ (L) Cα′ (L)

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Proof of Theorems 2.2 and 2.3. — We prove Theorems 2.2. Let F be the set of all a ∈ A of bounded type, that is, such that the orbit gR (a) is relatively compact in the moduli space M . The set F has full Hausdorﬀ dimension. For all a ∈ F , the function EM (a, t) is uniformly bounded over t ∈ R. By Corollary 11.3 for f = Zα α ∈ ΩR there exist constants Ca,α > 0 and δa,α > 0 such that we have vol (M \ Mα (t, ǫ)) ≤ Ca,α ǫδa,α . It follows that correlations on the set Mα (t, 2ǫ) can be estimated by the expression

1 (1 + t3/4 ) 3/4

(1 + t ) , ≤ 2

1 + Dt L1 (Mα (t,ǫ)) 1 + ǫ2 t

while the correlation on the complementary set M \Mα (t, 2ǫ) can be estimated simply ′ as follows: there exists a constant Ca,α > 0 such that ′ hh ◦ φVt , giL2 (M \Mα (t,2ǫ) ≤ Ca,α ǫδa,α khks (kgk0 + kZgk0 ).

By optimizing the above estimates we derive a bound for the decay of correlations of the following form: for every h, g ∈ W s (M ) of zero average, and for all t > 0, we have δ

a,α − 4(2+δ

hh ◦ φVt , giL2 (M ) ≤ Ca,α (1 + t)

a,α )

khks (kgk0 + kZgk0 ).

This finishes the proof of Theorem 2.2. For Theorem 2.3, by an analogous reasoning based on Corollary 12.2 we get that for a generic set of time-changes we have the following bound on correlations. For every δ > 1/2, there exists a constant Ca,α,δ > 0 such that, for all h, g ∈ W s (M ) of zero average, and for all t ∈ R, we have −

hh ◦ φVt , giL2 (M ) ≤ Ca,α,δ (1 + |t|)

1 1+logδ (1+|t|)

khks (kgk0 + kZgk0 ).

This finishes the proof of Theorem 2.3. References [1] A. Avila, G. Forni, R. D. & C. Ulcigrai – “Mixing for smooth time-changes of general nilﬂows”, preprint arXiv:1905.11628. [2] A. Avila, G. Forni & C. Ulcigrai – “Mixing for time-changes of Heisenberg nilﬂows”, J. Differential Geom. 89 (2011), p. 369–410. [3] A. Brudnyi – “On local behavior of analytic functions”, J. Funct. Anal. 169 (1999), p. 481–493. [4] J. A. Brudny˘ı & M. I. Ganzburg – “A certain extremal problem for polynomials in n variables”, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), p. 344–355. [5] A. Bufetov & G. Forni – “Limit theorems for horocycle ﬂows”, Ann. Sci. Éc. Norm. Supér. 47 (2014), p. 851–903. [6] A. I. Bufetov – “Limit theorems for translation ﬂows”, Ann. of Math. 179 (2014), p. 431–499. [7] A. I. Bufetov & B. Solomyak – “Limit theorems for self-similar tilings”, Comm. Math. Phys. 319 (2013), p. 761–789.

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[8] F. Cellarosi & J. Marklof – “Quadratic Weyl sums, automorphic functions and invariance principles”, Proc. Lond. Math. Soc. 113 (2016), p. 775–828. [9] D. Dolgopyat, Y. Pesin, M. Pollicott & L. Stoyanov (eds.) – Hyperbolic dynamics, fluctuations and large deviations, Proceedings of Symposia in Pure Mathematics, vol. 89, Amer. Math. Soc., 2015. [10] D. Dolgopyat & O. Sarig – “Temporal distributional limit theorems for dynamical systems”, J. Stat. Phys. 166 (2017), p. 680–713. [11] B. Fayad – “Polynomial decay of correlations for a class of smooth ﬂows on the two torus”, Bull. Soc. Math. France 129 (2001), p. 487–503. [12] B. R. Fayad – “Analytic mixing reparametrizations of irrational ﬂows”, Ergodic Theory Dynam. Systems 22 (2002), p. 437–468. [13] B. R. Fayad, G. Forni & A. Kanigowski – “Lebesgue spectrum for area preserving ﬂows on the two torus”, preprint arXiv:1609.03757. [14] L. Flaminio & G. Forni – “Equidistribution of nilﬂows and applications to theta sums”, Ergodic Theory Dynam. Systems 26 (2006), p. 409–433. [15] G. Forni & C. Ulcigrai – “Time-changes of horocycle ﬂows”, J. Mod. Dyn. 6 (2012), p. 251–273. [16] A. Gorodnik & N. Peyerimhoff (eds.) – Dynamics and analytic number theory, London Mathematical Society Lecture Note Series, vol. 437, Cambridge Univ. Press, 2016. [17] J. Griffin & J. Marklof – “Limit theorems for skew translations”, J. Mod. Dyn. 8 (2014), p. 177–189. [18] A. V. Kočergin – “The absence of mixing in special ﬂows over a rotation of the circle and in ﬂows on a two-dimensional torus”, Dokl. Akad. Nauk SSSR 205 (1972), p. 512–518, translated in Soviet Math. Dokl. 13 (1972), 949–952. [19]

, “Mixing in special ﬂows over a rearrangement of segments and in smooth ﬂows on surfaces”, Mat. Sb. (N.S.) 96(138) (1975), p. 471–502, 504.

[20] A. G. Kušnirenko – “Spectral properties of certain dynamical systems with polynomial dispersal”, Vestnik Moskov. Univ. Ser. I Mat. Meh. 29 (1974), p. 101–108. [21] B. Marcus – “Ergodic properties of horocycle ﬂows for surfaces of negative curvature”, Ann. of Math. 105 (1977), p. 81–105. [22] S. Marmi, P. Moussa & J.-C. Yoccoz – “Aﬃne interval exchange maps with a wandering interval”, Proc. Lond. Math. Soc. 100 (2010), p. 639–669. [23] M. Ratner – “The rate of mixing for geodesic and horocycle ﬂows”, Ergodic Theory Dynam. Systems 7 (1987), p. 267–288. [24] D. Ravotti – “Quantitative mixing for locally Hamiltonian ﬂows with saddle loops on compact surfaces”, Ann. Henri Poincaré 18 (2017), p. 3815–3861. [25]

, “Mixing for suspension ﬂows over skew-translations and time-changes of quasiabelian filiform nilﬂows”, Ergodic Theory Dynam. Systems 39 (2019), p. 3407–3436.

[26] D. Scheglov – “Absence of mixing for smooth ﬂows on genus two surfaces”, J. Mod. Dyn. 3 (2009), p. 13–34.

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[27] Y. G. Sina˘ı & K. M. Khanin – “Mixing of some classes of special ﬂows over rotations of the circle”, Funktsional. Anal. i Prilozhen. 26 (1992), p. 1–21, translated in: Functional Analysis and its Applications, 26:3 (1992), 155–169. [28] C. Ulcigrai – “Mixing of asymmetric logarithmic suspension ﬂows over interval exchange transformations”, Ergodic Theory Dynam. Systems 27 (2007), p. 991–1035. [29]

, “Absence of mixing in area-preserving ﬂows on surfaces”, Ann. of Math. 173 (2011), p. 1743–1778.

G. Forni, Department of Mathematics, University of Maryland, College Park, MD USA E-mail : [email protected] A. Kanigowski, Department of Mathematics, Pennsylvania State University, State College, PA USA • E-mail : [email protected]

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Astérisque 416, 2020, p. 301–320 doi:10.24033/ast.1117

STABLE ACCESSIBILITY WITH 2-DIMENSIONAL CENTER by Artur Avila & Marcelo Viana

Abstract. — For partially hyperbolic diﬀeomorphisms with 2-dimensional center, accessibility is C 1 -stable. Moreover, for center bunched skew-products (stable) accessibility is C ∞ -dense. Résumé (Accessibilité stable de dimension centrale 2). — L’accessibilité est une propriété C 1 -stable parmi les diﬀéomorphismes partiellement hyperboliques à dimension centrale 2. De plus, l’accessibilité (stable) est une propriété C 1 -dense dans le domaine des produits gauches satisfaisant la condition de regroupement central (‘center bunching’).

1. Introduction A diﬀeomorphism f : M → M of a compact manifold M is partially hyperbolic if there exist: a continuous splitting of the tangent bundle T M = E u ⊕ E c ⊕ E s invariant under the derivative Df (all three sub-bundles are assumed to have positive dimension); a Riemannian metric k · k on M ; and positive continuous functions ν, νˆ, γ, γˆ with ν, νˆ < 1 and ν < γ < γˆ −1 < νˆ−1 , such that kDf (p)vk < ν(p) if v ∈ E s (p), (1)

−1

γ(p) < kDf (p)vk < γˆ (p) −1

νˆ(p)

< kDf (p)vk

if v ∈ E c (p), if v ∈ E u (p)

for any unit vector v ∈ Tp M . This is an open property in the space of C 1 diﬀeomorphisms. We will denote d∗ = dim E ∗ , for ∗ ∈ {u, c, s}, and d = dim M . 2010 Mathematics Subject Classification. — 37D30; 37C86. Key words and phrases. — Partial hyperbolicity, central dimension, invariant foliation, accessibility. This work was partly conducted during the period A. A. served as a Clay Research Fellow and it was partly supported by Fondation Louis D. – Institut de France (project coordinated by M. Viana), CNPq and FAPERJ.

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The stable bundle E s and the unstable bundle E u are uniquely integrable and their integral manifolds form two quasi transverse continuous foliations, W u = Wfu and W s = Wfs , whose leaves are immersed submanifolds of the same class of diﬀerentiability as f . These are called the strong unstable and strong stable foliations of f . They are invariant under f , in the sense that f (W ∗ (x)) = W ∗ (f (x)) for any x ∈ M ∗ and ∗ ∈ {u, s}. Given ǫ > 0 and ∗ ∈ {u, s}, we represent by Wǫ∗ (x) = Wf,ǫ (x) the ∗ ǫ-neighborhood of x inside W (x). Given two points x, y ∈ M , we say that x is accessible from y if there exists a C 1 path that connects x to y and is tangent at every point to the union E u ∪ E s . The equivalence classes of this (equivalence) relation are called f -accessibility classes. The diﬀeomorphism f is called accessible if there exists a unique f -accessibility class, namely, the ambient M . Moreover, f is called stably accessible if it admits a C 1 neighborhood U such that every C 2 diﬀeomorphism g ∈ U is accessible. For any k ≥ 1, we denote by PH k the space of C k partially hyperbolic diﬀeomorphisms in M . Most of our results concern the subspace PH2k of diﬀeomorphisms f ∈ PH k with 2-dimensional center bundle, that is, such that dc = 2. Theorem A. — If f ∈ PH21 is accessible then f is stably accessible. We say that an f -accessibility class C is stable if for every compact set K ⊂ C there exists a C 1 neighborhood U = UK of f such that K is contained in a unique g-accessibility class for every C 2 diﬀeomorphism g ∈ U . In particular, f is stably accessible if, and only if, the ambient M is a stable f -accessibility class. Stable accessibility classes are open sets. Indeed, let p and q be two distinct points in C (for instance, in the same stable manifold). For any r ∈ M close to q, let h : M → M be a diﬀeomorphism C ∞ close to the identity, such that h(p) = p and h(r) = q. Then g = h ◦ f ◦ h−1 is close to f . Taking K = {p, q}, the assumption implies that p and q are in the same g-accessibility class. This means that p and q are in the same f -accessibility class, that is, r ∈ C. So, C contains a whole neighborhood of q. Here we prove that the converse is also true, at least when the center bundle is 2-dimensional: Theorem B. — If f ∈ PH21 then any open f -accessibility class is stable. Theorem A is a direct consequence of Theorem B. The main technical step in the proof of Theorem B is a result on approximation of general paths in accessibility classes by a certain class of paths for which a continuation exists for every nearby diﬀeomorphism. This result is stated in Section 4 (Theorem 4.1), where we also explain how it leads to Theorem B. In Sections 6–5 we state and prove a result about density of stable accessibility (Theorem 6.1), for a class of fibered partially hyperbolic diﬀeomorphisms with 2-dimensional center bundle. It contains a claim made in Section 7 of our paper [2], that

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was used for proving Theorem H in that paper. After our research had been completed, we learned from V. Horita and M. Sambarino that they had independently obtained a similar result, in a paper that appeared in [8]. When the center dimension dc = 1, the accessibility property is always stable [5]. The present work extends that fact to center dimension equal to 2. Recently, and also in the 2-dimensional case, J. Rodriguez-Hertz and C. Vasquez [13] proved that accessibility classes are immersed submanifolds, which implies Theorem A. When the center bundle is one-dimensional, the (stable) accessibility property is known to be C r dense among partially hyperbolic diﬀeomorphisms [3, 12]. Without any hypothesis on the dimension of the central bundle, Dolgopyat and Wilkinson [6] proved that stable accessibility is C 1 dense. 2. Deformations paths In this section, all maps are assumed to be C 1 and proximity is always meant in the C topology. We introduce a class of paths, that we call deformation paths, contained in accessibility classes and having a useful property of persistence under variation of the diﬀeomorphism and the base point. This also provides a kind of parametrization for accessibility classes: 1

Theorem 2.1. — For every f ∈ PH 1 , there exist k ≥ 1, a neighborhood V of f and a sequence Pl : V × M × Rk(du +ds )l → M of continuous maps such that, for any g ∈ V , 1. Pm (g, · , w) ◦ Pl (g, · , v) = Pl+m (g, · , (v, w)) for every l ≥ 1 and m ≥ 1 and v ∈ Rk(du +ds )l and w ∈ Rk(du +ds )m ; 2. ζ 7→ Pl (g, ζ, v) is a homeomorphism from M to M , with Pl (g, ·, 0) = id, for every l ≥ 1 and v ∈ Rk(du +ds )l ; S 3. n≥1 Pn ({(g, z)} × Rk(du +ds )n ) is the g-accessibility class of each z ∈ M .

A deformation path based on (f, z) is a (continuous) map γ : [0, 1] → M such that there exist l ≥ 1 and a continuous map Γ : [0, 1] → Rk(du +ds )l satisfying γ(t) = Pl (f, z, Γ(t)). Notice that any deformation path based on (f, z) is contained in the f -accessibility class of z. It follows immediately from the definition that deformation paths are persistent, in the following sense:

Corollary 2.2. — If γ : [0, 1] → M is a deformation path based on (f, z) then, for any g close to f and any w close to z, there exists a deformation path based on (g, w) that is uniformly close to γ. In the remainder of this section we prove Theorem 2.1. Let I = [−1, 1]. We need the following particular case of [7, Theorem 4.1]: Lemma 2.3. — For every f ∈ PH 1 and ζ ∈ M , there exists a neighborhood V of f and a continuous map ψ = ψf,ζ : V × I d → M such that for every g ∈ V , 1. ψ(g, 0) = ζ,

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2. x 7→ ψ(g, x) is a homeomorphism, 3. ψ(g, x, y) ∈ Wgu (ψ(g, 0, y)) for every x ∈ I du and y ∈ I d−du . Lemma 2.4. — For every f ∈ PH 1 there exist a neighborhood V of f , numbers k ≥ 1 and ǫ > 0 and continuous maps Φu : V ×M ×Rkdu → M and Φs : V ×M ×Rkds → M such that: 1. x 7→ Φu (g, x, v) is a homeomorphism, for every g ∈ V and v ∈ Rkdu ; u 2. Wg,ǫ (x) ⊂ Φu ({g} × {x} × Rkdu ) ⊂ Wgu (x) for every g ∈ V and x ∈ M ,

and analogously for Φs . Proof. — We will only go through the details of the construction of Φu , the case of Φs being analogous. Let ht : I → I be the ﬂow satisfying (dht /dt)(x) = 1 − ht (x)2 . Let H : I d → I be given by H(v) = (1 − vd2u +1 ) . . . (1 − vd2 ). For v ∈ Rdu , let hv : I d → I d be given by hv (x) = hH(x)v1 (x1 ), . . . , hH(x)vdu (xdu ), xdu +1 , . . . , xd . Pick points ζi ∈ M , 1 ≤ i ≤ k so that the interiors of the images ψi ({f } × I d ) cover M , where ψi = ψf,ζi : Vi × I d → M T are the maps given by Lemma 2.3. Let V be a neighborhood of f contained in i Vi and ǫ be a positive number such that for every g ∈ V and z ∈ M there exist i and y such that u Wg,ǫ (z) ⊂ ψi {g} × inter(I du ) × {y} .

Let Φi : V × M × Rdu → M be given by

Φi (g, ψi (g, ζ), v) = ψi (g, hv (ζ)) for ζ ∈ I d Φi (g, z, v) = z

if z ∈ / ψi ({g} × I d ).

Then define Φ(i) : V × M × Ridu → M , 1 ≤ i ≤ k by Φ(1) = Φ1

and

Φ(i+1) (g, · , (wi , w)) = Φi+1 (g, · , w) ◦ Φ(i) (g, ·, wi )

and take Φu = Φ(k) . Claim (1) follows from part (2) of Lemma 2.3, by composition. The lower bound in claim (2) follows from the choice of ǫ and the upper bound is a consequence of part (3) of Lemma 2.3. Proof of Theorem 2.1. — Define Pl : V × M × Rk(du +ds )l → M , l ∈ N by letting P1 (g, · , (wu , ws )) = Φs (g, · , ws ) ◦ Φu (g, · , wu ) for wu ∈ Rkdu and ws ∈ Rkds and Pl (g, · , (w1 , . . . , wl )) = P1 (g, · , wl ) ◦ · · · ◦ P1 (g, · , , w1 ) for w1 , . . . , wl ∈ Rk(du +ds ) . Property (1) in Theorem 2.1 is a direct consequence of this definition. Property (2) follows from part (2) of Lemma 2.3, by composition. Finally, S Lemma 2.4 gives that n≥1 Pn (g, z, Rk(du +ds )n ) is the g-accessibility class of z, as claimed in part (3) of the theorem.

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Theorem 2.5. — For every l ≥ 1 there exists m ≥ 1 and for every v ∈ Rk(du +ds )l there exists v ∗ ∈ Rk(du +ds )m , depending linearly on v, such that Pm (g, · , v ∗ ) = Pl (g, · , v)−1

for any g ∈ V .

The initial step in the proof of this theorem is: Lemma 2.6. —

(a) For every v ∈ Rkdu there exists v ∗ ∈ Rk(du +ds )k such that Pk (g, · , v ∗ ) = Φu (g, · , v)−1

for any g ∈ V .

(b) For every v ∈ Rkds there exists v ∗ ∈ Rk(du +ds )k such that Pk (g, · , v ∗ ) = Φs (g, · , v)−1 Proof. — Write v = (v1 , . . . , vk ) with vj ∈ R

du

for any g ∈ V . for j = 1, . . . , k. Then define

∗ v ∗ = (vk∗ , 0, vk−1 , 0, . . . , v1∗ , 0) ∈ Rk(du +ds )k ,

where 0 ∈ Rkds and each vj∗ ∈ Rkdu is defined by ∗ ∗ vj∗ = (vj,1 , . . . , vj,k ) with

∗ vj,i =

(

0 −vj

if i 6= j if i = j.

Then, by the definition of Φi , (2)

Φu (g, · , vi∗ ) = Φi (g, · , −vi ) = Φi (g, · , vi )−1 .

By the definition of Φu , (3)

Φu (g, · , v) = Φk (g, · , vk ) ◦ · · · ◦ Φ1 (g, · , v1 ).

By the definition of Pk , Pk (g, · , v ∗ ) = P1 (g, · , (v1∗ , 0)) ◦ · · · ◦ P1 (g, · , (vk∗ , 0)) (4) = Φu (g, · , v1∗ ) ◦ · · · ◦ Φu (g, · , vk∗ ). Claim (a) in the lemma is a direct consequence of (2) – (4). ∗ For claim (b), take v ∗ = (0, vk∗ , 0, vk−1 , . . . , 0, v1∗ ) ∈ Rk(du +ds )k , where 0 ∈ Rkdu and kds ∗ is defined just as before. Then argue as in the previous case. each vj ∈ R Proof of Theorem 2.5. — For l = 1, take m = 2k and for each v = (vu , vs ) ∈ Rk(du +ds ) take v ∗ = (vs∗ , vu∗ ) ∈ Rk(du +ds )m , where vu∗ and vs∗ are the vectors in Rk(du +ds )k given by Lemma 2.6. Then Pm (g, ·, v ∗ ) = Pk (g, · , vu∗ ) ◦ Pk (g, · , vs∗ ) = Φu (g, · , vu )−1 ◦ Φs (g, · , vs )−1 −1 = Φs (g, · , vs ) ◦ Φu (g, · , vu ) = P1 (g, · , v)−1 .

In general, for any l ≥ 1, take m = 2kl and for each v = (vu,1 , vs,1 , . . . , vu,l , vs,l ) ∗ ∗ ∗ ∗ , . . . , vs,1 , vu,1 ) in Rk(du +ds )m . Then in Rk(du +ds )l consider v ∗ = (vs,l , vu,l ∗ ∗ ∗ ∗ Pm (g, · , v ∗ ) = P2k (g, · , (vs,1 , vu,1 )) ◦ · · · ◦ P2k (g, · , (vs,l , vu,l )).

By the previous paragraph, we may rewrite the right-hand side of this equality as −1 P1 (g, · , v1 )−1 ◦ · · · ◦ P1 (g, · , vl )−1 = P1 (g, · , vl ) ◦ · · · ◦ P1 (g, · , v1 ) . SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2020

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It follows that Pm (g, · , v ∗ ) = Pl (g, · , v)−1 , as claimed.

3. An intersection property The following result lies at the heart of the proof of Theorem B: Theorem 3.1. — Let f be a partially hyperbolic diﬀeomorphism with 2-dimensional center. Let D be a 2-dimensional disk transverse to E s ⊕ E u and ηu , ηs be smooth paths in D intersecting transversely at some point. Then, for every C 1 diﬀeomorphism g close to f and any continuous paths γu , γs uniformly close to ηu , ηs , there are points xu , xs in the images of γu , γs such that Wgu (xu ) intersects Wgs (xs ). For the proof we need the following lemma. Let 0d denote the origin of Rd . Recall that if M, N, P are compact orientable smooth manifolds and f : M → P and g : N → P are continuous maps, then we can define the intersection number of f and g as the intersection number of the map (f, g) : M × N → P × P , (f, g)(x, y) = (f (x), g(y)) with the diagonal in P ×P . Clearly, the intersection number is a homotopy invariant of f and g. Lemma 3.2. — Let n, u, s ∈ N with n = u+2+s. There exists ǫ > 0 with the following property. Let W u and W s be foliations with C 1 leaves in Rn , tangent to continuous distributions E u and E s of u-and s-dimensional planes. Assume that Exu is ǫ-close to Ru × {02+s } and Exs is ǫ-close to {0u+2 } × Rs for every x in the unit ball B n of Rn . Let γu , γs : [−1, 1] → Rn be continuous paths ǫ-close to the paths ηu , ηs : [−1, 1] → Rn given by ηu (t) = (0u , t, 0, 0s ) and ηs (t) = (0u , 0, t, 0s ). Then there exist tu , ts ∈ (−1, 1) such that W u (γu (tu )) intersects W s (γs (ts )). Proof. — For k ∈ N, consider the sphere S k as the one point compactification of Rk . Let ρu = γu − ηu and ρˆu be a continuous extension of ρu to R, with compact support and kˆ ρu k0 = kρu k0 . Let φu : [−1/4, 1/4]u+1 → R2+s be the only continuous map such that φu (0u , t) = 0 and x 7→ (x, φu (x, t)) + γu (t) is a C 1 map from [−1/4, 1/4]u to W u (γu (t)), for every t ∈ [−1/4, 1/4]. Let φˆu be a continuous extension of φu to Ru+1 , with compact support and kφˆu k0 = kφu k0 . The map Ru × R → Ru × R × R × Rs ,

(x, t) 7→ (x, φˆu (x, t)) + (0u , t, 0, 0s ) + ρˆs (t)

coincides with (x, t) 7→ (x, t, 0, 0s ) outside some compact set and, thus, admits a continuous extension Φu : S u × S 1 → S u × S 1 × S 1 × S s . By construction, Φu (x, t) = (x, φu (x, t)) + γu (t) for every (x, t) ∈ [−1/4, 1/4]u+1 . Since ǫ is small, Φu is uniformly close (and, hence, homotopic) to the map (x, t) 7→ (x, t, 0, 0s ). Define analogous objects ρs , ρˆs , φs , φˆs and Φs . Then Φu and Φs have intersection number 1. Consequently, there exist (xu , tu ) ∈ S u × S 1 and (ts , xs ) ∈ S 1 × S s such that Φu (xu , tu ) = Φs (ts , xs ). Since Φu (xu , tu ) is close to (xu , tu , 0, 0s ) and Φs (ts , xs ) is

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close to (0u , 0, ts , xs ), all xu , tu , ts , xs are small. Thus, we have Φu (xu , tu ) = (xu , φu (xu , tu )) + γu (tu ) ∈ W u (γu (tu )), Φs (ts , xs ) = (φs (ts , xs ), xs ) + γs (ts ) ∈ W s (γs (ts )) and both coincide, giving the conclusion. Proof of Theorem 3.1. — Consider a local change of coordinates close to the transverse intersection of ηs and ηu and apply Lemma 3.2. 4. An approximation property The other main ingredient in the proof of Theorem B is: Theorem 4.1. — Let f : M → M be a partially hyperbolic diﬀeomorphism and U be an open f -accessibility class. Then, for every z ∈ U , the set of deformation paths based on (f, z) is dense in C 0 ([0, 1], U ). The proof of this theorem is contained in the three lemmas that follow. Let z ∈ M be such that the f -accessibility class U of z is open. Lemma 4.2. — There exist l0 ≥ 1 and v0 ∈ Rk(du +ds )l0 such that y0 = Pl0 (f, z, v0 ) is in the interior of Pl0 (f, z, V ) for every neighborhood V of v0 . k(du +ds )l Proof. — For l, m ≥ 1, let K(l, m) = Pl (f, z, [−m, m]S ). Each K(l, m) is compact and, hence, closed in M . Since the union U = l,m K(l, m) is an open set and M is a Baire space, there exist l0 , m0 ≥ 1 such that K(l0 , m0 ) has non-empty interior. Let l0 be fixed and consider the map ϕ : Rk(du +ds )l0 → M given by ϕ(v) = Pl0 (f, z, v). A point y ∈ M is a regular value if every v0 ∈ ϕ−1 (y) is a regular point, meaning that ϕ(v0 ) is in the interior of ϕ(V ) for every neighborhood V of v0 . Since the set of regular values is residual in M (see [1, Proposition 7.6]), there exists some regular value y0 ∈ ϕ(Rk(du +ds )l0 ). Any v0 ∈ ϕ−1 (y0 ) satisfies the conclusion of the lemma.

Lemma 4.3. — For any compact set K ⊂ U and any ǫ > 0 there is δ > 0 such that, given any y ′ , y ′′ ∈ K with d(y ′ , y ′′ ) ≤ δ, there exists a deformation path γ : [0, 1] → M , satisfying γ(0) = y ′ and γ(1) = y ′′ and diam γ([0, 1]) < ǫ. Proof. — Let l0 , v0 and y0 = Pl0 (f, z, v0 ) be as in Lemma 4.2. For each y ∈ K, choose q ≥ 1 and u ∈ Rk(du +ds )q such that Pq (f, y0 , u) = y. By Theorem 2.1, Pl0 +q (f, z, (v0 , u)) = y and the map y0′ 7→ Pq (f, y0′ , u) defines a homeomorphism from a neighborhood of y0 to a neighborhood of y. It follows that the image of any small ball around v0 under the map v0′ 7→ Pl0 +q (f, z, (v0′ , u)) has diameter less than ǫ and contains some neighborhood Wy of y. Let δ > 0 be a Lebesgue number for the cover {Wy : y ∈ K} of K obtained in this way. Given y ′ and y ′′ as in the statement, take y ∈ K such that Wy contains both y ′ and y ′′ and, hence, there are v0′ and v0′′ close to v0 such that Pl0 +q (f, z, (v0′ , u)) = y ′ and Pl0 +q (f, z, (v0′′ , u)) = y ′′ . Then define γ(t) = Pl0 +q (f, z, Γ(t)), with Γ(t) = ((1 − t)v0′ + tv0′′ , u)) for t ∈ [0, 1].

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Lemma 4.4. — The deformation paths based on (f, z) are dense in C 0 ([0, 1], U ). Proof. — Given c ∈ C 0 ([0, 1], U ) and ǫ > 0, consider K = c([0, 1]) and let δ > 0 be given by Lemma 4.3. We may take δ < ǫ, of course. Fix N ≥ 1 such that d(xi−1 , xi ) < δ for i = 1, . . . , N , where xi = c(i/N ). By Lemma 4.3, for each i = 1, . . . , N there exists li ≥ 1 and a continuous map Γi : [0, 1] 7→ Rk(du +ds )li such that the deformation path γi (t) = Pl (f, z, Γi (t)) satisfies γi (0) = xi−1 and γi (1) = xi and diam γi ([0, 1]) < ǫ. For each i = 1, . . . , N , let mi ≥ 1 be associated to li , in the sense of Theorem 2.5. Take γ : [0, 1] 7→ M to be defined by γ(t) = PL (f, z, Γ(t)), where L = l1 + m2 + l2 + · · · + mN + lN and ˆ 1 (t), Γ2 (0)∗ , Γ ˆ 2 (t), . . . , ΓN (0)∗ , Γ ˆ N (t)) Γ(t) = (Γ with

Γi (0) ˆ i (t) = Γ Γi (N t − i + 1) Γi (1)

if t ≤ (i − 1)/N for (i − 1)/N ≤ t ≤ i/N if t ≥ i/N.

ˆ i is continuous. We claim that Note that Γ is continuous, as each of the Γ (5)

γ(t) = γi (N t − i + 1) for every t ∈ [(i − 1)/N, i/N ] and i = 1, . . . , N .

By the properties of γi (t), this implies that d(γ(t), c(t)) ≤ 2ǫ for every t ∈ [0, 1]. Thus, the theorem will follow once we have proved our claim. To this end, observe that, for any t ∈ [(i − 1)/N, i/N ], Γ(t) = Γ1 (1), Γ2 (0)∗ , . . . , Γi (0)∗ , Γi (N t − i + 1), Γi+1 (0), Γi+1 (0)∗ , . . . , ΓN (0)∗ , ΓN (0) . By Theorem 2.1(1) and Theorem 2.5, Plj +mj f, z, (Γj (0)∗ , Γj (0)) = Plj f, Pmj (f, z, Γj (0)∗ ), Γj (0) = z for j = i + 1, . . . , N . Similarly,

Plj +mj+1 f, z, (Γj (1), Γj+1 (0)∗ ) = Pmj+1 f, Plj (f, z, Γj (1)), Γj+1 (0)∗ = Pmj+1 f, xj , Γj+1 (0)∗

= Pmj+1 f, Plj+1 (f, z, Γj+1 (0)), Γj+1 (0)∗ = z

for j = 1, . . . , i − 1. Then Theorem 2.1(1) gives that

γ(t) = PL (f, z, Γ(t)) = Pli (f, z, Γi (N t − i + 1)) = γi (N t − i + 1) for t ∈ [(i − 1)/N, i/N ], as we claimed. We are ready to conclude the proof of Theorem B: Proof of Theorem B. — It suﬃces to prove that for every x, y ∈ U there exists a neighborhood V of y and a neighborhood V of f such that z is in the g-accessibility class of x, for every z ∈ V and g ∈ V . Fix a small open 2-disk D ⊂ U through x transverse to E s ⊕ E u . Consider C 1 paths ηs and ηu in D intersecting transversely at a unique point. By Theorem 4.1, there exists a deformation path based on (f, x) which is uniformly close to ηs and there exists a deformation path based on (f, y)

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which is uniformly close to ηu . Then, by Theorem 2.1, for each g close to f and each z close to y there exists — a deformation path γs based on (g, x) which is still close to ηs and — a deformation path γu based on (g, z) which is still close to ηu . Applying Theorem 3.1, we find points xs , xu in the images of γs , γu such that Wgs (xs ) intersects Wgu (xu ). Since xs is in the g-accessibility class of x and xu is in the g-accessibility class of z, the conclusion follows. 5. Connected subgroups of surface diffeomorphisms The goal in the remainder of the paper is to prove that stable accessibility is dense inside certain classes of partially hyperbolic diﬀeomorphisms with 2-dimensional center bundle. For the sake of simplicity, we focus on the case of diﬀeomorphisms for which the center foliation is an invariant fibration. The precise statement will be given in Theorem 6.1. Towards the proof, in this section we analyze certain properties of connected subgroups of surface diﬀeomorphisms. Let M be a C 1 compact surface and G be a path-connected subgroup of the group Diff 1 (M ) of C 1 diﬀeomorphisms of M . Along the way, we will make a few additional assumptions on G. The G-orbit of a point x ∈ M is the set G(x) = {g(x) : g ∈ G}. It is said to be trivial if G(x) = {x}. Proposition 5.1. — Assume that no G-orbit is trivial. Then for every x ∈ M , either G(x) is open or every compact, connected and locally path-connected set Z ⊂ G(x) is either a point, a C 1 -embedded segment or a C 1 -embedded circle. Proof. — The strategy is borrowed from Rodriguez-Hertz [11, Section 5]. If G(x) is a neighborhood of some point, then using the obvious fact that G acts transitively on G(x), we conclude that it is a neighborhood of every point. In other words, if G(x) has non-empty interior then it is actually open. In what follows we assume that the interior of G(x) is empty. Fix δ < diam G(x). Then there exists ǫ ∈ (0, δ/2) such that diam G(y) > δ for every y in the ǫ-neighborhood of x. We claim that if Z is contained in the ǫ-neighborhood of x then Z is a tree, that is, for every two distinct z0 and z1 there exists a single embedded segment contained in Z that connects z0 to z1 . To prove the existence of such a segment, let γ : [0, 1] → Z be a path such that γ(0) = z0 and γ(1) = z1 . Let U be a maximal open subset of [0, 1] such that γ(t) is constant on the boundary of each connected component of U . Then γ([0, 1] \ U ) is a topologically embedded segment (it is a homeomorphic image of the quotient of [0, 1] by the map that collapses each connected component of U to a point) connecting z0 to z1 . To prove uniqueness, notice that if two such segments exist then their union contains some embedded circle C. Since C is contained in Z, which is contained in the δ/2-neighborhood of x, it follows that one of the connected components of M \ C is contained in the δ/2-neighborhood of x. Let D be this connected component. Then

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diam D < δ and, since D is also contained in the ǫ-neighborhood of x, diam G(y) > δ for every y ∈ D. It follows that G(y) intersects C ⊂ Z ⊂ G(x) for every y ∈ D, and so G(x) contains D. This contradicts the assumption that G(x) has no interior, and this contradiction completes the proof of the claim. Now assume that Z is contained in the ǫ/2-neighborhood of x. Then Z is a tree and so, either it is a point or an embedded segment, or it has a subset h(Y ) homeomorphic via h to the figure Y . We are going to exclude this third alternative. Let y be the 3-valent vertex of h(Y ). Let y1 , y2 , y3 be points in each of the three connected components I1 , I2 , I3 of h(Y ) \ {y} and g1 , g2 , g3 ∈ G be such that gj (y) = yj . We claim that gj may be chosen arbitrarily close to the identity if yj is close enough to y. This can be seen as follows. Let γj : [0, 1] → G be a continuous path with γj (0) = id and γj (1) = gj . Let τj ≥ 0 be maximum such that γj (τj )(y) = y and define βj (t) = γj (t) ◦ γj (τj )−1 for t ≥ τj . By construction, βj (t)(y) = γj (t)(y) ∈ Ij for t > τj and βj (t) → id when t decreases to τj . The claim follows, replacing gj and yj with βj (t) and βj (t)(y) for t close τj . Then Z ′ = Z ∪ g1 (Z) ∪ g3 (Z) is contained in the ǫ-neighborhood of x. Moreover, g1 (I3 ) is an embedded segment C 0 close to I3 and passing through y1 . Then, since Z ′ is a tree, g1 (I3 ) must be disjoint from I3 . Analogously, g3 (I1 ) is an embedded segment C 0 close to I1 , passing through y3 and disjoint from I1 . Then g1 (I3 ) and g3 (I1 ) must intersect, which means that Z ′ cannot be a tree. This contradiction proves that the third alternative above is indeed impossible. Now let Z be any compact locally path-connected subset of G(x). Every z ∈ Z has a compact path-connected neighborhood Uz inside Z with arbitrarily small diameter. Let gz ∈ G be such that gz (x) = z. Then Vz = gz−1 (Uz ) may be taken to be contained in the ǫ/2-neighborhood of x. By the previous considerations, it follows that Uz is either a point or an embedded segment. Since this holds for every z ∈ Z, and we also assume that Z is connected, it follows that Z is either a point, an embedded segment or an embedded circle. It remains to check that Z is C 1 -embedded in the case it is not reduced to a point. Consider Z ′ ⊃ Z defined as follows. If Z is an embedded segment with endpoints y, z, and w ∈ Z \ {y, z}, let gy and gz be elements of G such that gy (w) = y and gz (w) = z. Then Z ∪ gy (Z) ∪ gz (Z) contains an embedded open segment Z ′ ⊃ Z. If Z is an embedded circle, just let Z ′ = Z. The set Z ′ is locally compact and C 1 -homogeneous in the sense that for every y0 , y1 ∈ Z ′ there exists a C 1 -diﬀeomorphism g ∈ G and a neighborhood W of y0 inside Z ′ such that g(y0 ) = y1 and g(W ) is a neighborhood of y1 inside Z ′ . According to [10], this implies that Z ′ is a C 1 submanifold.

Theorem 5.2. — Assume that no G-orbit is trivial. Then M is the disjoint union of an open set U and a compact set K such that K supports a lamination L with C 1 leaves and every G-orbit is either a connected component of U or a leaf of L .

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Proof. — Let K be the set of x ∈ M whose G-orbit is not open. By Proposition 5.1, we can associate to each x ∈ K a line field l(x) tangent to the G-orbit at x. So to get the lamination structure we only have to prove that this line field is continuous. As a first step we claim that for every x ∈ M there exists a continuous path γx : [0, 1] → G such that γx ([0, 1]) is close to the identity, γx (0) = id and γx (1)(x) 6= x. That can be seen as follows. Since M is compact, the assumption implies that we may choose g1 , . . . , gk ∈ G such that for every x ∈ M there exists i such that gi (x) 6= x. Let us choose also paths γi : [0, 1] → G connecting id to gi . Then, for every x ∈ M and n ≥ 1, there exist i and 0 ≤ j ≤ n − 1 such that γi (j/n)(x) 6= γi ((j + 1)/n)(x). Notice that the path γi,j,n (t) = γi ((j + t)/n))−1 ◦ γi (j/n) is contained in a small C 1 -neighborhood of the identity if n is large. By construction, γi,j,n (0) = id and γi,j,n (1)(x) 6= x. This proves the claim. It is clear that this construction is stable: given n, we may choose i and j uniform in a neighborhood of every x ∈ M . Thus, by compactness, the previous considerations prove that for every C 1 -neighborhood N of the identity there exists δ > 0 such that for every x ∈ M there exists a continuous path γN ,x : [0, 1] → N such that d(x, γN ,x (1)(x)) > 3δ. Let ΓN ,x = γN ,x ([0, 1]) and IN ,x = ΓN ,x (x). For any y, z ∈ IN ,x , pick gy , gz ∈ ΓN ,x such that gy (x) = y and gz (x) = z. Then l(z) = Dg(y) · l(y), where g = gz ◦ gy−1 . In particular, l(y) is close to l(z) if N is small. By Proposition 5.1, IN ,x is a C 1 -embedded manifold everywhere tangent to l. It follows that IN ,x is an almost straight segment, of length greater than 3δ. Let w be a point of IN ,x at roughly the same distance from the two endpoints. Then let g ∈ ΓN ,x be such that g(x) = w, and define JN ,x = g −1 (IN ,x ). By construction, this is a C 1 -embedded segment passing through x, such that both components of JN ,x \ {x} have diameter at least δ, and such that l(y) is close to l(x) for every y ∈ JN ,x . Moreover, if y ∈ / JN ,x is close to x then l(y) is close to l(x) for otherwise JN ,y would intersect JN ,x , which would contradict Proposition 5.1. This proves that the line field l is indeed continuous. Lemma 5.3. — Let K be a compact subset of a surface supporting a lamination L by C 1 leaves L . Let xn , yn ∈ K be points in the same leaf Ln such that (xn )n and (yn )n converge to the same point p ∈ K. If the leafwise distance dn between xn and yn is bounded away from zero and infinity then the leaf L through p is an embedded circle of length at most lim inf dn . Proof. — Let ℓn be the leaf segment connecting xn to yn . Up to restricting to a subsequence, we may assume that the length of ℓn converges to inf n dn and, using AscoliArzela, (ℓn )n converges to a leaf segment ℓ of length inf n dn connecting p = limn xx to p = limn yn . This proves the claims. Let KS ⊂ Diff 1 (M ) denote the set of Kupka-Smale diﬀeomorphisms on M .

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Theorem 5.4. — Let G, K and L be as in Theorem 5.2. Assume that for every g ∈ G there exists a compact connected and locally connected set C ⊂ G such that g ∈ KS ∩C. If L ⊂ K is a non-isolated leaf of L , then there exists a G-invariant, leafwise continuous vector field tangent to L. Proof. — Let x0 ∈ L be arbitrary. It is enough to show that for every g0 ∈ G such that g0 (x0 ) = x0 we have Dg0 (x0 )|Tx0 L = id. Since G is connected, all the elements of G preserve an (arbitrary) orientation of L, and so Dg(x0 )|Tx0 L is the multiplication by some λ > 0. Suppose that λ is diﬀerent from 1. Let C be as in the hypothesis. Then C(x0 ) has bounded leafwise length in L. We claim that for every g1 ∈ C close to g0 there exists a fixed point x1 ∈ L leafwise close to x0 . It is clear that g1 (x0 ) ∈ L is close to x0 in the topology of M . Then, since C(x0 ) has bounded leafwise length, g1 (x0 ) must also be close to x0 in the topology of L. Since λ 6= 1, there is a small open segment around x inside L strictly invariant under g0 or g0−1 . This segment is still strictly invariant under g1 or g1−1 , and so it must contain a fixed point x1 of g1 . This proves the claim. Choose g1 ∈ KS ∩C and let x1 ∈ L be a fixed point close to x0 , as above. Since L is not isolated, we can choose a sequence zn ∈ K \ L converging to x1 . The distance from zn to wn = g1 (zn ) along the corresponding leaf Ln cannot go to infinity. Here we use the local connectivity of C to break into finitely many connected pieces Cj so that Cj (z) has small diameter, and hence, by connectivity, small leafwise diameter, for all z ∈ K. The leafwise distance from zn to wn cannot go to zero either, for otherwise x1 would be accumulated by fixed points of g1 , contradicting the KupkaSmale hypothesis (1). By Lemma 5.3, it follows that the leaf L is a circle. Then either every leaf through any point near L is a circle of bounded length (close to either the length of L or twice the length of L), or there is a leaf that spirals around L along one direction. The first case can be excluded because each such circle would have a fixed point under g1 near x1 , which would also contradict the Kupka-Smale condition. So from now on we ˜ spiraling around L. assume that there exists a leaf L ˜ in the direction of the spiraling, that is, Let v be a unit vector field tangent to L such that the corresponding ﬂow Ft is such that Ft (z) gets close to L as t → +∞ ˜ We can extend this vector field to L in a unique way so that there for any z ∈ L. ˜ if tn → +∞ and is forward continuity, that is, in such a way that for every z ∈ L Ftn (z) → w ∈ L then v(Ftn (z)) → v(w). ˜ → R such For any g ∈ G, there exists a unique continuous function φg : L that g(z) = Fφg (z) (z). Then φg extends to L in a unique way, with forward conti˜ and hence also on G × L. nuity. Note that (g, z) 7→ φg (z) is continuous on G × L (1)

Indeed, since g1 is Kupka-Smale, Dg1 (x1 ) is either a contraction or an expansion in the direction of L. Thus, by continuity, either g1 or g1−1 contracts the leaf segment joining zn and wn . Then the iterates, either forward or backward, of this segment must accumulate on a fixed point of g1 close to x1 in the same leaf as zn and, thus, distinct from x1 .

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Now suppose g is a Kupka-Smale element of g and x is a fixed point of g | L with ˜ and tn → ∞ such that Ft (z) → x. Then some small intervals φg (x) = 0. Select z ∈ L n around Ftn (z) are strictly invariant under either g or g −1 . This implies that x is accumulated by fixed points of g, which contradicts the Kupka-Smale. We may not be able to apply this argument directly to g = g1 and x = x1 because φg1 (x1 ) need not be zero: it may be any integer multiple of the length T of L (= minimal period of x1 under the ﬂow Ft ). However, we are going to show that there exist γn ∈ G such that, for every n large, x1 is a fixed point of γn g1n with φγn g1n (x1 ) = 0 and the derivative D(γn g1n | L) is far from 1. Then any Kupka-Smale g ∈ G near γn g1n has a fixed point x near x1 with φg (x) = 0 and derivative far from 1. In this way, the general case will be reduced to the setting handled in the previous paragraph. We are left to find (γn )n . We claim that supg∈G φg (z) = +∞ for every z ∈ L. Indeed, since G(z) is nontrivial, we can always find g ∈ G such that φg (z) is non-zero and, up to considering the inverse, it is no restriction to assume that φg (z) > 0. By the compactness of L, it follows that inf z∈L supg∈G φg (z) > 0. Then, using the relation φg2 ◦g1 (w) = φg2 (g1 (w)) + φg1 (w), and composing appropriately, we get the claim. Considering the inverse, it follows that inf g∈G φg (z) = −∞. Thus G ∋ g 7→ φg (z) ∈ R is surjective, for every z ∈ L. Note that maxz,w∈L |φg (z) − φg (w)| ≤ T , since g : L → L is a homeomorphism. It follows that there is h ∈ G such that φh has irrational translation number φhn (z) α = lim (uniformly over z ∈ L). n→±∞ nT To see this, take ǫ > 0, z˜ ∈ L and g˜ ∈ G such that φg˜ (˜ z ) > T +ǫ. Then inf z∈L φg˜ (z) > ǫ and so the the translation number of φg˜ is at lest ǫ. Join the identity to g˜ by some path in G. The translation numbers along the path must cover [0, ǫ] and so they do take irrational values. Then limn→±∞ | log Dhn |/n = 0 uniformly on z ∈ L. Let γ : [0, 1] → G be a continuous path with γ(0) = id and γ(1) = h, and extend it to a (continuous) path γ : R → G with γ(t + 1) = γ(t)h for every t ∈ R. Writing γ(t) = γ(t − [t])h[t] we see that | log Dγ(t)| | log Dhn | (6) lim = lim =0 t→±∞ n→±∞ t n and |φγ(t) (z) − tαT | ≤ |φγ(t−[t]) (h[t] (z))| + t − [t] αT + |φh[t] (z) − [t]αT |

is bounded. The latter ensures that we may select tn ∈ R such that φγ(tn ) (x1 ) = −nφg1 (x1 ), that is, φγ(tn )g1n (x1 ) = 0. Them x1 is a fixed point of γ(tn )g1n . Observe that tn → ±∞, depending on the sign of φg1 (x1 ), which at this point we may take to be non-zero. So, (6) implies that the derivative of γ(tn )g1n at the point x1 is far from 1 if n is large. To conclude it suﬃces to take γn = γ(tn ).

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Corollary 5.5. — Let G, K and L be as in Theorem 5.4, and G′ ⊂ G be the set of commutators. Then G′ acts trivially on any non-isolated leaf. Proof. — Let L be a non-isolated leaf, and v be the vector field given by Theorem 5.4. Then every element of G acts on L as time-t(g) map of the ﬂow of v. In particular, any commutator restricts to the identity on L. Corollary 5.6. — Let G, K and L be as in Theorem 5.4, and assume further that some commutator has only isolated fixed points. Then K is a finite union of circles. Proof. — Let g be a Kupka-Smale commutator. If there were a non-isolated leaf L, the previous corollary would give that g | L = id, which would violate the Kupka-Smale condition. So all leaves are isolated, and the result follows. For any periodic point p of g ∈ Diff 1 (M ) which is a hyperbolic attractor or repeller, let λ(g, p) ∈ [1, ∞) be the quotient of the logarithms of the norms of the eigenvalues of Dg n (p), where n is the period of p. We say that g ∈ Diff 1 (M ) is non-resonant if whenever λ(g, p) = λ(g, q), either the periods of p and q are distinct, or p and q belong to the same orbit. We say that g ′ ∈ Diff 1 (M ) is transverse to g ∈ Diff 1 (M ) if the stable manifolds of periodic saddles of g do not contain periodic points of g ′ . Theorem 5.7. — G, K and L be such that g and g k f , k ≥ 1 are points. Assume further that the ery g ∈ G there exist g ′ , g ′′ ∈ G there is only one G-orbit.

as in Theorem 5.4. Assume that there exist f, g ∈ G Kupka-Smale and f gf −1 g −1 has only isolated fixed non-resonant elements are dense in G, and for evarbitrarily close to g with g ′′ transverse to g ′ . Then

Proof. — By Corollary 5.6, K consists of finitely many circles. We are going to show that under the current assumptions it is actually empty. Suppose otherwise and let L be any of the leaves of K. Let g0 = g and gk = g k f for k ≥ 1. We claim that gk | L has a periodic point for some k ≥ 0. This can be seen as follows. Let µ be any g-invariant probability measure supported in L. If g0 = g has no periodic points in L then supp µ is either a Cantor set or the whole L. Keep in mind that all the elements of G preserve an (arbitrary) orientation of L. Since µ is g-invariant, the measure of a segment [y, g(y)] ⊂ L is independent of y. If µ is f -invariant, then µ([y, f (y)]) is also independent of y. Then µ [x, gf (x)] = µ [x, f (x)] + µ [f (x), gf (x)] = µ [g(x), f g(x)] + µ [x, g(x)] = µ [x, f g(x)] . This implies that gf (x) = f g(x) for every x ∈ supp µ, and so f gf −1 g −1 = id on the support of µ. That contradicts the hypothesis on the fixed points of f gf −1 g −1 , so µ cannot be f -invariant. In particular, there exists a segment J ⊂ L with µ(f (J)) < µ(J). We may choose J such that the endpoints are recurrent under g.

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Then we may find k ≥ 1 in such a way that g k (f (J)) ⊂ J, and so gk = g k ◦ f has a fixed point in J. This proves our claim. By perturbing gk we conclude that there exists some element h1 ∈ G which is nonresonant and has a non-zero number of periodic points in L which are all hyperbolic. If h1 has a saddle, let h = h1 or h = h−1 1 so that the stable manifold of some periodic point p ∈ L of h is contained in L. Let h′ and h′′ be diﬀeomorphisms close to h such that h′′ is transverse to h′ . Then the stable manifold of the continuation of p for h′ contains a definite neighborhood of p in L, and this neighborhood must also contain the continuation of p for g ′′ . This contradicts transversality, so the periodic points of h1 in L cannot be saddles. Now assume that all periodic points of h1 are hyperbolic attractors or repellers, and let n be their period. Let p be an attractor and q be a repeller for h1 . Let h2 ∈ G be such that h2 (p) = q. Using that h1 is non-resonant, we get that for suitable choices nk nl of k, l ∈ Z the diﬀeomorphism h = h−1 2 h1 h2 h1 has a hyperbolic saddle at p whose stable manifold is contained in L. Then we can use the argument in the previous paragraph to reach a contradiction. This completes the proof that K is empty. Hence all the G-orbits are open. By connectedness, it follows that there is a single orbit.

6. Density of stable accessibility Let f : M → M be a partially hyperbolic diﬀeomorphism of class C k , with k ≥ 2. Following Burns, Wilkinson [4], we say that f is center bunched if the functions ν, νˆ, γ, γˆ in (1) may be chosen to satisfy (7)

ν < γˆ γ

and

νˆ < γˆ γ.

Then the strong-stable (respectively, strong-unstable) holonomy maps of f inside each center-stable (respectively, center-unstable) leaf are C 1 (see [9]). We say that f is fibered, if — the quotient space M/W c is compact and Hausdorﬀ, and the canonical projection π : M → M/W c is a fiber bundle with C 1 fibers; — the map fc : M/W c → M/W c induced by f in the quotient space is a hyperbolic homeomorphism (in the sense of [14]). By the stability theorem of Hirsch, Pugh, Shub [7, Theorem 6.8], this is a robust property, that is, any C 1 -nearby diﬀeomorphism g : M → M is still fibered. Furthermore, g is topologically conjugate to a skew-product (gc , (gx )x ), where gc is a hyperbolic homeomorphism and each gx is a diﬀeomorphism between two fibers. Moreover, this conjugacy varies continuously in a neighborhood of f . Theorem 6.1. — Let f ∈ PH2k be a center bunched and fibered C k diﬀeomorphism with 2-dimensional center bundle. Then stably accessible diﬀeomorphisms are C k -dense in some neighborhood U of f .

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Let p be a periodic point of fc and Sp be the corresponding fiber. Let Gp be the group of contractible su-loops in M/W c based at p. Contractibility ensures that Gp is path-connected. The holonomy of the strong stable and strong unstable foliations of f yields a representation of Gp as a subgroup of Diff 1 (Sp ). We call an su-loop γ ∈ Gp simple if its corners are either periodic points or heteroclinic points associated to periodic points, and at least one of the corners is crossed only once by the loop. Similarly, we say that a pair of su-loops γ1 , γ2 ∈ Gp is simple if each one is simple and at least one of the corners is crossed only once by the union of the two loops. Proposition 6.2. — There exists a C 1 -neighborhood U of f and, for each 1 ≤ k ≤ ∞ there exists a C k -residual subset R ⊂ U ⊂ Diff k (M ) such that for g ∈ R , 1. every simple su-loop corresponds to a Kupka-Smale diﬀeomorphism; 2. if p and q are fixed attractors or repellers of a simple loop γ then λ1 (p)/λ2 (p) 6= λ1 (q)/λ2 (q), where λ1 , λ2 denote the Lyapunov exponents, in decreasing order; 3. for every simple pair (γ1 , γ2 ) of su-loops, the stable manifolds of saddle points of γ1 do not contain periodic points of γ2 ; 4. for every simple pair, the fixed points of the commutator are isolated. Proof. — Note that the holonomy associated to an su-loop can be decomposed as φ ◦ ψ, where ψ is the holonomy corresponding to the loop segment from p to q and φ is the holonomy over the loop segment from q to p. By considering perturbations localized around the fiber over q, the corresponding holonomy gets changed to φ◦h◦ψ, where h is an arbitrary smooth perturbation of the identity. Similarly, when considering a simple pair, we can perturb the dynamics so that one of the holonomy maps is unchanged, while the other changes from φ ◦ ψ to φ ◦ h ◦ ψ, where h is an arbitrary smooth perturbation of the identity. The conclusion follows then from usual transversality arguments, but there is a caveat. Transversality statements usually show that a C r map can be perturbed to another C r map so to obtain, say, the Kupka-Smale condition. However here we have perturbations of a slightly more special type. Recall that a periodic point p of period κ ≥ 1 of a diﬀeomorphism ϕ is said to be non-degenerate if Dϕκ (p) − id is an isomorphism. The main point in the proof is to show that, by arbitrarily small perturbations of the dynamics, one can ensure that the holonomies have only non-degenerate periodic points. We start with some abstract considerations. Let M be a compact smooth manifold of dimension d. Consider any Riemannian metric on M , and let d(·, ·) the associated distance function. It is no restriction to assume that the diameter is 1. The volume measure associated to the Riemannian metric will be denoted as vol or dx, indiﬀerently.

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Lemma 6.3. — If g : M → M is a C 1 map and p ∈ M is a degenerate fixed point then there exists an increasing function ω : [0, 1] → [0, 1] such that ω(δ)/δ d → ∞ when δ → 0 and vol({x ∈ U : d(g(x), x)k ≤ δ}) ≥ ω(δ) for all δ ∈ (0, 1). Moreover, ω may be chosen depending only on a modulus of continuity for Dg and an upper bound for its norm. Proof. — Let v ∈ Tp M be a unit vector with Dg(p)v = v and γ : (−1, 1) → M be the geodesic with γ(0) = p and γ ′ (0) = v. Let C > 1 be an upper bound for kDgk and φ : [0, 1] → [0, ∞) be a modulus of continuity for t 7→ Dg(γ(t))γ(t) ˙ − γ(t) ˙ (interpret this expression in local coordinates). For each δ > 0 small, let r be given by φ(r)r = δ/2. It is clear that r → 0 when δ → 0, and so r/δ = 1/(2φ(r)) → ∞ when δ → 0. Define ω(δ) = 2r(δ/(4C))d−1 . Let V be the tubular neighborhood of radius δ/(4C) around the geodesic segment γ([−r, r]). On the one hand, kDg(γ(t))γ(t) ˙ − γ(t)k ˙ ≤ φ(r) for every t ∈ [−r, r], and so kg(γ(t)) − γ(t)k ≤ φ(r)|t| ≤ δ/2 for every t ∈ [−r, r]. Thus, by the triangle inequality, kg(x) − xk ≤ C(δ/4C) + δ/2 + (δ/4C) < δ for every x ∈ V. On the other hand, vol(V ) ≥ 2r(δ/(4C))d−1 = ω(δ). Observe also that the property in the conclusion implies that there exists a decreasing homeomorphism α : [1, ∞) → (0, 1] such that Z ∞ Z ∞ (8) ω(α(s)) ds = ∞ and α(s)d ds < ∞. 1

1

This can be seen as follows. Let δj ∈ (0, 1), j ≥ 1, be a sequence such that δ1 = 1, d δj+1 < δj /10 and ω(δj+1 ) ≥ 2j δj+1 . Denote ǫj = j −2 δj−d for each j ≥ 1. Notice that ǫj+1 ≥ 2ǫjR. Now let α be a homeomorphism mapping each [ǫj , ǫj+1 ) to (δj+1 , δj ] ǫ and such that ǫjj+1 α(s)d ds is within a factor of two of the infimum for such homeomorphisms. Then Z ∞ ∞ ∞ X X 1 d < ∞. (ǫj+1 − ǫj ) δj+1 ≤2 α(s)d ds ≤ 2 (j + 1)2 1 j=1 j=1 On the other hand, Z ∞ ∞ ∞ X X ω(α(s))ds ≥ (ǫj+1 − ǫj ) ω(δj+1 ) ≥ 1

j=1

j=1

2j = ∞. 2(j + 1)2

Let X1 , . . . , Xr be smooth vector fields spanning T M , and Φt1 , . . . , Φtr be the corresponding ﬂows. For each t = (t1 , . . . , tr ) ∈ Rr , denote Φt = Φtrr ◦ · · · ◦ Φt11 . Fix ǫ > 0 small enough that the map t 7→ Φt (x) from P = [−ǫ, ǫ]r to M is a submersion for every x ∈ M . The Lebesgue measure on P is denoted by m or dt, indiﬀerently. Lemma 6.4. — Let ψ : M → M be a C 1 diﬀeomorphism. Then the fixed points of Φt ◦ ψ −1 are non-degenerate for almost every t ∈ P .

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Proof. — Fix ω : [0, 1] → [0, 1] satisfying the conclusion of Lemma 6.3 for every g = Φt ◦ ψ −1 : this is possible because the derivatives of these maps are uniformly bounded and admit a uniform modulus of continuity. Then let α : [1, ∞) → (0, 1] be as in (8) and β = α−1 : (0, 1] → [1, ∞). Define Z β(d(Φt (x), ψ(x)) dx for each t ∈ P. E(t) = M

The conclusion follows directly from the observations in the next two paragraphs. Since ψ is a diﬀeomorphism, there exists c1 > 0 such that, for any δ > 0, vol x : d(Φt (x), ψ(x)) ≤ δ ≥ c1 vol x : d(Φt (ψ −1 (y)), y) ≤ δ .

So, if t ∈ P is such that Φt ◦ ψ −1 has some degenerate fixed point then, Z ∞ Z vol x : β(d(Φt (x), ψ(x))) ≥ s ds β(d(Φt (x), ψ(x)) dx = E(t) = 1 M Z ∞ t −1 vol y : β(d(Φ (ψ (y)), y))) ≥ s ds ≥ c1 Z ∞ Z1 ∞ ω(α(s)) ds = ∞. vol y : d(Φt (ψ −1 (y)), y) ≤ α(s) ds ≥ c1 ≥ c1 1

1

Since t 7→ Φ (x) is a submersion, there exists C2 > 0 such that m t : d(Φt (x), ψ(x)) ≥ δ ≤ C2 δ d t

for every x ∈ M and δ > 0. Thus, Z Z Z β(d(Φt (x), ψ(x)) dt dx E(t) dt = P ZM ZP∞ m t : β(d(Φt (x), ψ(x))) ≥ s ds = Z ZM Z1 ∞ m x : d(Φt (x), ψ(x)) ≤ α(s) ds ≤ = M

In particular,

R

P

1

C2 α(s)d ds.

M

E(t) dt is finite.

At this stage, we can conclude that given a simple loop there is a perturbation that makes the holonomy have only non-degenerate fixed points. Such fixed points are finitely many, and they undergo no bifurcations under small perturbations. It is then easy to do an additional perturbation that makes the fixed points hyperbolic. In order to deal with more general periodic points, we proceed by induction. We assume that we have already obtained a perturbation whose periodic orbits of period at most n are hyperbolic. We now consider periodic orbits of period n + 1. These must be far from periodic orbits of period n (since those are hyperbolic). We proceed in a similar way as before, with more localized perturbations (by considering vector fields with small support). More precisely, given some small open set V which does not intersect its first n iterates, and a compact set K ⊂ V we can consider vector fields supported in V and which span the tangent space over K. Then the iterate n+1

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near K decomposes as a diﬀeomorphism φ ◦ h ◦ ψ, where h is controlled by the vector fields, while φ and ψ remain fixed. It thus follows, as before, that we can eliminate degenerate periodic points of period n + 1 over K. By a covering argument, we can thus eliminate degenerate periodic points of period n + 1 over all of M (and make then hyperbolic by a further perturbation). Now we can easily obtain the first item by applying again localized perturbations and covering arguments to eliminate tangencies between stable and unstable manifolds by the usual technique. Indeed, in charts the issue reduces to being given two graphs of C 1 functions R → R and being able to move smoothly one of them in order to achieve transversality with respect to the other: this can be achieved simply by vertical translations, applying Sard’s Theorem to the diﬀerence of the two functions. The second item presents no diﬃculties, using localized perturbations in the usual way. As for the third, we use a preliminary perturbation in order to make the sets of periodic points for the two diﬀeomorphisms (which are countable, by the first item) disjoint, then perturb again to move either the stable manifolds associated to periodic points of γ1 or the periodic points of γ2 so that they avoid each other. For the fourth item, we may first perturb to ensure that γ1 maps fixed points of γ2 to non-fixed points of γ2 and vice-versa. We can now make localized perturbations to make the fixed points of the commutator non-degenerate with an argument similar to the one we used to show non-degeneracy of the periodic points of a single simple loop above (since the localized perturbation will aﬀect only one step of the composition).

Proof of Theorem 6.1. — Take U and R k as in Proposition 6.2. The proposition ensures that for g ∈ R the representation of Gp in Diff 1 (Sp ) satisfies the conditions of Theorem 5.7. Hence, Sp is contained in an accessibility class. Saturating by stable and unstable leaves, we conclude that g is accessible. By Theorem B it is stably accessible. Let us conclude by pointing out that the same arguments also yield a version of Theorem 6.1 restricted to the subspace of volume-preserving diﬀeomorphisms. Actually, the definition of Kupka-Smale in the volume-preserving setting is slightly diﬀerent, allowing for elliptic periodic points. However, that need not concern us here, because we only deal with fixed points contained in some invariant leaf, and those cannot be elliptic.

References [1] A. Avila, J. Santamaria & M. Viana – “Holonomy invariance: rough regularity and applications to Lyapunov exponents”, Astérisque 358 (2013), p. 13–74. [2] A. Avila & M. Viana – “Extremal Lyapunov exponents: an invariance principle and applications”, Invent. math. 181 (2010), p. 115–189.

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[3] K. Burns, F. R. Hertz, M. A. R. Hertz, A. Talitskaya & R. Ures – “Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center”, Discrete Contin. Dyn. Syst. 22 (2008), p. 75–88. [4] K. Burns & A. Wilkinson – “On the ergodicity of partially hyperbolic systems”, Ann. of Math. 171 (2010), p. 451–489. [5] P. Didier – “Stability of accessibility”, Ergodic Theory Dynam. Systems 23 (2003), p. 1717–1731. [6] D. Dolgopyat & A. Wilkinson – “Stable accessibility is C 1 dense”, in Geometric methods in dynamics II, Astérisque, vol. 287, 2003, p. 33–60. [7] M. W. Hirsch, C. Pugh & M. Shub – Invariant manifolds, Lecture Notes in Math., vol. 583, Springer, 1977. [8] V. Horita & M. Sambarino – “Stable ergodicity and accessibility for certain partially hyperbolic diffeomorphisms with bidimensional center leaves”, Comment. Math. Helv. 92 (2017), p. 467–512. [9] C. Pugh, M. Shub & A. Wilkinson – “Hölder foliations”, Duke Math. J. 86 (1997), p. 517–546. [10] D. Repovš, A. B. Skopenkov & E. V. Ščepin – “C 1 -homogeneous compacta in Rn are C 1 -submanifolds of Rn ”, Proc. Amer. Math. Soc. 124 (1996), p. 1219–1226. [11] F. Rodriguez Hertz – “Stable ergodicity of certain linear automorphisms of the torus”, Ann. of Math. 162 (2005), p. 65–107. [12] F. Rodriguez Hertz, M. A. Rodriguez Hertz & R. Ures – “Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle”, Invent. math. 172 (2008), p. 353–381. [13] J. Rodriguez Hertz & C. Vásquez – “Structure of accessibility classes”, preprint arXiv:1706.01156. [14] M. Viana – “Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents”, Ann. of Math. 167 (2008), p. 643–680.

A. Avila, Institut für Mathematik, Universität Zürich Winterthurerstrasse 190, CH-8057 Zürich, Switzerland & IMPA – Estrada D. Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro – Brazil. E-mail : [email protected] M. Viana, IMPA – Estrada D. Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro – Brazil. • E-mail : [email protected]

ASTÉRISQUE 416

Astérisque 416, 2020, p. 321–340 doi:10.24033/ast.1118

ATTRACTED BY AN ELLIPTIC FIXED POINT by Bassam Fayad, Jean-Pierre Marco & David Sauzin

À notre mentor et ami Jean-Christophe Yoccoz

Abstract. — We give examples of symplectic diﬀeomorphisms of R6 for which the origin is a non-resonant elliptic fixed point which attracts an orbit. Résumé (Attiré par un point fixe elliptique). — Nous donnons des exemples de diﬀéomorphismes symplectiques de R6 pour lesquels l’origine est un point fixe elliptique non résonant qui attire une orbite.

1. Introduction Consider a symplectic diﬀeomorphism of R2n (for the canonical symplectic form) with a fixed point at the origin. We say that the fixed point is elliptic of frequency vector ω = (ω1 , . . . , ωn ) ∈ Rn if the linear part of the diﬀeomorphism at the fixed point is conjugate to the rotation map Sω : (R2 )n ý,

Sω (s1 , . . . , sn ) := (Rω1 (s1 ), . . . , Rωn (sn )).

Here, for β ∈ R, Rβ stands for the rigid rotation around the origin in R2 with rotation number β. We say that the frequency vector ω is non-resonant if for any k ∈ Zn − {0} we have (k, ω) ∈ / Z, where (· , ·) stands for the Euclidean scalar product. It is easy to construct symplectic diﬀeomorphisms with orbits attracted by a resonant elliptic fixed point. For instance, the time-1 map of the ﬂow generated by the Hamiltonian function H(x, y) = y(x2 + y 2 ) in R2 has a saddle-node type fixed point, at which the linear part is zero, which attracts all the points on the positive part of the x-axis. The situation is much subtler in the non-resonant case. 2010 Mathematics Subject Classification. — 37C75; 37J12. Key words and phrases. — Dynamical system, symplectomorphism, elliptic fixed point, instability, Gevrey map.

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Our goal in this paper is to construct an example of symplectic diﬀeomorphism with an orbit converging to an elliptic non-resonant fixed point. Note that, in such an example, the inverse symplectomorphism has a Lyapunov unstable fixed point. The Anosov-Katok construction [1] of ergodic diﬀeomorphisms by successive conjugations of periodic rotations of the disk gives examples of smooth area preserving diﬀeomorphisms with non-resonant elliptic fixed points at the origin that are Lyapunov unstable. The method also yields examples of ergodic symplectomorphisms with non-resonant elliptic fixed points in higher dimensions. These constructions obtained by the successive conjugation technique have totally degenerate fixed points since they are C ∞ -tangent to a rotation Sω at the origin. In the non-degenerate case, R. Douady gave examples in [4] of Lyapunov unstable elliptic points for smooth symplectic diﬀeomorphisms for any n ≥ 2, for which the Birkhoﬀ normal form has non-degenerate Hessian at the fixed point but is otherwise arbitrary. Prior examples for n = 2 were obtained in [5] (note that by KAM theory, a non-resonant elliptic fixed point of a smooth area preserving surface diﬀeomorphism that has a non zero Birkhoﬀ normal form is accumulated by invariant quasi-periodic smooth curves—see [14]—, hence, for n = 1, non-degeneracy implies that the point is Lyapunov stable). In both of the above examples, there is no claim about the existence of an orbit converging to the fixed point for the forward or backward dynamics. In fact, in the Anosov-Katok examples, a sequence of iterates of the diﬀeomorphism converges uniformly to Identity, hence every orbit is recurrent and no forward orbit can converge to the origin, besides the origin itself. As for the non-degenerate examples of Douady and Le Calvez, their Lyapunov instability is deduced from the existence of a sequence of points that converge to the fixed point and whose orbits travel along a simple resonance away from the fixed point, not from the existence of one particular orbit. In this paper, we will construct an example of a Gevrey diﬀeomorphism possessing an orbit which converges to a fixed point.Recall that, given a real α ≥ 1, Gevrey-α regularity is defined by the requirement that the partial derivatives exist at all (multi)orders ℓ and are bounded by CM |ℓ| |ℓ|!α for some C and M (when α = 1, this simply means analyticity); upon fixing a real L > 0 which essentially stands for the inverse of the previous M , one can define a Banach algebra Gα,L (R2n ), k . kα,L . We set X := (R2 )3 and denote by U α,L the set of all Gevrey-(α, L) symplectic diﬀeomorphisms of X which fix the origin and are C ∞ -tangent to Id at the origin. We refer to Appendix for the precise definition of U α,L and of a distance dist(Φ, Ψ) = kΦ − Ψkα,L which makes it a complete metric space. We will prove the following.

Theorem A. — Fix α > 1 and L > 0. For each γ > 0, there exist a non-resonant vector ω ∈ R3 , a point z ∈ X, and a diﬀeomorphism Ψ ∈ U α,L such that kΨ − Idkα,L ≤ γ and T = Ψ ◦ Sω satisfies T n (z) −→ 0. n→+∞

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We do not know how to produce real analytic examples. After the first version of the present work was completed, the first real analytic symplectomorphisms with Lyapunov unstable non-resonant elliptic fixed points were constructed in [6] (but with no orbits asymptotic to the fixed point). For other instances of the use of Gevrey regularity with symplectic or Hamiltonian dynamical systems, see e.g., [15], [11], [12], [13], [10], [3]. Our construction easily extends to the case where X = (R2 )n with n ≥ 3, however we do not know how to adapt the method to the case n = 2. As for the case n = 1, there may well be no regular examples at all. Indeed if the rotation frequency at the fixed point is Diophantine, then a theorem by Herman (see [7]) implies that the fixed point is surrounded by invariant quasi-periodic circles, and thus is Lyapunov stable. The same conclusion holds by Moser’s KAM theorem if the Birkhoﬀ normal form at the origin is not degenerate [14]. In the remaining case of a degenerate Birkhoﬀ normal form with a Liouville frequency, there is evidence from [2] that the diﬀeomorphism should then be rigid in the neighborhood of the origin, that is, there exists a sequence of integers along which its iterates converge to Identity near the origin, which clearly precludes the convergence to the origin of an orbit. Similar problems can be addressed where one searches for Hamiltonian diﬀeomorphisms (or vector fields) with orbits whose α-limit or ω-limit have large Hausdorﬀ dimension (or positive Lebesgue measure) and in particular contain families of nonresonant invariant Lagrangian tori instead of a single non-resonant fixed point. A specific example for Hamiltonian ﬂows on (T × R)3 is displayed in [9], while a more generic one has been announced in [8]. In these examples, the setting is perturbative and the Hamiltonian ﬂow is non-degenerate in the neighborhood of the tori. The methods involved there are strongly related to Arnold diﬀusion and are completely diﬀerent from ours. 2. Preliminaries and outline of the strategy From now on we fix α > 1 and L > 0. We also pick an auxiliary L1 > L. For z ∈ R2 and ν > 0, we denote by B(z, ν) the closed ball relative to k . k∞ centered at z with radius ν. Since α > 1, we have Lemma 2.1. — There is a real c = c(α, L1 ) > 0 such that, for any z ∈ R2 and ν > 0, there exists a function fz,ν ∈ Gα,L1 (R2 ) which satisfies (a) 0 ≤ fz,ν ≤ 1, (b) fz,ν ≡ 1 on B(z, ν/2), (c) fz,ν ≡ 0 on B(z, ν)c , 1 (d) kfz,ν kα,L1 ≤ exp(c ν − α−1 ). Proof. — Use Lemma 3.3 of [12]. We now fix an arbitrary real R > 0 and pick an auxiliary function ηR ∈ Gα,L1 (R) which is identically 1 on the interval [−2R, 2R], identically 0 outside [−3R, 3R], and

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everywhere non-negative. We then define gR :R2 → R by the formula (2.1)

gR (x, y) := xy ηR (x) ηR (y).

The following diﬀeomorphisms will be of constant use in this paper: Definition 2.1. — For i 6= j ∈ {1, 2, 3}, z ∈ R2 and ν > 0, we denote by Φi,j,z,ν the 2 time-one map of the Hamiltonian ﬂow generated by the function exp(−c ν − α−1 )fz,ν ⊗i,j gR , where fz,ν ⊗i,j gR :X → R stands for the function s = (s1 , s2 , s3 ) 7→ fz,ν (si )gR (sj ). In the above definition, our convention Hamiltonian vector field generP ∂H ∂ for the ∂H ∂ ated by a function H is XH = (− ∂y + ∂xi ∂yi ). Note that the Hamiltonian i ∂xi 2

exp(−c ν − α−1 )fz,ν ⊗i,j gR can be viewed as a compactly supported function of si and sj , hence it generates a complete vector field and Definition 2.1 makes sense. Actually, any H ∈ Gα,L1 (X) has bounded partial derivatives, hence XH is always complete; the ﬂow of XH is made of Gevrey maps for which estimates are given in Appendix A.2. In the case of Φi,j,z,ν , for ν small enough we have (2.2)

Φi,j,z,ν ∈ U α,L

and

1

kΦi,j,z,ν − Idkα,L ≤ K exp(−c ν − α−1 ),

with K := CkgR kα,L1 , where C is independent from i, j, z, ν and stems from (A.6). Here are the properties which make the Φi,j,z,ν ’s precious. To alleviate the notations, we state them for Φ2,1,z,ν but similar properties hold for each diﬀeomorphism Φi,j,z,ν . Lemma 2.2. — Let z ∈ R2 and ν > 0. Then Φ2,1,z,ν satisfies: (a) For every (s1 , s2 , s3 ) ∈ X such that s2 ∈ B(z, ν)c , Φ2,1,z,ν (s1 , s2 , s3 ) = (s1 , s2 , s3 ). (b) For every x1 ∈ R, s2 ∈ R2 and s3 ∈ R2 , Φ2,1,z,ν ((x1 , 0), s2 , s3 ) = ((e x1 , 0), s2 , s3 ) with |e x1 | ≤ |x1 | . (c) For every x1 ∈ [−2R, 2R], s2 ∈ B(z, ν/2) and s3 ∈ R2 , Φ2,1,z,ν ((x1 , 0), s2 , s3 ) = ((e x1 , 0), s2 , s3 ) with |e x1 | ≤ κ |x1 | , where κ := 1 −

1 2

2

exp(−c ν − α−1 ).

Hence, a map like Φ2,1,z2 ,ν2 will preserve the x1 -axis and “descend" orbits towards the origin on this axis, while keeping the other two variables frozen (item (b)). However, it is only when the second variable is inside the ball of radius ν2 around z2 that Φ2,1,z2 ,ν2 will eﬀectively bring down a point of the x1 -axis towards the origin (item (c)). Let us roughly summarize this by saying that Φ2,1,z2 ,ν2 acts as an elevator on the first x-axis, that never goes up and that eﬀectively goes down when the second variable is in some given ball, that we call “activating”. Moreover, if the second variable is securely outside the activating ball, then Φ2,1,z2 ,ν2 is completely inactive

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(Identity). Finally, our elevator is close to Identity when the parameter ν2 is close to zero. Proof of Lemma 2.2. — The dynamics of the ﬂow generated by fz,ν ⊗2,1 gR can easily be understood from those of the ﬂows generated by fz,ν alone on the second factor R2 and by gR alone on the first factor R2 . Indeed, for any functions f and g on R2 , Φf ⊗2,1 g (s1 , s2 , s3 ) = Φf (z2 )g (s1 ), Φg(z1 )f (s2 ), s3 , where Φh denotes the time one map associated to the Hamiltonian h, hence Φ2,1,z,ν (s1 , s2 , s3 ) = Φδfz,ν (s2 )gR (s1 ), ΦδgR (s1 )fz,ν (s2 ), s3 2

with δ := exp(−c ν − α−1 ). This immediately yields (a). e1 (τ ), 0 with Suppose s1 = (x1 , 0). We get gR (s1 ) = 0 and Φτ gR (x1 , 0) = x x1 x e1 (0) = x1 and de x1 ηR (e x1 ), hence |e x1 (τ )| ≤ |x1 | for any τ ≥ 0, and (b) follows dτ = −e using τ := δfz,ν (s2 ). If moreover x1 ∈ [−2R, 2R], then x ˜1 (τ ) = e−τ x1 . We conclude by observing that s2 ∈ B(z, ν/2) implies τ = δfz,ν (s2 ) ≤ 1, whence e−τ ≤ 1 − 12 τ . From now on, we denote simply by | . | the k . k∞ norm in R2 or in X = R6 , and by B(s, ρ) the corresponding closed ball centered at s with radius ρ (the context will tell whether it is in R2 or R6 ). Heuristics of the synchronized attraction scheme. We describe now the attraction mechanism towards the origin, that will be carried out in Section 3. It is based on the use of longer and longer compositions of regularly alternating ‘elevators’, more precisely compositions of a large number of maps of the form Φ1,3,z1 ,ν1 ◦ Φ3,2,z3 ,ν3 ◦ Φ2,1,z2 ,ν2 (with an inductive choice of the parameters zi and νi ) followed by rigid rotations S (ωn ) , with an appropriate sequence of resonant frequencies ωn . Suppose that z2 is inside the activating ball of some elevator Φ2,1 , which is hence actively descending z1 on the x1 -axis. Suppose also that, simultaneously, some Φ3,2 is descending z2 . At some point, z2 will exit the activating ball of Φ2,1 , which then becomes completely inactive. The variable z1 stops its descent and will just be rotating due to the rotation S (ωn ) . A Φ1,3 that is active at this height of z1 can then be used to descend z3 . As z3 goes down, Φ3,2 becomes inactive and z2 will henceforth only rotate. This allows to introduce a new Φ2,1 which is active at this new height of z2 . An alternating procedure of the three types of elevators can thus be put in place. At each moment in the attraction procedure, one variable is just rotating and, each time it enters an activating ball, it drives down strictly another variable. The third variable in the meantime must just not go up. This is the content of Lemma 3.1 below, where we see that, when a composition T1 of Φ2,1 and Φ1,3 is active, we have z1 strictly going down, z3 just not going up, and z2 just rotating.

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However, for this description to hold, a fine synchronization between the three components ωi of the frequency of the rotation is required, guaranteeing for example that when one variable enters an activating ball, the corresponding variable that it should bring down indeed happens to be on its own x-axis. For example, when we use T1 , we take the ωi ’s rational, with the denominator of ω2 being a multiple of the denominator of ω1 that is itself a multiple of the denominator of ω3 . Of course, this constrains us to deal with resonant frequencies, which is why we use a sequence of resonant ω (n) ’s. They will be chosen so as to suit the fine-tuned alternating attraction mechanism that we just tried to convey, while converging to a non-resonant frequency. Observe that if ω1 and ω2 were rationals with the same denominator, then it would be possible to get an attraction scheme by just alternating maps of the form Φ2,1 and Φ1,2 . With non-resonant frequencies however, we could not put up the attraction scheme with just two variables and our method, as is, does not yield a statement similar to Theorem A on R4 , let alone R2 . In summary, Theorem A is obtained by an inductive construction of the required Ψ, z and ω: — The diﬀeomorphism Ψ in Theorem A will be obtained as an infinite product (for composition) of diﬀeomorphisms of the form Φi,j,z,ν , with smaller and smaller values of ν so as to derive convergence in U α,L from (2.2). — On the other hand, the initial condition z will be obtained as the limit of a sequence contained in the ball B(0, R) ⊂ X. — As for the non-resonant frequency vector ω in Theorem A, it will be obtained as a limit of vectors with rational coordinates with larger and larger denominators, so as to make possible a kind of “orbit synchronization” at each step of the construction.

3. The attraction mechanism From now on, for any three integers q1 , q2 , q3 , we use the notation “q1 | q2 ” to indicate the existence of k ∈ Z such that q1 k = q2 , and “q1 | q2 | q3 ” when q1 | q2 and q2 | q3 . Starting from a point z = ((x1 , 0), (x2 , 0), (x3 , 0)), the mechanism of attraction of the point to the origin is an alternation between bringing closer to zero the x1 ,x2 or x3 coordinates when all the coordinates of the point come back to the horizontal axes. The main ingredient is the following lemma, where we use shortcut notation Φi,j,x,ν for Φi,j,(x,0),ν . Lemma 3.1. — Let ω = (p1 /q1 , p2 /q2 , p3 /q3 ) ∈ Q3 with pi , qi coprime positive integers and z = ((x1 , 0), (x2 , 0), (x3 , 0)) ∈ B(0, R).

ASTÉRISQUE 416

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327

Set T1 = Φ2,1,x2 ,q−3 ◦ Φ1,3,x1 ,q−3 ◦ Sω 2

1

T2 = Φ3,2,x3 ,q−3 ◦ Φ2,1,x2 ,q−3 ◦ Sω 3

2

T3 = Φ1,3,x1 ,q−3 ◦ Φ3,2,x3 ,q−3 ◦ Sω . 1

3

Then the following properties hold. I) If q3 | q1 | q2 , and x1 ≥ 1/q1 ,

x2 ≥ 0,

x3 ≥ 1/q3 ,

then there exists N ≥ 1 such that T1N (z) = ((ˆ x1 , 0), (ˆ x2 , 0), (ˆ x3 , 0)) with 0≤x ˆ1 ≤ x1 /2,

0≤x ˆ2 = x2 ,

0≤x ˆ3 ≤ x3 ,

and |T1m (z)i | ≤ xi for all m ∈ {0, . . . , N }. II) If q1 | q2 | q3 , and x1 ≥ 0,

x2 ≥ 1/q2 ,

x3 ≥ 1/q3 ,

then there exists N ≥ 1 such that T2N (z) = ((ˆ x1 , 0), (ˆ x2 , 0), (ˆ x3 , 0)) with 0≤x ˆ1 ≤ x1 ,

0≤x ˆ2 ≤ x2 /2,

0≤x ˆ3 = x3 ,

and |T2m (z)i | ≤ xi for all m ∈ {0, . . . , N }. III) If q2 | q3 | q1 , and x1 ≥ 1/q1 ,

x2 ≥ 0,

x3 ≥ 1/q3 ,

then there exists N ≥ 1 such that T3N (z) = ((ˆ x1 , 0), (ˆ x2 , 0), (ˆ x3 , 0)) with 0≤x ˆ1 = x1 , and

|T3m (z)i |

0≤x ˆ2 ≤ x2 ,

0≤x ˆ3 ≤ x3 /2,

≤ xi for all m ∈ {0, . . . , N }.

Proof. — We will prove the lemma for T2 since it will be the first map that we will use in the sequel. The proof for the maps T1 and T3 follows exactly the same lines. The hypothesis x2 ≥ 1/q2 implies that the orbit of z2 = (x2 , 0) under the rotation Rω2 enters the q2−3 neighborhood of z2 only at times that are multiples of q2 . Moreover Rωℓq22 (z2 ) = z2 . A similar remark holds for z3 . Since q3 ≥ q2 , we consider the action of T := Φ2,1,x2 ,q−3 ◦ Sω first. Since q1 | q2 , if 2 s = (s1 , s2 , s3 ) with s1 = (u1 , 0) and s2 = (u2 , 0), by Lemma 2.2 (a)–(b): T

m

(s) = (s1,m , Rωm2 (s2 ), Rωm3 (s3 )) for all m ∈ N,

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with |s1,m | ≤ |s1 | . Consider now the orbit of z under the full diﬀeomorphism T2 . Since q2 | q3 , the previous remark shows that one has to take the eﬀect of Φ3,2,x3 ,q−3 into account only 3 for the iterates of order m = ℓq3 . One therefore gets T2m (z) = (z1,m , z2,m , Rωm3 (z3 )),

for all m ∈ N,

where in particular z2,ℓq3 = (x2,ℓq3 , 0) with 6

0 < x2,(ℓ+1)q3 ≤ (1 −

1 2

exp(−cq3α−1 ))x2,ℓq3 ,

and where ′

z2,ℓq3 +ℓ′ = Rωℓ 2 (z2,ℓq3 ), |z1,m | ≤ x1 ,

1 ≤ ℓ′ ≤ q3 − 1,

for all m ∈ N.

We let L be the smallest integer such that 0 < x2,Lq3 ≤ x2 /2 and get the conclusion with N = Lq3 .

4. Proof of Theorem A The proof is based on an iterative process (Proposition 4.3), which itself is based on the following preliminary result. Proposition 4.1. — Let ω = (p1 /q1 , p2 /q2 , p3 /q3 ) ∈ Q3+ with q3 | q1 | q2 and z = ((x1 , 0), (x2 , 0), (x3 , 0)) ∈ B(0, R) with x1 , x2 , x3 > 0 and x2 ≥ 1/q2 . Then, for any η > 0, there exist (a) ω = (p1 /q 1 , p2 /q 2 , p3 /q 3 ) such that q 3 | q 1 | q 2 , the orbits of the translation of vector ω on T3 are η-dense and |ω − ω| ≤ η; (b) z¯ = ((x1 , 0), (x2 , 0), (x3 , 0)) such that 0 < xi ≤ xi /2 for every i ∈ {1, 2, 3} and x2 ≥ 1/q 2 ; (c) z ′ ∈ X, x b1 ∈ (x1 + q13 , x1 ) and N ≥ 1, such that |z ′ − z| ≤ η and the diﬀeo1 morphism T

= Φ2,1,x2 ,q−3 ◦ Φ1,3,bx1 ,q−3 ◦ Φ3,2,x3 ,q−3 ◦ Φ2,1,x2 ,q−3 ◦ Sω 2

1

3

satisfies T

and |T

m

N

(z ′ ) = z¯

(z ′ )i | ≤ (1 + η)xi for m ∈ {0, . . . , N }.

Moreover, q 1 , q 2 and q 3 can be taken arbitrarily large. The proof of Proposition 4.1 will require the following

ASTÉRISQUE 416

2

ATTRACTED BY AN ELLIPTIC FIXED POINT

329

Lemma 4.2. — Given η, Q > 0 and p/q ∈ Q with p and q coprime integers, q ≥ 1, there exists pb/b q ∈ Q with pb and qb coprime integers such that pb p − < η. q | qb, qb > Q, qb q Proof of Lemma 4.2. — According to Dirichlet’s Theorem on Primes in Arithmetic Progressions, there are infinitely many prime numbers of the form ℓp + 1 with ℓ ∈ N∗ . We can thus find an integer ℓ > max{Q, 1/η} such that pb := ℓp + 1 is prime, and the 1 conclusion then holds with qb := ℓq since pqbb − pq = ℓq .

Proof of Proposition 4.1. — We divide the proof into three steps.

1. First use Lemma 4.2 to choose coprime integers pb3 and qb3 with qb3 large multiple of q2 , so that 1 1 1 x3 ≥ ,

1 , qe1

1 x b1 > 3. 2 qe1

(4.3) Set

qb3 < η, qe1

f3 = Φ T e. 1,3,b x1 ,e q −3 ◦ Φ3,2,x3 ,b q −3 ◦ Sω 1

3

e ≥ 1 and ze = ((e By Lemma 3.1 III), there exist N x1 , 0), (e x2 , 0), (e x3 , 0)) such e N

f3 (b z ) = ze with that T (4.4)

x e1 = x b1 ,

x e2 ≤ x b2 ≤ x2 /2,

x e3 ≤ x b3 /2 = x3 /2,

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e }. bi for all m ∈ {0, . . . , N and Te3m (b z )i ≤ x Define now T = Φ1,3,bx1 ,eq−3 ◦ Φ3,2,x3 ,bq−3 ◦ Φ2,1,x2 ,q−3 ◦ Sωe . 1

3

2

Choosing qe1 in (4.2) large enough and pe1 properly (using Lemma 4.2), one can assume that ω e is arbitrarily close to ω b , so that Sωe−1 is arbitrarily C 0 -close to Sωb−1 on the ball B = B(0, |z| + 1), and moreover that the inverse of Φ1,3,bx1 ,eq−3 is arbitrarily 1 C 0 -close to Id on B. As a consequence, one can assume that T−1 is arbitrarily C 0 -close to Tb2−1 on B. Hence one can choose ω e with |e ω − ω| < η such that there exists z with |z − z| < η which satisfies b

TN (z) = zb,

|Tm (z)i | ≤ (1 + η)xi

b }. for all m ∈ {0, . . . , N

Moreover, using Lemma 2.2, one proves by induction that: f3 m (b e }. Tm (b z )2 ∈ B(x2 , qb2−3 )c , Tm (b z) = T z ) for all m ∈ {0, . . . , N

As a consequence

b e f3 Ne (b TN +N (z) = T z ) = ze m b e }. and |T (z)i | < (1 + η)xi for all m ∈ {0, . . . , N + N

3. It remains now to perturb T in the same way as above to bring the first component of ze closer to the origin. Use again Lemma 4.2 and consider coprime integers p2 and q 2 such that qe1 x2 ≥ 1/q 2 , x e2 ≥ 1/q 2 , < η, (4.5) qe1 | q 2 , q2 and such that the vector (4.6) satisfies |ω − ω| < η. Set now T

ω = (e p1 /e q1 , p2 /q 2 , pb3 /b q3 )

= Φ2,1,ex2 ,q−3 ◦ Φ1,3,bx1 ,eq−3 ◦ Φ3,2,x3 ,bq−3 ◦ Φ2,1,x2 ,q−3 ◦ Sω . 2

1

3

2

As above, a proper choice of p2 and q 2 satisfying (4.5) makes T −1 arbitrarily 0 C -close to T−1 and yields the existence of a z ′ ∈ X such that |z ′ − z| < η, satisfying T

b +N e N

Set

(z ′ ) = ze,

|T

m

(z ′ )i | < (1 + η)xi

b +N e }. for all m ∈ {0, . . . , N

T 1 = Φ2,1,ex2 ,q−3 ◦ Φ1,3,bx1 ,q−3 ◦ Sω . 2

1

Using Lemma 2.2 and Lemma 3.1 I), one proves by induction that now for m ≥ 0: T

m

c (e z )2 ∈ B(x2 , q −3 2 ) ,

T

m

c (e z )3 ∈ B(x3 , q −3 3 ) ,

T

m

By Lemma 3.1 I) there exists N such that N

T 1 (e z ) = z¯ = ((x1 , 0), (x2 , 0), (x3 , 0)) with x1 ≤ x e1 /2 ≤ x1 /2, ASTÉRISQUE 416

x2 = x e2 ≤ x2 /2,

m

(e z ) = T 1 (e z ).

x3 ≤ x e3 ≤ x3 /2,

ATTRACTED BY AN ELLIPTIC FIXED POINT

331

m ei ≤ xi for all m ∈ {0, . . . , N }. As a consequence, setting and (T 1 (e z )i ≤ x b e N = N + N + N: T

N

(z ′ ) = z¯,

|T

m

(z ′ )i | ≤ (1 + η)xi

for all m ∈ {0, . . . , N }.

We finally change the notation of (4.6) and write ω = (ω 1 , ω 2 , ω 3 ) = (p1 /q 1 , p2 /q 2 , p3 /q 3 ), so that in particular qe1 = q 1 , qb3 = q 3 and q 3 | q 1 | q 2 . Hence the orbits of Sω are q 2 -periodic. Moreover, from (4.3) and the equality x b1 = x e1 , one deduces x b1 − x1 >

1 . q 31

Note finally that the last conditions in (4.1), (4.2) and (4.5) now read 1 < η, q3

q3 < η, q1

q1 < η. q2

Fix (θ1 , θ2 , θ3 ) ∈ T3 and recall that q 3 | q 1 | q 2 . By the first inequality one can first find ℓ3 ∈ N such that Rωℓ33 (0) is η-close to θ3 . Then, by the second inequality ℓ q +ℓ3

there is an ℓ1 ∈ N such that Rω11 3

(0) is η-close to θ1 . Finally, by the last inℓ q +ℓ1 q 3 +ℓ3

equality there is an ℓ2 ∈ N such that Rω22 1

(0) is η-close to θ2 . This proves

ℓ q +ℓ q +ℓ Sω2 1 1 3 3 (0, 0, 0) 3

is η-close to (θ1 , θ2 , θ3 ), so that the orbits of Sω are η-dense that on T . This concludes the proof. Definition 4.1. — Given z = (z1 , z2 , z3 ) ∈ X, we say that a diﬀeomorphism Φ of X is z-admissible if Φ ≡ Id on {s ∈ X : |si | ≤

11 10

|zi | , i = 1, 2, 3}.

Proposition 4.3. — Let ω = (p1 /q1 , p2 /q2 , p3 /q3 ) ∈ Q3+ with q3 | q1 | q2 and z = ((x1 , 0), (x2 , 0), (x3 , 0)) ∈ B(0, R) with x1 , x2 , x3 > 0 and x2 ≥ 1/q2 . Suppose Φ ∈ U α,L is z-admissible and kΦ2,1,x2 ,q−3 ◦ Φ − Idkα,L < ǫ, where ǫ is a positive 2 constant depending only on α, L and L1 , and let T := Φ2,1,x2 ,q−3 ◦ Φ ◦ Sω . 2

Assume that z0 ∈ X and M ≥ 1 are such that T M (z0 ) = z. Then, for any η > 0, there exist (a) ω = (p1 /q 1 , p2 /q 2 , p3 /q 3 ) such that q 3 | q 1 | q 2 , the orbits of the translation of vector ω on T3 are η-dense and |ω − ω| ≤ η; (b) z¯ = ((x1 , 0), (x2 , 0), (x3 , 0)) such that 0 < xi ≤ xi /2 for every i ∈ {1, 2, 3} and x2 ≥ 1/q 2 ; ¯ ∈ U α,L z¯-admissible, so (c) z¯0 ∈ X such that |¯ z0 − z0 | ≤ η, and M > M , and Φ that the diﬀeomorphism ¯ ◦ Sω T := Φ2,1,x2 ,q−3 ◦ Φ 2

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m z0 )i ≤ (1 + η)xi for all m ∈ M, . . . , M . (¯ z0 ) = z¯ and T (¯ ¯ −Φ (d) Moreover, kΦ2,1,x2 ,q−3 ◦ Φ 2,1,x2 ,q −3 ◦ Φkα,L ≤ η. satisfies T

M

2

2

b1 as in Proposition 4.1 and let Proof of Proposition 4.3. — Take ω, z¯, N, z , x ′

T

= Φ2,1,x2 ,q−3 ◦ Φ1,3,bx1 ,q−3 ◦ Φ3,2,x3 ,q−3 ◦ Φ2,1,x2 ,q−3 ◦ Sω 2

so that T

N

(z ) = z¯ and |T ′

1

m

3

2

(z )i | ≤ (1 + η)xi for all m ∈ {0, . . . , N }. If we define ′

T = Φ2,1,x2 ,q−3 ◦ Φ1,3,bx1 ,q−3 ◦ Φ3,2,x3 ,q−3 ◦ Φ2,1,x2 ,q−3 ◦ Φ ◦ Sω 2

1

3

2

m

then, since Φ is z-admissible and |z − z | < η, we get T (z ′ ) = T (z ′ ) for all N m m ∈ {0, . . . , N }, hence T (z ′ ) = z¯ and T (z ′ )i ≤ (1 + η)xi for all m ∈ {0, . . . , N }. m

′

Let

¯ := Φ Φ 1,3,b x1 ,q −3 ◦ Φ3,2,x3 ,q −3 ◦ Φ2,1,x2 ,q −3 ◦ Φ,

(4.7)

1

3

2

¯ ◦ Sω . Notice that we can write Φ ¯ = so that, indeed, T = Φ2,1,x2 ,q−3 ◦ Φ ◦Φ 2,1,x2 ,q −3 2 2 u1 u2 u3 Φ ◦ Φ ◦ Φ ◦ Ψ (notation of Lemma A.2), where Ψ = Φ2,1,x2 ,q−3 ◦ Φ and the 2 Gevrey-(α, L1 ) norms of u1 , u2 , u3 are controlled by Lemma 2.1; we thus get (d) by taking ǫ as in Lemma A.2 with n = 3 and applying (A.8), choosing q 1 , q 2 , q 3 suﬃciently large. Comparing T and T in C 0 -norm in the ball B(0, |z0 | + 1), since we can take ω arbitrarily close to ω and the q i ’s arbitrarily large, we can find z¯0 ∈ X such M

z0 ) = z ′ . We thus take M = M + N , so that T that |¯ z0 − z0 | ≤ η and T (¯ m and T (¯ z0 )i ≤ (1 + η)xi for all m ∈ M, . . . , M .

M

(¯ z0 ) = z¯

¯ ∈ U α,L and Φ ¯ is z¯-admissible since To finish the proof of (c), just observe that Φ −3 −3 xi ≤ xi /2 and q 1 ≤ x b1 /10, q 3 ≤ x3 /10 (possibly increasing q 1 and q 3 if necessary). Clearly, Proposition 4.3 is tailored so that it can be applied inductively. The gain obtained when going from T to T is twofold : on the one hand the orbit of the new initial point z¯0 is pushed further close to the origin, and on the other hand the rotation vector at the origin is changed to behave increasingly like an non-resonant vector. Proof of Theorem A. — Let γ > 0. We pick (0)

(0)

(0)

(0)

(0)

(0)

ω (0) = (p1 /q1 , p2 /q2 , p3 /q3 ) ∈ Q3+ (0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

with q3 | q1 | q2 , and x1 , x2 , x3 > R/4 so that x2 ≥ 1/q2

and

(0) (0) (0) (0) z0 := ((x1 , 0), (x2 , 0), (x3 , 0)) ∈ B(0, R/2).

Let Φ(0) := Id and M (0) := 0. Define T (0) := Ψ(0) ◦ Sω(0)

with

Ψ(0) := Φ2,1,x(0) ,1/(q(0) )3 ◦ Φ(0) . 2

ASTÉRISQUE 416

2

333

ATTRACTED BY AN ELLIPTIC FIXED POINT

(0)

Choosing q2 suﬃciently large, we have kΨ(0) − Idkα,L ≤ min{ǫ/2, γ/2} by (2.2). The (0) hypotheses of Proposition 4.3 hold for z (0) = z0 . We apply Proposition 4.3 repeatedly by choosing inductively a sequence (η (n) )n≥1 such that ∞ o n ǫ X γ R η (n) (n) η (k) ≤ (n) η ≤ min n+1 , n+1 , n+3 , 1/10 , 2 2 2 q 2

k=n+1

(n) q2

(where is determined at the nth step of the induction). We get sequences (n) (n) (ω )n≥0 , (z0 )n≥0 , (z (n) )n≥0 , (T (n) )n≥0 , (M (n) )n≥0 , with (n)

(n)

(n)

(n+1)

z (n) = ((x1 , 0), (x2 , 0), (x3 , 0)),

0 < xi

(n)

≤ xi /2

and T (n) = Ψ(n) ◦ Sω(n) with Ψ(n) = Φ2,1,x(n) ,1/(q(n) )3 ◦ Φ(n) ∈ U α,L , so that 2

2

|ω (4.8)

(n+1)

−ω

(n+1)

− z0 | ≤ η (n+1) ,

|z0

(n)

| ≤ η (n+1) ,

(n)

kΨ(n+1) − Ψ(n) kα,L ≤ η (n+1) . We also have m

(n+1)

(4.9) |(T (n+1) (z0

(j)

))i | ≤ 1.1xi

for all m ∈ {M (j) , . . . , M (j+1) } with j ≤ n.

(n)

In view of (4.8), the sequences (z0 ), (ω (n) ) and (Ψ(n) ) are Cauchy. We denote their limits by z0∞ , ω ∞ and Ψ∞ . Notice that kΨ∞ − Idkα,L ≤ γ and z0∞ 6= 0 (because (0) |z0∞ − z0 | ≤ R/8). We now check that T := Ψ∞ ◦ Sω∞ satisfies |Tm (z0∞ )| −→ 0. When restricted m→+∞

to B(0, R),

(n) Sω(n)

converges uniformly to Sω∞ (by compactness) thus T (n) converges (n)

uniformly to T, moreover B(0, R) is invariant by T (n) and contains all the points z0 ; hence we can use the following elementary lemma (the verification of which is left to the reader): Lemma 4.4. — Let E be a metric space and (Tn ) a sequence of self-maps which converges uniformly to a limit T . Then, for any sequence (z (n) ) which converges to a point z in E, we have Tnm (z (n) ) −→ Tm (z) for each m ∈ N. n→+∞

m

(n)

We thus get T (n) (z0 ) −→ Tm (z0∞ ) for each m. Letting n tend to ∞ in (4.9), n→+∞ (j)

we get |(Tm (z0∞ ))i | ≤ 1.1xi for all j and m such that M (j) ≤ m ≤ M (j+1) . Since (j) xi ↓ 0 and M (j) ↑ ∞ as j tends to ∞, this yields |Tm (z0∞ )| −→ 0. m→+∞

(n)

The orbits of the translation of vector ω (n) on T3 being η (n) -dense and q2 -periodic, we see that ω ∞ defines a minimal translation on T3 . Indeed, given θ ∈ T3 and ǫ > 0,

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2020

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(n)

we can choose n, m ∈ N so that η (n) ≤ ǫ/2, dist(mω (n) − θ, Z3 ) ≤ η (n) and m < q2 . Then, (n)

dist(mω ∞ − θ, Z3 ) ≤ η (n) + m|ω ∞ − ω (n) | ≤ η (n) + q2

∞ X

η (k) ≤ 2η (n) ,

k=n+1

which is ≤ ǫ. Hence the orbit of 0 under the translation of vector ω ∞ is ǫ-dense for every ǫ, which entails that ω ∞ is non-resonant. The proof of Theorem A is thus complete.

Appendix Gevrey functions, maps and flows A.1. Gevrey functions and Gevrey maps. — We follow Section 1.1.2 and Appendix B of [10], with some simplifications stemming from the fact that here we only need to consider functions satisfying uniform estimates on the whole of a Euclidean space. The Banach algebra of uniformly Gevrey-(α, L) functions. — Let N ≥ 1 be integer and α ≥ 1 and L > 0 be real. We define Gα,L (RN ) := {f ∈ C ∞ (RN ) | kf kα,L < ∞}, kf kα,L :=

X L|ℓ|α k∂ ℓ f kC 0 (RN ) . α ℓ! N

ℓ∈N

We have used the standard notations |ℓ| = ℓ1 + · · · + ℓN , ℓ! = ℓ1 ! . . . ℓN !, ∂ ℓ = ∂xℓ11 . . . ∂xℓNN , and N := {0, 1, 2, . . .}. The space Gα,L (RN ) turns out to be a Banach algebra, with (A.1)

kf gkα,L ≤ kf kα,L kgkα,L

for all f, g ∈ Gα,L (RN ), and there are “Cauchy-Gevrey inequalities”: if 0 < L′ < L, ′ then all the partial derivatives of f belong to Gα,L (RN ) and, for each p ∈ N, X p!α (A.2) k∂ m f kα,L′ ≤ kf kα,L (L − L′ )pα N m∈N ; |m|=p

(see [11]). The Banach space of uniformly Gevrey-(α, L) maps. — Let N, M ≥ 1 be integer and α ≥ 1 and L > 0 be real. We define Gα,L (RN , RM ) := {F ∈ C ∞ (RN , RM ) | kF kα,L < ∞}, kF kα,L := kF[1] kα,L + · · · + kF[M ] kα,L . This is a Banach space.

ASTÉRISQUE 416

ATTRACTED BY AN ELLIPTIC FIXED POINT

335

When N = M = 2n, we denote by Id + Gα,L (R2n , R2n ) the set of all maps of the form Ψ = Id + F with F ∈ Gα,L (R2n , R2n ). This is a complete metric space for the distance dist(Id + F1 , Id + F2 ) = kF2 − F1 kα,L . We use the notation dist(Ψ1 , Ψ2 ) = kΨ2 − Ψ1 kα,L as well. We then define U

α,L

⊂ Id + Gα,L (R2n , R2n )

as the subset consisting of all Gevrey-(α, L) symplectic diﬀeomorphisms of R2n which fix the origin and are C ∞ -tangent to Id at the origin. This is a closed subset of the complete metric space Id + Gα,L (R2n , R2n ). Composition with close-to-identity Gevrey-(α, L) maps. — Let N ≥ 1 be integer and α ≥ 1 and L > 0 be real. We use the notation (NN )∗ := NN r {0} and define X L|ℓ|α ∗ Nα,L (f ) := k∂ ℓ f kC 0 (RN ) , ℓ!α N ∗ ℓ∈(N )

so that kf kα,L = kf kC 0 (RN ) +

∗ Nα,L (f ).

Lemma A.0. — Let L1 > L. There exists ǫc = ǫc (N, α, L, L1 ) such that, for any f ∈ Gα,L1 (RN ) and F = (F[1] , . . . , F[N ] ) ∈ Gα,L (RN , RN ), if ∗

∗

Nα,L (F[1] ), . . . , Nα,L (F[N ] )

≤ ǫc ,

then f ◦ (Id + F ) ∈ Gα,L (RN ) and kf ◦ (Id + F )kα,L ≤ kf kα,L1 . Proof. — Since L < L1 , we can pick µ > 1 such that µLα < Lα ; we then choose α−1 1 α−1 a > 0 such that (1 + a) ≤ µ and set λ := N (1 + 1/a) . We will prove the α − µL )/λ. lemma with ǫc := (Lα 1 Let f and F be as in the statement, and g := f ◦ (Id + F ). We first derive a formula 1 for ∂ k g by computing the Taylor expansion of g(x + h) = f (x + h + F (x + h)) k! at h = 0: since the Taylor expansion of xi + hi + F[i] (x + h) is X 1 ∂ k F[i] (x)hk , xi + F[i] (x) + hi + Si with Si := k! N ∗ k∈(N )

by composition of the Taylor series, we obtain that the formal series X 1 ∂ k g(x)hk ∈ R[[h1 , . . . , hN ]] k! N k∈N

is given by

(h1 + S1 )r1 · · · (hN + SN )rN = ∂ r f x + F (x) r 1 ! · · · rN ! N

X

r∈N

X

n,ℓ∈N

hn1 S ℓ1 · · · hnNN SNℓN ∂ n+ℓ f x + F (x) 1 1 . n1 !ℓ1 ! · · · nN !ℓN ! N

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2020

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B. FAYAD, J.-P. MARCO & D. SAUZIN

Writing SiL =

P

K 1 ,...,K L ∈(NN )∗

1 L 1 hK +···+K K 1 !···K L !

X 1 X ∂ n+ℓ f x + F (x) k k ∂ g(x)h = k! n!ℓ! N N

k∈N

n,ℓ∈N

with P :=

N Q

p

∂ K F[i] (x), we get

Q

1≤p≤L 1

X

k1 ,...,k|ℓ| ∈(NN )∗

|ℓ|

hn+k +···+k P k 1 ! · · · k |ℓ| !

p

∂ k F[i] (x). Thus, for each k ∈ NN ,

Q

i=1 ℓ1 +···+ℓi−1