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*English*
*Pages 274
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*Year 2020*

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

- Categories
- Mathematics
- Dynamical Systems

415

ASTÉRISQUE 2020

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

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Astérisque est un périodique de la Société Mathématique de France. Numéro 415, 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]

AMS P.O. Box 6248 Providence RI 02940 USA http://www.ams.org

Tarifs Vente au numéro : 50 e ($ 75) Abonnement Europe : 665 e, hors Europe : 718 e ($ 1 077) Des conditions spéciales sont accordées aux membres de la SMF. Secrétariat Astérisque Société Mathématique de France Institut Henri Poincaré, 11, rue Pierre et Marie Curie 75231 Paris Cedex 05, France Fax: (33) 01 40 46 90 96 [email protected] • http://smf.emath.fr/ © Société Mathématique de France 2020 Tous droits réservés (article L 122–4 du Code de la propriété intellectuelle). Toute représentation ou reproduction intégrale ou partielle faite sans le consentement de l’éditeur est illicite. Cette représentation ou reproduction par quelque procédé que ce soit constituerait une contrefaçon sanctionnée par les articles L 335–2 et suivants du CPI.

ISSN: 0303-1179 (print) 2492-5926 (electronic) ISBN 978-2-85629-916-6 doi:10.24033/ast.1096 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 I 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). — 00, 05A05, 05A16, 11J06, 20C30, 28A78, 30F30, 30F60, 32Q65, 34D08, 37B, 37B20, 37C40, 37C60, 37C85, 37C99, 37D20, 37D25, 37E30, 37E99, 37F10, 37F15, 37F45, 57S25, 58F08, 76A99; 05, 37C55, 37D35, 53D42.

Mots-clefs. — Actions de groupes paraboliques, automorphismes polynomiaux de C2 , Birkhoff, cocycles projectivisés, combinatoire, compacité, courbe analytique réelle, courbes férales, courbes pseudoholomorphiques, decomposition de Thurston-Nielsen, diagramme de cordes, diagramme de Rauzy, diagramme de séparatrices, dimensions, distribution de Cauchy, distribution uniforme, éclatement, ensembles de Cantor réguliers, espace des modules des différentielles abéliennes, exposants de Lyapunov, graphe distance héréditaire, groupe nilpotent, lemme de fermeture ergodique, mécanique des fluides numérique, mésures invariantes, nombre d’enlacement, nombre de rotation, opérade, Poincaré, point fixe, quasi-périodicité, récurrence par chaînes, revêtement infini cyclique, rotations du cercle, singularité, spectres dynamiques de Markov et Lagrange, surface à petits carreaux, systèmes dynamiques, théorèmes limites “annealed”, théorèmes limites pour les moyennes temporelles, volume de Masur-Veech.

Keywords. — Annealed limit theorems, Birkhoff, blow-up, Cauchy distribution, chain-recurrence, chord diagram, circle rotations, combinatorics, compactness, computational fluid mechanics, distance hereditary graphs, dynamical systems, ergodic closing lemma, feral curves, fixed point, fractal dimensions, fractales, infinite cyclic covering, invariant measures, linking number, Lyapunov exponents, Markov and Lagrange dynamical spectra, Masur-Veech volume, moduli space of Abelian differentials, nilpotent group, operad, parabolic group actions, Poincaré, polynomial automorphisms of C2 , projective cocycles, pseudoholomorphic curves, quasi-periodicity, Rauzy diagram, real analytic curve, regular Cantor sets, resolution, rotation number, separatrix diagram, singularity, square-tiled surface, temporal distributional limit theorems, Thurston-Nielsen decomposition, uniform distribution.

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

Abstract. — This is the first 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 premier 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

Étienne Ghys & Christopher-Lloyd Simon — On the topology of a real analytic curve in the neighborhood of a singular point . . . . . . . . . . . . . . . . . . . Statement of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The genesis of this paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Analytic chord diagrams: an algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Polynomial interchanges: algorithmic description . . . . . . . . . . . . . . . . . 1.2. Chord diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. A necessary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Proof of the fundamental lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. More non-analytic diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. With a computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7. Marked chord diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8. Let us bound the number of chord diagrams . . . . . . . . . . . . . . . . . . . . . 2. Analytic chord diagrams: interlace graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Polynomial interchanges: permutation graph . . . . . . . . . . . . . . . . . . . . . 2.2. Collapsible graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Distance hereditary and treelike graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Some proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Appendix: completely decomposable graphs . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 4 5 5 8 10 12 13 15 17 17 19 23 27 29 32

Romain Dujardin — A closing lemma for polynomial automorphisms of C2 1. Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The atomic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The non-atomic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 35 37 37 37 38 39 42

Carlos Gustavo T. de A. Moreira — On the minima of Markov and Lagrange Dynamical Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries from dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 47 47 50 57

Dmitry Dolgopyat & Omri Sarig — Quenched and annealed temporal limit theorems for circle rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Known results on spatial limit theorems: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Known results on temporal limit theorems: . . . . . . . . . . . . . . . . . . . . . . . . . . . This paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heuristic overview of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions with more than one discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The main steps in the proofs of Theorems 2.1, 2.2 . . . . . . . . . . . . . . . . . . . . Step 1: Identifying the resonant harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . Step 2: An identity for the sum of resonant terms . . . . . . . . . . . . . . . . . . . . Step 3: Limit theorems for resonant harmonics . . . . . . . . . . . . . . . . . . . . . . . 4. Proofs of the Key Steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Step 1 (Propositions 3.1 and 3.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Step 2 (Proposition 3.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Step 3 (Proposition 3.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Proof of Proposition 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Cauchy and Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60 60 60 61 61 62 62 62 62 65 66 67 68 70 70 72 78 80 82 83 84

Joel W. Fish & Helmut Hofer — Exhaustive Gromov compactness for pseudoholomorphic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Direct limit manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Pseudoholomorphic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Convergence of pseudoholomorphic curves . . . . . . . . . . . . . . . . . . . . . . . . 3. Proof of exhaustive Gromov compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Formula for arithmetic genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 90 90 91 93 95 97 101 110 111

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Sebastião Firmo & Patrice Le Calvez & Javier Ribón — Fixed points of nilpotent actions on R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Rotation number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Blowed up annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Linking function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Proof of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Compactly covered Nielsen classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Nielsen classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Existence of classes with non-vanishing Lefschetz number . . . . . . . . 3.3. A minimality property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. A particular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Rotational properties of homeomorphisms of the plane . . . . . . . . . . . . . . . 5.1. The function Link and Turn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Rotation numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Privileged isotopies for groups of plane homeomorphisms . . . . . . . . . . . . . 6.1. Properties (P2)′ , (Q)′ , (R), (S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Property (P2) for abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Rotational properties for groups of plane diﬀeomorphisms . . . . . . . . . . . 7.1. Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Consequences of property (R)j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Proof of Proposition 46 in case supp(µ) ⊂ Fix(φ) . . . . . . . . . . . . . . . . 7.4. Proof of Proposition 46 in case µ(Fix(φ)) = 0 . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christian Bonatti & Alex Eskin & Amie Wilkinson — Projective cocycles over SL(2, R) actions: measures invariant under the upper triangular group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Applications and the irreducibility criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Linear representations of G-lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. The suspension construction and a criterion for simplicity . . . 2.1.2. Foliated geodesic and horocyclic ﬂows . . . . . . . . . . . . . . . . . . . . . . . 2.2. The Kontsevich-Zorich cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Construction of an invariant subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The forward and backward ﬂags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The forward ﬂag and the unstable horocycle ﬂow: defining the inert ﬂag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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113 113 117 117 117 118 121 122 123 125 125 126 128 130 134 137 137 138 141 141 142 147 148 148 148 149 151 155

157 158 160 161 161 162 164 168 168 170

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3.3. The inert ﬂag is G-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Relaxing the definition of the components of the inert ﬂag . . . . . . . 4. Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Proof of Theorem 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 172 173 174 177

Kristian Bjerklöv & L. Håkan Eliasson — Positive fibered Lyapunov exponents for some quasi-periodically driven circle endomorphisms with critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. One dimensional models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Skew-products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Our model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Building expansion for good ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The functions ϕk and the sets Ask . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Scales and some arithmetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Good frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Base case for the induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Inductive step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Positive Lyapunov exponents for all good frequencies . . . . . . . . . . . . 3. Parameter exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Dependence on ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Parameter exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 181 181 182 183 183 184 184 184 185 185 186 188 189 189 191 193

Albert Fathi — Recurrence on infinite cyclic coverings . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The displacement function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Pageault barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The ρ+ and ρ− functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The fundamental proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 195 198 202 204 206 208 214

Dennis Sullivan — Lattice Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Introduction to the “momentum model” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The ideas of the construction and definitions . . . . . . . . . . . . . . . . . . . . . . . . . The lattice vector field VL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The face velocity vectors and face normal components VF , vF . . . . . . The model proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The derivative outside the nonlinear term . . . . . . . . . . . . . . . . . . . . . . . . . .

215 215 217 218 218 219 219 219

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The nonlinear term as a lattice vector field . . . . . . . . . . . . . . . . . . . . . . . . The nonlinear term as a one chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Lattice Vector calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volume preserving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Divergence operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gradient of a lattice scalar field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laplacian of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curl of a lattice vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Lattice topology, the Laplacian and the Hodge decomposition . . . . . . . . 6. The “potential term” and the “friction term” . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vincent Delecroix & Élise Goujard & Peter Zograf & Anton Zorich — Contribution of one-cylinder square-tiled surfaces to MasurVeech volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Siegel-Veech constants and Masur-Veech volumes . . . . . . . . . . . . . . . . . . Equidistribution of square-tiled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contribution of 1-cylinder square-tiled surfaces and large genus asymptotics of Masur-Veech volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Siegel-Veech constants and Masur-Veech volumes of strata of meromorphic quadratic diﬀerentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Equidistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Strata of Abelian diﬀerentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Strata of quadratic diﬀerentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Contribution of 1-cylinder square-tiled surfaces to Masur-Veech volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Jenkins-Strebel diﬀerentials. Critical graphs (separatrix diagrams) 2.2. Contribution of 1-cylinder diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Asymptotics in large genera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Application: experimental evaluation of the Masur-Veech volumes 2.5. Contribution of a single 1-cylinder separatrix diagram: computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choice of cyclic ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abelian versus quadratic diﬀerentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contribution of each individual 1-cylinder separatrix diagram . . . . . . 2.6. Counting 1-cylinder diagrams for strata of Abelian diﬀerentials based on Frobenius formula and Zagier bounds . . . . . . . . . . . . . . . . . . Frobenius formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Alternative counting of 1-cylinder separatrix diagrams . . . . . . . . . . . . . . . . 3.1. Approach based on recursive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

219 219 219 219 219 219 220 220 220 221 222

223 224 224 224 225 226 227 227 227 227 230 232 232 234 238 241 241 242 242 243 246 249 252 252

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Strata of Abelian diﬀerentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strata of quadratic diﬀerentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Approach based on Rauzy diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The example of Q(13 , −13 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Impact of the choice of the integer lattice on diagram-bydiagram counting of Masur-Veech volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. (by Philip Engel) Square-tiled surfaces with one horizontal cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Sur la topologie des courbes analytiques réelles au voisinage des points singuliers Étienne Ghys & Christopher-Lloyd Simon . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Le but de cet article est de décrire la topologie des courbes analytiques réelles planes au voisinage d’un point singulier. Localement, une telle courbe est constituée d’un certain nombre de branches qui coupent un petit cercle centré sur la singularité en deux points. La topologie locale est décrite par un diagramme de cordes : un nombre pair de points sur un cercle, associés deux par deux. Nous montrons que la plupart des diagrammes de cordes ne proviennent pas de singularités. Quand c’est le cas nous les qualifions d’analytiques. Nous proposons d’abord une description récursive des diagrammes analytiques. Puis nous caractérisons ces diagrammes analytiques comme étant ceux qui ne contiennent pas un certain nombre de sous-diagrammes que nous explicitons. Un lemme de fermeture pour les automorphismes polynomiaux de C2 Romain Dujardin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

Nous montrons que pour un automorphisme polynomial dissipatif de C2 , le support de toute mesure invariante est contenu dans ladhérence de lensemble des points selles, à lexception de quelques cas bien compris. Sur les minima des spectres dynamiques de Markov et Lagrange Carlos Gustavo T. de A. Moreira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Nous considérons des spectres dynamiques typiques de Lagrange et Markov associés aux fers à cheval des surfaces. Pour un grand ensemble de fonctions définies sur la surface à valeurs réelles, nous montrons que les minima des spectres dynamiques de Markov et Lagrange coïncident, sont isolés et sont l’image par la fonction d’un point périodique de la dynamique. Cela répond à une question de Jean-Christophe Yoccoz. Théorèmes limites temporels modifiés pour les rotations du cercle

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Dmitry Dolgopyat & Omri Sarig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Pn−1 1 Soit h(x) = {x} − 2 . On étudie la distribution de k=0 h(x + kα) pour x fixé et n tiré au hasard uniformément dans {1, . . . , N }, quand N → ∞. Beck a montré dans [2, 3] que pour x = 0 et α irrationnel quadratique, ces distributions convergent, après un changement d’échelle approprié, vers une distribution gaussienne. Nous montrons que l’ensemble des α pour lesquels la distribution limite après changement d’échelle existe est de mesure de Lebesgue nulle, mais qu’on a le théorème limite modifié suivant : soit (α, n) choisi au hasard uniformément Pn−1 dans R/Z × {1, . . . , N }, alors la distribution de k=0 h(kα) converge après un changement d’échelle approprié quand N → ∞ vers la distribution de Cauchy. Compacité de Gromov pour courbes pseudoholomorphes au sens exhaustif Joel W. Fish & Helmut Hofer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 On s’intéresse à la convergence de courbes pseudo-holomorphes vers une variété au sens de Gromov et dans un sens local. On étend cette notion de convergence au cas où la variété cible n’est pas supposée compacte mais recouvrable par une suite de voisinages compacts. On suppose que dans chaque voisinage compact, les courbes considérées ont une aire et un genre borné (ce qui n’est pas le cas au sens global) et on prouve l’existence d’une sous-suite qui converge au sens « exhaustif » de Gromov. Points fixes des actions nilpotentes de R2 Sebastião Firmo & Patrice Le Calvez & Javier Ribón . . . . . . . . . . . . . 113 Nous montrons plusieurs résultats d’existence de points fixes pour des groupes nilpotents de diﬀéomorphismes de classe C 1 du plan R2 . Les cas principaux sont ceux de groupes de diﬀéomorphismes de la sphère fixant le point à l’infini, de groupes de diﬀéomorphismes fixant un compact donné du plan, et finalement de groupes de diﬀéomorphismes préservant une mesure de probabilité. Cocycles projectifs au-dessus d’actions de SL(2, R) : mesures invariantes au-dessus du groupe triangulaire supérieur Christian Bonatti & Alex Eskin & Amie Wilkinson . . . . . . . . . . . . . . . . 157 Nous considérons l’action de SL(2, R) sur un fibré vectoriel H, préservant une mesure de probabilité ergodique ν sur la base X. Soit νˆ un relevé quelconque de ν qui est une mesure de probabilité sur le fibré projectivisé P(H), invariante sous l’action du sous-groupe triangulaire supérieur. Sous une hypothèse d’irréductibilité de l’action, nous prouvons que toute mesure νˆ comme ci-dessus est supportée par le projectivisé P(E1 ) de l’espace de Lyapunov associé à l’exposant de Lyapunov le plus grand pour l’action du semi-groupe diagonal positif. Nous en déduisons deux applications : Premièrement, les exposants du cocycle de Kontsevich-Zorich dépendent continûment des mesures aﬃnes, ce qui répond à une question de [57]. Deuxièmement, soit P(V ) un fibré projectif irréducible, plat, au dessus d’une surface fermée hyperbolique Σ, et soit F le feuilletage à

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feuilles hyperboliques, tangent à la connection plate ; alors le ﬂot horocyclique sur T 1 (F ) est uniquement ergodique sous l’hypothèse que le plus grand exposant de Lyapunov du ﬂot géodésique est simple. Ceci généralise le résultat principal de [13] en dimension arbitraire. Exposants de Lyapunov fibrés pour certains produits-croisés d’endomorphismes du cercle avec points critiques et force quasi-périodique Kristian Bjerklöv & L. Håkan Eliasson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 2 Dans cet article, nous donnons des exemples de produits croisés T : T → T2 de forme T : (x, y) = (x + ω, x + f (y)), où f : T → T est un endomorphisme C 1 explicite de degré deux avec un point critique unique et où ω appartient à un ensemble de mesures positives, dont l’exposant de Lyapunov fibré est positif pour βs − (log s)2

presque tout (x, y) ∈ T2 . Le point critique est de type f ′ (±e−s ) ≈ e tout s grand, où β > 0 est une petite constante numérique.

pour

Récurrence sur les revêtements infinis cycliques Albert Fathi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Nous étudions les propriétés de récurrence par chaînes du relevé d’un homéomorphisme à un revêtement infini cyclique. Cette étude est connectée au théorème de Poincaré-Birkhoﬀ, voir les travaux de John Franks [2, 3]. Hydrodynamique sur les réseaux Dennis Sullivan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Nous utilisons la combinatoire de deux réseaux cubiques à faces centrées s’interpénétrant pour construire un calcul vectoriel discret qui permet la construction de deux modèles de l’hydrodynamique incompressible de l’espace tridimensionel tripériodique, l’une fondée sur la conservation de l’élan et l’autre sur le principe du transport de la vorticité. Sans passer à la limite diﬀérentielle, on arrive néanmoins dans le langage du calcul vectoriel exactement aux formulations du modèle continu, d’abord celle de Jean Leray, où la dérivation agit à l’exterieur du terme nonlinéaire, et ensuite celle plus habituelle où elle agit à l’intérieur. Des études numériques montrent que ces deux modèles diﬀèrent au niveau discret. Contribution des surfaces à petits carreaux à un cylindre aux volumes de Masur-Veech Vincent Delecroix & Élise Goujard & Peter Zograf & Anton Zorich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Nous établissons une formule pour la contribution des surface à petits carreaux formées d’un seul cylindre horizontal au volume de Masur-Veech des strates de différentielles abéliennes. Nous en déduisons le comportement asymptotique lorsque le genre des surfaces grandit. À la lumière des résultats récents de Aggarwal et Chen-Möller-Zagier sur l’asymptotique des volumes de Masur-Veech, nous en déduisons que la contribution relative est de l’ordre de 1/d où d est la dimension de la strate. De manière similaire, nous donnons une formule pour la contribution

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des surfaces à petits carreaux formées d’un seul cylindre horizontal au volume de Masur-Veech des strates de diﬀérentielles quadratiques. En combinant cette formule avec nos résultats récents sur l’équidistribution des surface à un cylindre horizontal, nous proposons une méthode empirique pour le calcul des volumes de Masur-Veech des strates de diﬀérentielles quadratiques. Cette dernière s’avère être eﬃcace en petites dimensions.

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On the topology of a real analytic curve in the neighborhood of a singular point Étienne Ghys & Christopher-Lloyd Simon . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

The purpose of this paper is to describe the topology of real analytic planar curves in the neighborhood of a singular point. Locally, such a curve consists of a number of branches that intersect a small circle centered on the singularity at two points. The local topology is described by a chord diagram: an even number of points on a circle, associated two by two. We show that most chord diagrams do not come from singularities. When this is the case, we call them analytical diagrams. First, we propose a recursive description of analytical diagrams. Then we characterize these analytical diagrams as those that do not contain a number of subdiagrams that we describe explicitly. A closing lemma for polynomial automorphisms of C2 Romain Dujardin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2

We prove that for a dissipative polynomial diﬀeomorphism of C , the support of any invariant measure is, apart from a few well-understood cases, contained in the closure of the set of saddle periodic points. On the minima of Markov and Lagrange Dynamical Spectra Carlos Gustavo T. de A. Moreira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

We consider typical Lagrange and Markov dynamical spectra associated to horseshoes on surfaces. We show that for a large set of real functions on the surface, the minima of the corresponding Lagrange and Markov dynamical spectra coincide, are isolated, and are given by the image of a periodic point of the dynamics by the real function. This solves a question by Jean-Christophe Yoccoz. Quenched and annealed temporal limit theorems for circle rotations Dmitry Dolgopyat & Omri Sarig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Pn−1 1 Let h(x) = {x} − 2 . We study the distribution of k=0 h(x + kα) when x is fixed, and n is sampled randomly uniformly in {1, . . . , N }, as N → ∞. Beck proved in [2, 3] that if x = 0 and α is a quadratic irrational, then these distributions converge, after proper scaling, to the Gaussian distribution. We show

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that the set of α where a distributional scaling limit exists has Lebesgue measure zero, but that the following annealed limit theorem holds: Let (α, n) be chosen Pn−1 randomly uniformly in R/Z × {1, . . . , N }, then the distribution of k=0 h(kα) converges after proper scaling as N → ∞ to the Cauchy distribution.

Exhaustive Gromov compactness for pseudoholomorphic curves Joel W. Fish & Helmut Hofer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

Here we extend the notion of target-local Gromov convergence of pseudoholomorphic curves to the case in which the target manifold is not compact, but rather is exhausted by compact neighborhoods. Under the assumption that the curves in question have uniformly bounded area and genus on each of the compact regions (but not necessarily global bounds), we prove that a subsequence converges in an exhaustive Gromov sense.

Fixed points of nilpotent actions on R2 Sebastião Firmo & Patrice Le Calvez & Javier Ribón . . . . . . . . . . . . .

113

We show several results providing global fixed points for nilpotent groups of orientation-preserving C 1 diﬀeomorphisms of the plane R2 . The main cases are namely groups of diﬀeomorphisms of the sphere such that ∞ is a global fixed point, groups of diﬀeomorphisms preserving a non-empty compact set and finally groups of diﬀeomorphisms preserving a probability measure.

Projective cocycles over SL(2, R) actions: measures invariant under the upper triangular group Christian Bonatti & Alex Eskin & Amie Wilkinson . . . . . . . . . . . . . . . . 157 We consider the action of SL(2, R) on a vector bundle H preserving an ergodic probability measure ν on the base X. Under an irreducibility assumption on this action, we prove that if νˆ is any lift of ν to a probability measure on the projectivized bunde P(H) that is invariant under the upper triangular subgroup, then νˆ is supported in the projectivization P(E1 ) of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on aﬃne measures, answering a question in [57]. Second, if P(V) is an irreducible, ﬂat projective bundle over a compact hyperbolic surface Σ, with hyperbolic foliation F tangent to the ﬂat connection, then the foliated horocycle ﬂow on T 1 F is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic ﬂow is simple. This generalizes results in [13] to arbitrary dimension.

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Positive fibered Lyapunov exponents for some quasi-periodically driven circle endomorphisms with critical points Kristian Bjerklöv & L. Håkan Eliasson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 In this paper we give examples of skew-product maps T : T2 → T2 of the form T (x, y) = (x + ω, x + f (y)), where f : T → T is an explicit C 1 -endomorphism of degree two with a unique critical point and ω belongs to a set of positive measure, for which the fibered Lyapunov exponent is positive for a.e. (x, y) ∈ T2 . The 2 critical point is of type f ′ (±e−s ) ≈ e−βs/(ln s) for all large s, where β > 0 is a small numerical constant. Recurrence on infinite cyclic coverings Albert Fathi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195

We study the chain-recurrence properties of the lift of a homeomorphism to an infinite cyclic cover. This is related to the Poincaré-Birkhoﬀ theorem, see the work of John Franks [2, 3]. Lattice Hydrodynamics Dennis Sullivan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215

We construct a lattice model of 3D incompressible hydrodynamics on triply periodic three space. It is not possible to have a discrete finite dimensional model of exterior d, wedge product of forms and hodge star [equivalently of vector calculus] satisfying all of the identities used to manipulate the NSE = Navier Stokes Equation. This means the same discretization method applied to diﬀerent but equivalent versions of NSE should be considered to be diﬀerent when the identities used to prove the equivalence do not all hold for the discrete replacements. Thus we do not derive the lattice model by directly discretizing the NSE, but rather use the derivation of the NSE based on momentum transfer and creation in small regions of fixed size as in a finite volume method (see reference). Besides the perspective on discretization mentioned above the new point and the main point is to express the finite scale calculation in terms of optimally chosen operations of combinatorial topology that are discrete analogs of the continuum ones. Such exist in this special combinatorics consisting of eight translated cubical decompositions of edge size twice the spacing of the basic lattice. The discrete analogs of d, wedge and star will satisfy all of the familiar identities except that the Leibniz rule for d acting on a product is deformed. The discrete operators are the coboundary of algebraic topology, the Poincare dual cell operator and the lattice site wedge product. The calculus limit need not be taken and the model closure point happens when one replaces the average of a product [the velocity at a face times its

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normal component] by the product of these averages. At a physical scale where the velocity is not varying the model would be physically correct. The derived ODE for the lattice velocity vector field resembles the Leray form of NSE with the derivative outside the nonlinear term, but it cannot be manipulated into the other familiar form with the derivative inside the nonlinear term because Leibniz has changed. Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes Vincent Delecroix & Élise Goujard & Peter Zograf & Anton Zorich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

223

We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian diﬀerentials. The resulting count is particularly simple and eﬃcient in the large genus asymptotics. Using the recent results of Aggarwal and of ChenMöller-Zagier on the long-standing conjecture about the large genus asymptotics of Masur-Veech volumes, we derive that the relative contribution is asymptotically of the order 1/d, where d is the dimension of the stratum. Similarly, we evaluate the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes of low-dimensional strata in the moduli space of quadratic diﬀerentials. We combine this count with our recent result on equidistribution of one-cylinder square-tiled surfaces translated to the language of interval exchange transformations to compute empirically approximate values of the Masur-Veech volumes of strata of quadratic diﬀerentials of all small dimensions.

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JEAN-CHRISTOPHE YOCCOZ (1957–2016)

Jean-Christophe Yoccoz was without a doubt one of the great mathematicians of his generation. His work in the field of dynamical systems is of considerable importance and depth and it has shed new light on many problems of this theory. Together with his thesis advisor, Michel Herman, he helped carry the French school of dynamical systems to its highest level yet. His results covered an impressive number of areas: his construction with P. Arnoux of exotic pseudo-Anosov dynamics (“Arnoux-Yoccoz pseudo-Anosov systems”), his relentless quest for optimal linearization conditions for circle diﬀeomorphisms and holomorphic germs, his development of a revolutionary combinatorial concept (the “Yoccoz puzzles”) for the fine study of the topology of the Mandelbrot set related to Douady and Hubbard’s MLC conjecture, his detailed investigation with J. Palis and C. G. Moreira of the dynamics of systems obtained by homoclinic and heteroclinic bifurcations, a series of works with S. Marmi and P. Moussa describing the surprising similarities and diﬀerences between the linearization problems of circle diﬀeomorphisms and generalized interval exchange maps. . . His ease of calculation and the power of his combinatorial analysis were enthralling. His major achievements earned him the Fields Medal in 1994, elected him to the Académie des Sciences, and subsequently led to a professorship at the Collège de France. But honors and distinctions never changed his simple, humble, generous and optimistic view on life. Throughout his career, he developed close ties with numerous colleagues around the world. Following his military service in Brazil, he would very regularly visit the Instituto de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro. He also continuously inﬂuenced the dynamical systems community through his courses in Orsay and at the Collège de France, through the seminar he hosted with H. Eliasson, through the many conferences he organized, and finally through the advice he always gave so generously. He was looking forward to having his 60th birthday celebrated by a conference, and to have the proceedings on topics that were dear to him appear as a special volume of Astérisque. His premature death forced a modification of these plans: his memory was honored in May 2017 during a colloquium organized at the Collège de France. A special issue of the Gazette des Mathématiciens published in May 2018 pays tribute to this Mathematician through various texts of reminiscences as well as texts presenting his most important scientific works.

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The present two volumes of proceedings extend the scope of the conference and gather research articles which we hope would have pleased Jean-Christophe. S. Crovisier R. Krikorian C. Matheus S. Senti

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Research articles 1. J.-C. Y., Centralisateur d’un diﬀéomorphisme du cercle dont le nombre de rotation est irrationnel. C.R.A.S. 291 (1980), A523–A526. 2. J.-C. Y., Sur la disparition de propriétés de type Denjoy-Koksma en dimension 2. C.R.A.S. 291 (1980), A655–A658. 3. P. Arnoux, J.-C. Y., Construction de diﬀéomorphismes pseudo-Anosov. C.R.A.S. 292 (1981), 75–78. 4. A. Fathi, M. Herman, J.-C. Y., A proof of Pesin’s stable manifold theorem. Geometric dynamics (Rio de Janeiro, 1981), 177–215. Lecture Notes in Math. 1007, Springer, Berlin, 1983. 5. M. Herman, J.-C. Y., Generalizations of some theorems of small divisors to non-Archimedean fields. Geometric dynamics (Rio de Janeiro, 1981), 408–447. Lecture Notes in Math. 1007, Springer, Berlin, 1983. 6. J.-C. Y., C 1 -conjugaison des diﬀéomorphismes du cercle. Geometric dynamics (Rio de Janeiro, 1981), 814–827. Lecture Notes in Math. 1007, Springer, Berlin, 1983. 7. J.-C. Y., Il n’y a pas de contre-exemple de Denjoy analytique. C.R.A.S. 298 (1984), 141–144. 8. J.-C. Y., Conjugaison diﬀérentiable des diﬀéomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne. Ann. Sci. École Norm. Sup. (4) 17 (1984), 333–359. 9. J.-C. Y., Bifurcations de points fixes elliptiques. (d’après A. Chenciner). Séminaire Bourbaki, Vol. 1985/86, Exp. 668. Astérisque 145-146 (1987), 313–334. 10. J.-C. Y., Linéarisation des germes de diﬀéomorphismes holomorphes de (C, 0). C.R.A.S. 306 (1988), 55–58. 11. J.-C. Y., Non-accumulation de cycles limites. Séminaire Bourbaki, Vol. 1987/88, Exp. 690. Astérisque 161-162 (1988), 87–103. 12. J. Palis, J.-C. Y., Rigidity of centralizers of diﬀeomorphisms. Ann. Sci. École Norm. Sup. (4) 22 (1989), 81–98.

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13. J. Palis, J.-C. Y., Centralizers of Anosov diﬀeomorphisms on tori. Ann. Sci. École Norm. Sup. (4) 22 (1989), 99–108. 14. J. Palis, J.-C. Y., Diﬀerentiable conjugacies of Morse-Smale diﬀeomorphisms. Bol. Soc. Brasil. Mat. (N.S.) 20 (1990), 25–48. 15. J.-C. Y., Polynômes quadratiques et attracteur de Hénon. Séminaire Bourbaki, Vol. 1990/91, Exp. 734. Astérisque 201-203 (1991), 143–165. 16. J.-C. Y., Travaux de Herman sur les tores invariants. Séminaire Bourbaki, Vol. 1991/92, Exp. No. 754. Astérisque 206 (1992), 311–344. 17. J.-C. Y., An introduction to small divisors problems. From number theory to physics (Les Houches, 1989), 659–679, Springer, Berlin, 1992. 18. J. Palis, J.-C. Y., Homoclinic tangencies for hyperbolic sets of large Hausdorﬀ dimension. Acta Math. 172 (1994), 91–136. 19. L. Carleson, P. Jones, J.-C. Y., Julia and John. Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), 1–30. 20. R. Perez-Marco, J.-C. Y., Germes de feuilletages holomorphes à holonomie prescrite. Complex analytic methods in dynamical systems (Rio de Janeiro, 1992). Astérisque 222 (1994), 345–371. 21. J.-C. Y., Introduction to hyperbolic dynamics. Real and complex dynamical systems (Hillerød, 1993), 265–291. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 464, Kluwer Acad. Publ., Dordrecht, 1995. 22. J.-C. Y., Petits diviseurs en dimension 1. Astérisque 231 (1995), 89–242. 23. J.-C. Y., Recent developments in dynamics. Proceedings of the International Congress of Mathematicians, Vol. I. (Zürich, 1994), 246–265, Birkhäuser, Basel, 1995. 24. S. Marmi, P. Moussa, J.-C. Y., The Brjuno functions and their regularity properties. Comm. Math. Phys. 186 (1997), 265–293. 25. J. Palis, J.-C. Y., On the arithmetic sum of regular Cantor sets. Ann. Inst. H. Poincare Anal. Non Linéaire 14 (1997), 439–456. 26. P. Le Calvez, J.-C. Y., Un théorème d’indice pour les homéomorphismes du plan au voisinage d’un point fixe. Ann. of Math. (2) 146 (1997), 241–293. 27. J.-C. Y., Dynamique des polynômes quadratiques. Notes prepared by Marguerite Flexor. Dynamique et géométrie complexes (Lyon, 1997). Panor. Synthèses 8 (1999), 187–222. Translated in SMF/AMS Texts and Monographs 10 (2003). 28. R. Douady, J.-C. Y., Nombre de rotation des diﬀéomorphismes du cercle et mesures automorphes. Regul. Chaotic Dyn. 4 (1999), 19–38. 29. S. Marmi, P. Moussa, J.-C. Y., Complex Brjuno functions. J. Amer. Math. Soc. 14 (2001), 783–841.

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30. C. G. Moreira, J.-C. Y., Stable intersections of regular Cantor sets with large Hausdorﬀ dimensions. Ann. of Math. (2) 154 (2001), 45–96. Corrigendum: Ann. of Math. (2) 154 (2001), 527. 31. J. Palis, J.-C. Y., Implicit formalism for aﬃne-like maps and parabolic composition. Global analysis of dynamical systems, 67–87, Inst. Phys., Bristol, 2001. 32. J. Palis, J.-C. Y., Fers à cheval non uniformément hyperboliques engendrés par une bifurcation homocline et densité nulle des attracteurs. C.R.A.S. 333 (2001), 867–871. 33. J.-C. Y., Analytic linearization of circle diﬀeomorphisms. Dynamical systems and small divisors (Cetraro, 1998), 125–173. Lecture Notes in Math. 1784 (2002). 34. S. Marmi, J.-C. Y., Some open problems related to small divisors. Dynamical systems and small divisors (Cetraro, 1998), 175–191. Lecture Notes in Math. 1784 (2002). 35. S. Marmi, P. Moussa, J.-C. Y., On the cohomological equation for interval exchange maps. C.R.A.S. 336 (2003), 941–948. 36. J.-C. Y., Some questions and remarks about SL(2, R) cocycles. Modern dynamical systems and applications, 447–458, Cambridge Univ. Press, Cambridge, 2004. 37. S. Marmi, P. Moussa, J.-C. Y., The cohomological equation for Roth-type interval exchange maps. J. Amer. Math. Soc. 18 (2005), 823–872. 38. J.-C. Y., Continued fraction algorithms for interval exchange maps: an introduction. Frontiers in number theory, physics, and geometry. I, 401–435, Springer, Berlin, 2006. 39. S. Marmi, P. Moussa, J.-C. Y., Some properties of real and complex Brjuno functions. Frontiers in number theory, physics, and geometry. I, 601–623, Springer, Berlin, 2006. 40. A. Avila, S. Gouezel, J.-C. Y., Exponential mixing for the Teichmüller ﬂow. Publ. Math. Inst. Hautes Études Sci. 104 (2006), 143–211. 41. J.-C. Y., Ensembles de Julia de mesure positive et disques de Siegel des polynômes quadratiques (d’après X. Buﬀ et A. Chéritat), Séminaire Bourbaki. vol. 2005/2006, exp. no 966, Astérisque 311 (2007), 385–401. 42. J. Palis, J.-C. Y., Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles. Publ. Math. Inst. Hautes Études Sci. 110 (2009), 1–217. 43. J.-C. Y., Échanges d’intervalles et surfaces de translation, Séminaire Bourbaki. vol. 2007/2008, exp. no 996, Astérisque 326 (2009), 387–409. 44. C. G. Moreira, J.-C. Y., Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale. Ann. Sci. Éc. Norm. Super. (4) 43 (2010), 1–68.

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45. S. Marmi, P. Moussa, J.-C. Y., Aﬃne interval exchange maps with a wandering interval. Proc. Lond. Math. Soc. (3) 100 (2010), 639–669. 46. J.-C. Y., Interval exchange maps and translation surfaces. Homogeneous ﬂows, moduli spaces and arithmetic, 1–69. Clay Math. Proc. 10, Amer. Math. Soc., Providence, RI, 2010. 47. A. Avila, J. Bochi, J.-C. Y., Uniformly hyperbolic finite-valued SL(2, R)-cocycles. Comment. Math. Helv. 85 (2010), 813–884. 48. C. Matheus, J.-C. Y., The action of the aﬃne diﬀeomorphisms on the relative homology group of certain exceptionally symmetric origamis. J. Mod. Dyn. 4 (2010), 453–486. 49. K.-T. Kim, J.-C. Y., CR manifolds admitting a CR contraction. J. Geom. Anal. 21 (2011), 476–493. 50. J.-C. Y., Small divisors: number theory in dynamical systems. In: An invitation to mathematics. From competitions to research. D. Schleicher, M. Lackmann (Eds). Springer, Berlin, 2011. 43–54. 51. S. Marmi, P. Moussa, J.-C. Y., Linearization of generalized interval exchange maps. Ann. of Math. (2) 176 (2012), 1583–1646. 52. A. Avila, C. Matheus, J.-C. Y., SL(2, R)-invariant probability measures on the moduli spaces of translation surfaces are regular. Geom. Funct. Anal. 23 (2013), 1705–1729. 53. C. Matheus, J.-C. Y., D. Zmiaikou, Homology of origamis with symmetries. Ann. Inst. Fourier (Grenoble) 64 (2014), 1131–1176. Corrigendum: Ann. Inst. Fourier (Grenoble) 66 (2016), 1279–1284. 54. C. Matheus, M. Möller, J.-C. Y., A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces. Invent. Math. 202 (2015), 333–425. 55. S. Marmi, J.-C. Y., Hölder regularity of the solutions of the cohomological equation for Roth type interval exchange maps. Comm. Math. Phys. 344 (2016), 117–139. 56. A. Avila, C. Matheus, J.-C. Y., Zorich conjecture for hyperelliptic Rauzy-Veech groups. Math. Ann. 370 (2018), 785–809. 57. A. Avila, C. Matheus, J.-C. Y., The Kontsevich-Zorich cocycle over VeechMcMullen family of symmetric translation surfaces. J. Mod. Dyn. 14 (2019), 21–54. 58. P. Berger, J.-C. Y., Strong regularity. Astérisque 410 (2019), 1–12. 59. J.-C. Y., A proof of Jakobson’s theorem. Astérisque 410 (2019), 15–52. 60. C. Matheus, J. Palis, J.-C. Y., Stable sets of certain non-uniformly hyperbolic horseshoes have the expected dimension. To appear in J. Inst. Math. Jussieu. 61. S. Marmi, C. Ulcigrai, J.-C. Y., On Roth type conditions, duality and central birkhoﬀ sums for i.e.m. In this volume.

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Editorial works 62. M. Flexor, P. Sentenac, J.-C. Y. (Eds), Géométrie complexe et systèmes dynamiques – Colloque en l’honneur d’Adrien Douady (Orsay, 1995). Astérisque 261 (2000). 63. W. de Melo, M. Viana, J.-C. Y. (Eds), Geometric Methods in Dynamics (I and II). Volumes in honor of Jacob Palis. Astérisque 286 and 287 (2003). 64. A. Fathi, J.-C. Y. (Eds), Michael Robert Herman memorial volume. Ergodic Theory and Dynamical Systems 24 (2004). 65. J.-C. Y. (Ed.), Les mathématiques dans le monde scientifique contemporain. Éditions Tec & Doc (2005). 66. M. Herman, Notes inachevées – sélectionnées par Jean-Christophe Yoccoz. Documents Mathématiques 16. SMF, Paris, 2018. Other texts Jean-Christophe also participated actively in the eﬀort to explain the world of mathematics to more general audiences. Among these works we cite the following ones. 67. J.-C. Y., Idées géométriques en systèmes dynamiques. In Chaos et déterminisme. A. Dahan Dalmedico, J.-L. Chabert, K. Chemla (Eds.). Seuil (1992). 68. J.-C. Y., La place des mathématiques dans l’éducation: L’univers mathématique et l’univers physique. Le langage de la science, l’apprentissage du raisonnement. L’outil de sélection solaire. . . Gaz. Math. 77 (1998), 55–58. 69. J.-C. Y., Une erreur féconde du mathématicien Henri Poincaré. Gaz. Math. 107 (2006), 19–26. 70. J.-C. Y., Hyperbolicité et Quasipériodicité. In Chaos. Systèmes dynamiques – éléments pour une épistémologie. S. Franceschelli, M. Paty, T. Roque (Eds). Hermann (2007). 71. J.-C. Y., Un oncle bienveillant. Gaz. Math. 113 (2007), 57–58. 72. J.-C. Y., Poincaré et la théorie des systèmes dynamiques (CD). De vive voix (2012).

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Astérisque 415, 2020, p. 1–33 doi:10.24033/ast.1097

ON THE TOPOLOGY OF A REAL ANALYTIC CURVE IN THE NEIGHBORHOOD OF A SINGULAR POINT by Étienne Ghys & Christopher-Lloyd Simon

Dedicated to the memory of Jean-Christophe Yoccoz Abstract. — The purpose of this paper is to describe the topology of real analytic planar curves in the neighborhood of a singular point. Locally, such a curve consists of a number of branches that intersect a small circle centered on the singularity at two points. The local topology is described by a chord diagram: an even number of points on a circle, associated two by two. We show that most chord diagrams do not come from singularities. When this is the case, we call them analytical diagrams. First, we propose a recursive description of analytical diagrams. Then we characterize these analytical diagrams as those that do not contain as subdiagrams those which belong to a collection that we describe explicitly. Résumé (Sur la topologie des courbes analytiques réelles au voisinage des points singuliers) Le but de cet article est de décrire la topologie des courbes analytiques réelles planes au voisinage d’un point singulier. Localement, une telle courbe est constituée d’un certain nombre de branches qui coupent un petit cercle centré sur la singularité en deux points. La topologie locale est décrite par un diagramme de cordes : un nombre pair de points sur un cercle, associés deux par deux. Nous montrons que la plupart des diagrammes de cordes ne proviennent pas de singularités. Quand c’est le cas nous les qualifions d’analytiques. Nous proposons d’abord une description récursive des diagrammes analytiques. Puis nous caractérisons ces diagrammes analytiques comme étant ceux ne contenant pas comme sous-diagramme ceux qui appartiennent à une famille que nous décrivons explicitement.

Statement of the main result In this paper, we propose a complete description of the topology of real analytic planar curves in the neighborhood of a singular point. Denote by R{x, y} the factorial ring of germs of real analytic functions defined in some neighborhood of (0, 0) ∈ R2 . The germ of a real analytic planar curve is defined 2010 Mathematics Subject Classification. — 00; 05. Key words and phrases. — Real analytic curve, singularity, resolution, blow-up, combinatorics, chord diagram, distance hereditary graphs, operad.

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by an equation F (x, y) = 0, where F ∈ R{x, y} vanishes at the origin. If F is an irreducible element in this ring, the topology of the curve CF defined by F is well known. Either it only contains the origin (as for x2 + y 2 = 0) or there is a local homeomorphism of the plane, mapping CF to (the germ of) a straight line (as for instance with x3 − y 2 = 0). In this second case, CF intersects small circles centered at the origin in exactly two points. In general F is a product F1n1 · · · Fknk of irreducible non-associated factors Fi . Our curve CF is therefore the union of the CFi , which are usually called the branches of CF . Since we are only interested in the topology of CF , we can discard those Fi ’s such that CFi only contains the origin. Two distinct factors Fi yield two branches which only intersect at the origin. Hence the analytic curve CF intersects small circles centered at the origin in an even number of points, grouped in pairs, each pair being associated to a branch. This yields a chord diagram which is by definition an even number of distinct points on the circle, grouped in pairs, up to an orientation preserving homeomorphism of the circle. Such a diagram is pictured by a certain number of chords with distinct endpoints in a circle. See for instance [4] for a detailed description of the role of chord diagrams in topology.

b a

a c

c

b

Figure 1. A curve with three branches and its associated chord diagram

The main theorem of this paper characterizes the chord diagrams arising from some analytic curve CF . Theorem. — A chord diagram is associated to some analytic curve if and only if it does not contain one of the “forbidden diagrams” shown in Figure 2 as a sub-chord diagram.

The genesis of this paper Consider four distinct polynomials P1 , P2 , P3 , P4 in R[x]. Order them in such a way that P1 (x) < P2 (x) < P3 (x) < P4 (x) for small negative values of x. Then define the permutation π on {1, 2, 3, 4} such that Pπ(1) (x) < Pπ(2) (x) < Pπ(3) (x) < Pπ(4) (x) for small positive values of x. In 2009, Maxim Kontsevich explained to the first author EG that among the 24 permutations on {1, 2, 3, 4} exactly two cannot be obtained

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ON THE TOPOLOGY OF A REAL ANALYTIC CURVE

Figure 2. Forbidden diagrams:

,

,

3

, and Cn (n ≥ 5)

by this construction: (1, 2, 3, 4) 7→ (2, 4, 1, 3) or (3, 1, 4, 2). EG easily generalized this to any number of polynomials and proved that a permutation on {1, . . . , n} can be obtained from n polynomials if and only if it does not “contain” one of Kontsevich’s permutations. This was published as an elementary paper [7]. We will give a diﬀerent proof later in Section 2. It was then very natural to look at the topological configurations of the branches of a real analytic curve in the neighborhood of a singular point. Trying to solve this problem, EG found an explicit algorithm determining if a given chord diagram is analytic, i.e., is associated to the branches of some real analytic singular point. In particular, it followed that the above forbidden chord diagrams were indeed not analytic. One can always delete some branches of an analytic curve, so that a subchord diagram of an analytic diagram is of course analytic. In particular, a diagram containing one of the forbidden examples is non-analytic. The question of knowing whether these examples were the only “minimal” forbidden configurations remained open. Since this proof was enjoyable and involved classical methods, EG decided to write a book proposing a leisurely promenade towards this partial result, intended for undergraduate students. The second author CS was such a student and read a preliminary draft of that book. He proposed to look at the problem from another side, explained below, and this new point of view enabled both authors to complete the proof of the above theorem in a joint eﬀort. Therefore the final version of the book contains an additional chapter, describing this result [8]. The present paper contains two sections. The first provides an algorithmic description of the analytic chord diagrams and the second uses the first to prove the main result. This paper is very close to the corresponding chapters of the book. We essentially “compressed” these chapters in order to get more eﬃciently to the main goal.

1. Analytic chord diagrams: an algorithm In this section, we get an algorithmic description of the analytic chord diagrams, that we defined as those which are determined by the branches of planar real analytic curves.

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1.1. Polynomial interchanges: algorithmic description. — The only purpose of this subsection is to discuss quickly the much simpler situation of permutations arising from polynomials in R[x] which were the starting point of this paper. This serves as a motivation and gives a pattern for the general strategy, somewhat diﬀerent from that in [7]. Let π be a permutation of {1, . . . , n} (n ≥ 2). We say that π is a polynomial interchange if there exist n polynomials P1 , . . . , Pn in R[x] such that P1 (x) < P2 (x) < · · · < Pn (x) for small negative x and Pπ(1) (x) < Pπ(2) (x) < · · · < Pπ(n) (x) for small positive x. We describe an elementary algorithm that determines if a given permutation is a polynomial interchange. In the next section, we will characterize polynomial interchanges as those permutations which do not contain the two forbidden Kontsevich permutations. Lemma. — For any polynomial interchange, at least two consecutive integers have consecutive images. The proof is easy. Denote by v(P ) ∈ N ∪ {∞} the valuation (at 0) of a polynomial P ∈ R[x], i.e., the lowest degree of a non zero monomial in P (and ∞ if P = 0). Choose polynomials P1 , · · · , Pn as above. For every integer N , the relation v(Pi − Pj ) ≥ N is an equivalence relation RN on {1, . . . , n}. Each equivalence class I ⊂ {1, . . . , n} is an interval. Indeed, suppose that i < j < k and that i, k ∈ I. We know that Pi (x) < Pj (x) < Pk (x) for small negative x. It follows that v(Pj − Pi ) ≥ v(Pk − Pi ) ≥ N so that j ∈ I. The same argument, for small positive x, implies that π(I) is also an interval. Let N0 be the largest value of N for which equivalence classes of RN are not reduced to singletons. Let I be an equivalence class of RN0 with at least two elements. Since all the valuations v(Pi − Pj ) are equal to N0 for i, j in I, the permutation π is either increasing or decreasing from I to π(I), depending on the parity of N0 . The lemma follows if one chooses two consecutive elements in I. Note in particular that the two permutations (1, 2, 3, 4) 7→ (2, 4, 1, 3) or (3, 1, 4, 2) are not polynomial interchanges. Theorem. — The following algorithm decides if a permutation π is a polynomial interchange: 1. If no pair of consecutive integers have consecutive images then π is not a polynomial interchange. 2. If there is such a pair, merge it to a singleton. This produces a permutation with one object less. Continue. 3. If you end up with the trivial permutation on one object, then the original permutation was a polynomial interchange.

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Figure 3. Merging a pair of consecutive elements

If π is a polynomial interchange and if {i, i + 1}, {π(i), π(i + 1)} are merged into singletons, we produce a permutation π ′ on n − 1 objects which is obviously another polynomial interchange associated to the polynomials P1 , . . . , Pi , Pi+2 , . . . , Pn . Conversely, given a polynomial interchange associated to P1 , . . . , Pi , . . . , Pn−1 , one can define Pi′ (x) = Pi (x) + (−x)N for a suﬃciently large value of N , even or odd, and consider the permutation associated to the n polynomials P1 , . . . , Pi , Pi′ , . . . , Pn−1 . This shows that if the merged permutation is a polynomial interchange, then so was the initial permutation. 1.2. Chord diagrams. — One can think of a chord diagram as a cyclic word of length 2n in which every letter occurs exactly twice (the labels of the letters being irrelevant). To be more pedantic (and precise), we are discussing fixed-point free involutions on Z/2nZ up to conjugacies by cyclic permutations. We can also draw n chords in a circle. The total number of chord diagrams of length 2n has been studied in many papers, like in [13] with strong motivations from knot theory. The problem would be easy if, instead of a cyclic word, we looked for standard (non-cyclic) words of length 2n in which every letter occurs exactly twice and whose names are irrelevant. Indeed, write the first letter of the word and then choose any of the remaining 2n − 1 locations for the other letter which is identical to the first. Then write the second letter in the first available free place and choose the other identical letter in any of the 2n−3 remaining locations. Etc. Therefore the total number of these words is (2n − 1) · (2n − 3) · · · 3 · 1. These numbers are sometimes called double factorials and denoted by (2n − 1)!!. It would be tempting to divide (2n − 1)!! by 2n to take into account the cyclic permutations, but some words admit symmetries and this makes the exact combinatorics more subtle. In any case, it follows from these considerations that the number of chord diagrams of length 2n grows super-exponentially in n. We will see that a very tiny proportion of chord diagrams are analytic, in the sense that they arise from the singularity of a planar analytic curve. 1.3. A necessary condition. — We will say that a chord in a diagram is solitary if it connects two consecutive points of the diagram as in the first picture in Figure 4. Two chords are parallel (resp. antiparallel ) if they are as in the second (resp. third) picture, i.e., if the corresponding letters a, b occur in the cyclic word as · · · ab · · · ba · · · (resp. · · · ab · · · ab · · · ). Finally, two chords as in the fourth picture constitute a pitchfork

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(· · · a · · · bab · · · ). Letter a is the stick and letter b is the fork. With these notations we can state the fundamental lemma.

Figure 4. Solitary, parallel, antiparallel and pichfork

Fundamental lemma. — Any analytic chord diagram (with at least two chords) contains a solitary chord, a pair of parallel or an antiparallel chords, or a pitchfork. As a simple corollary, there is no singular analytic plane curve whose branches intersect a small circle as in Figure 5.

A

B

C

E

A D

B

C D

E

Figure 5. A forbidden diagram with five chords

Start with some singular point of some analytic curve in the (real) plane. Blow it up a first time (1). The result is a curve in some Moebius band, whose singular points are on the exceptional divisor, core of the band. If things go well, the singular point splits into several singular points, presumably simpler. Let us blow up all of them. It could happen that after one blow up, there is still a unique singular point on the divisor. Then, blow it up a second time. Let us continue the blowing up process as many times as necessary. We know that after some time, the singularity will be resolved. This means that the strict transform of the initial curve is now a collection of n disjoint smooth analytic curves intersecting transversally the exceptional divisor. This exceptional divisor is a union of real projective lines which are circles intersecting transversally. Consider the graph whose vertices are these projective lines and where an edge connects two vertices if the projective lines intersect. The inductive process of desingularization shows that this graph is a tree. Indeed, at each step we (1)

For basics about resolutions of real analytic singularities, see the corresponding chapter of [8].

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blow up a point which can be either a smooth point of the exceptional divisor, or an intersection of two projective lines. In the first case, a new leaf is grafted to a tree and in the second case, an edge is split into two edges. The first projective line, coming from the first blow up, can be chosen as the root of this tree.

Figure 6. Exceptional divisor made of six projective lines and the associated tree

In order to make sure that at the end of the process each projective line contains at most one point of the strict transform, it will be convenient to blow up once more each of the n points on the exceptional divisor, if necessary, introducing new projective lines. We can even suppose that the components of the divisor meeting the strict transform are leaves of the desingularization tree. Let us sum up. Given some analytic curve C defined in a neighborhood of (0, 0) in R2 by some equation F (x, y) = 0, we can construct the following objects. — A surface S with a connected oriented boundary. — An exceptional divisor E ⊂ S, consisting of a certain number of circles intersecting transversally, each pair meeting at most once. The associated intersection graph is a rooted tree. The embedding E ⊂ S is a homotopy equivalence. — The strict transform of C : a finite disjoint union Cˆ of smooth analytic arcs β1 , . . . , βn in S intersecting transversally E. The intersection of Cˆ with a component of E is empty if this component is not a leaf of the tree, and contains at most one point if it is a leaf. We can assume moreover that Cˆ is transversal to the boundary of S and that each arc βi intersects the boundary in two points. — A blowing down analytic map Ψ : S → R2 , collapsing E to the origin, which is a diﬀeomorphism from S \ E onto some small punctured disk, and which collapses Cˆ to our singular curve C . Each loop in S can be orienting or disorienting. Let γ be a closed immersed curve in a surface, passing once through a point x. When the surface is blown up at x, the selfintersection modulo 2 of the strict transform of γ (in the blown up surface) is equal to the self-intersection of γ (in the original surface) plus 1. In the inductive construction, when a projective line appears for the first time in the exceptional divisor, it is the core of a Moebius band, of self-interSection 1. Later on, some of its points may be blown up. Each blowing up permutes the orienting/disorienting status of a component

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β1

β2

Figure 7. A part of the blown-up surface S

in the divisor. Figure 6, is obtained after six blowing ups: six lines, a tree with six vertices, and six circles. Some components of E intersect the desingularized curve Cˆ: they define some leaves in the desingularization tree. We say that those leaves are colored. Observe that some leaves might not be colored. Note that if we choose some orientation for each component of E, the corresponding tree is planar so that the children of any node are linearly ordered. Changing the orientation reverses this order. Look for example at the necklace in Figure 8.

Figure 8. The surface S, neighborhood of the exceptional divisor

Six blow ups produced six bands, two are orientable and four are not. The exceptional divisor consists of the six cores of the corresponding bands. The desingularized curve is made up from the three red arcs, labeled a, b, c, each intersecting the boundary of S in two points. On top, we see in black the strict transform of the y axis. Going around the boundary of S we can read the corresponding analytic chord diagram. Just follow the arrow and read abacbc. 1.4. Proof of the fundamental lemma. — First, observe that deleting a chord in an analytic diagram transforms it into some other analytic diagram. This corresponds to deleting a branch. Now, start with an analytic chord diagram w and consider the desingularization tree of some associated planar singularity. There is a projection ρ of the surface S

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onto the exceptional divisor E which is a homotopy equivalence. The fiber ρ−1 (x) of a point x ∈ E is an arc connecting two points of the boundary if x is a regular point, and a cross if x is the intersection of two circles.

Figure 9. Projection of S onto the exceptional divisor

Let L be a node of the tree, i.e., one of the projective lines that constitute the exceptional divisor E. There is a unique chain of nodes going from L to the root. Cut two disjoint arcs in S as in Figure 10, in order to disconnect L from the root in S. The four endpoints of these arcs decompose the circle boundary of S into four intervals. Two of them (colored in red) correspond to “what is below L” in the tree. Going around the boundary of S and reading the chord diagram, we therefore find two disjoint intervals of letters, below L, whose union is stable under the involution sending each occurrence of a letter to the other. Note that these intervals could be empty if there were no colored leaves below L. However, if there is a colored leaf below L, then at least one of the two intervals isn’t empty, though it could happen that only one is not empty.

L

to the root

to the root

Figure 10. Pruning

In summary, every node L in the tree defines a chord diagram w(L) which is a sub-diagram of the original diagram w and which is “connected" in the sense that its letters form one or two intervals in w.

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Think of a rooted tree as a genealogy tree, the root being the founding member of the family. Each node has a certain number of descendants, some of them being colored leaves. Let L be one of the youngest members of the family having at least two colored leaves as descendants. Among the children of L, let L1 , . . . , Lk be the list of those having at least one colored descendant (ordered in this way along L). We have k ≥ 2 since otherwise one of the children of L would have at least two colored descendants. For the same reason, each Li has a unique colored descendant.

L

L1

L1

L2

L2

Figure 11. Pruning colored leaves

Now cut the surface S as in Figure 11, disconnecting L1 and L2 from the root and from all other colored leaves. As before, this defines two (green) intervals on the boundary of S whose union contains exactly four points of our initial diagram, associated to two chords. Two chords in two intervals can be organized in the following fifteen ways.

Figure 12. Fifteen cases

In each case, there is a solitary chord, a pitchfork, or a pair of parallel or antiparallel chords. This ends the proof of the fundamental lemma. 1.5. More non-analytic diagrams. — We have observed that deleting letters in analytic chord diagrams preserves analycity. A diagram is called basic non-analytic if it

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is not analytic but all of its proper sub chord diagrams are. Clearly a chord diagram is analytic if and only if it does not contain a basic non-analytic chord diagram. Theorem. — There is an infinite number of basic non-analytic chord diagrams. Here is an example that will be denoted by Cn (n ≥ 5). Consider the 2n points of Z/2nZ naturally ordered on the circle. The chord diagram pairs 2k and 2k + 3 for k = 1, . . . , n. For n = 5, this is our previous example of non-analytic diagram with five chords. This diagram Cn (n ≥ 5) is not analytic by the fundamental lemma. We still have to show that if one letter is deleted, the remaining diagram is analytic. For this, we need a suﬃcient criterion of analyticity. Theorem. — The following algorithm decides if a chord diagram is analytic: 1. If there is no solitary chord, no pitchfork, and no pair of parallel or antiparallel chords, the diagram is not analytic. 2. If there is a solitary chord, delete it and continue. If there is a pitchfork, delete the fork and keep the stick. If there is a pair of parallel or antiparallel chords, delete one of them and continue. 3. If you end up with the one-chord diagram, then the original one was analytic. The proof is easy. If w is analytic, w is also analytic since it corresponds to deleting a branch. For the converse, we have to show that if w is a diagram and if w is the new diagram obtained after one step of the algorithm with one chord less, then w is analytic if w is analytic. So we have to add an additional branch.

1

2

3

4

Figure 13. Adding a branch

Choose a desingularization S of a singular point associated to w. A chord corresponds to some smooth arc γ connecting two points on the boundary, transverse to the divisor at a point x. As in 1/, add a new analytic smooth (red) curve γ ′ in S, very close and transverse to γ as well as to the divisor. The blowing down of S produces a new singular point with one more branch. Clearly, the new diagram has one more chord which is parallel or antiparallel to the initial (blue) chord, depending on the orientations of the boundary. Now choose γ ′ as in 2/, with a quadratic tangency with γ at x and you get the other parallel or antiparallel situation. To create a pitchfork with a given (blue) handle, just add a smooth curve γ ′ close to γ with a quadratic

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tangency with the divisor as in 3/. Finally, if you want to add a solitary chord right after some given letter, proceed as in 4/. Let us use our algorithm to proove that Cn (n ≥ 5) is basic non-analytic. Deleting one chord, we get a non cyclic chain of n − 1 chords. The first two chords u, v define a pitchfork, so we may delete the first chord and continue until there is only one chord left. This diagram is therefore analytic. So indeed, the set of basic non-analytic diagrams is infinite. u v u

v

Figure 14. A chain of chords

1.6. With a computer. — In order to count analytic chord diagrams, we can use a computer to test small values of n. We start by listing all possible words of length 2n in which each letter occurs twice. The only subtlety is to take into account the cyclic character of the word under consideration. Here is the result for n ≤ 7, in the following table: 2

n

3

4

5

6

7

Words 3 15 105 945 10395 135135 Chord diagrams 2 5 18 105 902 9749 Up to symmetry 2 5 17 79 554 5283 – Words means “linear words of length 2n in which each letter occurs twice”. There are (2n − 1)!! of those. – Chord diagrams, as we have defined them, are words up to cyclic permutations. – The image of a chord diagram by a symmetry with respect to some diameter is another diagram, which may be the same diagram or not. The item “up to symmetry” counts the number of cyclic words up to these dihedral symmetries. We can then count the number of analytic diagrams using the algorithm that was described earlier. The result is:

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n

2 3

4

5

Analytic diagrams

2 5 18 102 817 7641

Up to symmetry

2 5 17

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C5 Figure 15. The 3 basic non-analytic chord diagrams with 5 chords

It follows that for n ≤ 4, all diagrams are analytic. Among the 105 5-chord diagrams, only the 3 examples in Figure 15 are not analytic. The first diagram is already familiar, under the name C5 . We denote the others by and . It wasn’t diﬃcult to guess the first but the other two were discovered by our computer. Among the 902 diagrams with 6 chords, 85 are not analytic. However, the nonanalyticity of most of them is due to the fact that one of their sub-diagrams is not analytic. Only two 6-chord diagrams are basic non-analytic.

C6 Figure 16. The 2 basic non-analytic chord diagrams with 6 chords

Observe that the first one is the member C6 of the infinite familly we exhibited. It corresponds to Z/12Z where every even number k (mod 12) is connected to k + 3 . (mod 12). The second will be denoted by Among the 9749 diagrams with 7 chords, 2108 are not analytic. The only basic non-analytic example is C7 . Later, we will show that the computer did indeed find all basic non-analytic diagrams. 1.7. Marked chord diagrams. — It will be convenient to introduce a slight strengthening of the notion of analytic chord diagrams. When we proved that our algorithm decides if a diagram is analytic, the key point was the possibility of inserting a new branch. It turns out that more complicated singularities can also be inserted, as we now explain. Consider a desingularization of some curve C as before, so that we have a surface S, a divisor E, and a collection

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of smooth curves β1 , . . . , βn intersecting E transversally at p1 , . . . , pn , where n is the number of real branches. Choose one of these points, say p1 . Choose now some other singular curve C1 , with n1 real branches, and assume that it does not contain the y axis. Delete β1 and replace it by a copy of C1 in the surface S, in such a way that the y axis for C1 is mapped into the divisor E and the singular point of C1 is mapped to p1 . We can now blow down the union of this copy of C1 and β2 , . . . , βn . The result is a new singular point, with n + n1 − 1 branches: one of the branches of C has been replaced by a copy of C1 .

β1

Figure 17. Inserting a singular point

Let us examine the eﬀect of this operation on the associated chord diagram. Looking at the diagram associated to C1 , we see that the y axis decomposes the word of length 2n1 in two components, Left and Right. In the new chord diagram with 2(n + n1 − 1) letters, one pair of identical letters from the old diagram with 2n letters has been replaced by two intervals, which are Left and Right. We should bear in mind that during this process, the orders of the letters in Left and Right might have been reversed. Indeed, the two intersections of the oriented boundary of S with β1 might be of diﬀerent signs. Moreover, the insertion of C1 in S can be done in four ways since S is not orientable and E is not oriented. a1

a2

a2 a3 a3 a1 Figure 18. Right = a1 a2 , Left = a1 a2 a3 a3

Note that Left or Right could be empty. In an equation F (x, y) = 0, we can replace (x, y) by (−x, y) or (x, −y) or (x, xy). The transformation (x, y) 7→ (x, xy) preserves each vertical line, collapses the axis x = 0 to the origin, and reverses the orientation

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for x < 0. This is no surprise: it is a blow down map. The square of this transformation preserves the orientation on each vertical line (for x 6= 0). Of course, we can proceed in the same way with all other branches of C , using other singular curves C2 , . . . , Cn . All these remarks suggest the following definition. Definition. — A marked chord diagram is a collection of 2n distinct points ±1 a±1 in the union of two opposite sides of a square {−1, 1} × [−1, 1] 1 , . . . , an (up to orientation preserving homeomorphisms of each side). Note the additional features if one compares with standard diagrams. Marked diagrams have a right and a left part. Moreover, each chord ai is now labeled with a number i from 1 to n and is oriented from a−1 to a+1 i i . Let us denote by M the set of those marked chord diagrams which are analytic, i.e., which arise from some analytic curve F (x, y) = 0 which does not contain the y-axis. Note that the analyticity of a marked diagram depends neither on the orientation of the chords nor on the labeling. The role of the labelings and orientations is simply to give the relevant information about which marked diagram is inserted in each chord, and in which way. a1-1

a2-1

a2+1 a3-1 a3+1

a1+1

Figure 19. Inserting marked chord diagrams

Let w be some analytic marked chord diagram with n chords. Given n analytic marked chord diagrams w1 , . . . , wn , with k1 , . . . , kn chords, define the action of w on (w1 , . . . , wn ) in the following way. Draw w and thicken each chord ai of w, creat+1 ing rectangles. Use the a−1 i ’s and the ai ’s as the left and right sides of these new rectangles. Now, insert w1 , .., wn in these rectangles respecting the labels and the orientations. Rename the chords, from 1 to k1 + k2 + · · · + kn using the lexicographic ordering. The result is another marked analytic chord diagram since this operation corresponds to the previously described insertion of analytic curves. This defines an operad structure on M though we will not make use of it in this paper. 1.8. Let us bound the number of chord diagrams. — It would be great to have some precise information on the number an of analytic diagrams with n chords. For instance, P an explicit formula for the generating series an tn would give the exact exponential growth rate of an . Unfortunately, we were not able to compute this function. In this subsection, we show at least that the fundamental lemma provides a reasonable bound.

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Consider a finite planar rooted binary tree. Equip each of its interior nodes (including the root) with one of the six examples of marked diagrams with two chords represented in Figure 20. 1 2

1

2

2

1

2

2

1 1

1 2

Figure 20. Six marked diagrams with two chords

By recursive insertions of the diagrams of the siblings into the diagram of their parent, this produces a marked analytic diagram and hence an analytic diagram, forgetting the labels, the orientations, and the two sides of the square. We claim that all analytic diagrams with n ≥ 2 chords are produced by this recipe. This is true for n = 2 since both diagrams with 2 chords (linked of not linked) appear when one forgets the marking in the six examples. Now, let w be some analytic diagram with n + 1 chords, and apply the fundamental lemma. Therefore, we find · · · aa · · · , · · · ab · · · ba · · · , · · · ab · · · ab · · · or · · · b · · · aba · · · in the diagram. In the case of aa, call b the letter which comes before a in the cyclic order. Our algorithm deletes a and produces an analytic diagram w ¯ with n chords, for which we can apply the induction. This means that w ¯ can be decorated with labels and orientations in such a way that it is produced by a binary tree, as above. Our diagram w is obtained from w ¯ by replacing one chord by two chords. It is easy to check that our six examples are suﬃcient to realize this duplication using an insertion in the operad. Hence w is constructed from a binary tree with n + 1 leaves with the same recipe. 1 2 2

1

2 1

2

1

1 2

Figure 21. Recursive construction of marked diagrams

A rooted binary tree with n leaves has n − 1 interior nodes (including the root) so that there are 6n−1 possible labels on the interior nodes. The number of planar binary trees with n leaves is given by the (n − 1)-th Catalan number. Therefore, we get the following rough estimate. Theorem. — The number an of analytic chord diagrams with n chords is less than 6n−1 times the (n − 1)-st Catalan number Cn−1 .

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The growth of Calalan numbers is well known: n1 log Cn converges to log 4 as n goes to infinity. Therefore 1 lim sup log an ≤ log(24). n 2. Analytic chord diagrams: interlace graphs Let us begin by a definition. Definition. — The interlace graph of a chord diagram is the graph whose vertices are the chords and such that an edge connects two chords if they intersect. Not every graph comes from a chord diagram and a graph might come from several chord diagrams. Nevertheless, the interlace graphs coming from analytic chord diagrams turn out to be easy to analyze. The icing on the cake is that these graphs have been introduced forty years ago in a totally diﬀerent context and are very well understood. Thanks to this new perspective, we will get the complete description of basic non-analytic chord diagrams. 2.1. Polynomial interchanges: permutation graph. — Before discussing analytic chord diagrams, we come back to the easy example of polynomial interchanges, in order to describe our general strategy. Definition. — The permutation graph G(π) associated to a permutation π of {1, . . . , n} has {1, . . . , n} as vertices, and an edge connects i and j if π reverses the order of (i, j).

1 2 3 4

1

2

3

4

Figure 22. A permutation and its graph

Note that a permutation of {1, . . . , n} defines a (marked) chord diagram with n chords, as shown in Figure 23. The graph G(π) is nothing more than the interlace graph of this chord diagram. Recall that for any polynomial interchange, one can find two consecutive integers with consecutive images. The corresponding chords have therefore the property that any chord intersecting one of them intersects the other. In terms of the graph G(π), this suggests the following definition.

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π(i)

i

Figure 23. A permutation and its graph

Definition. — Two vertices x, y in a graph are called twins if they have the same neighbors (diﬀerent from x or y). They are called true or false twins (2) depending on the existence of an edge connecting them.

Figure 24. False and true twins

Twins can be merged in a single vertex, producing a smaller graph with one vertex less. The graph G(π) coming from a polynomial interchange π contains at least two twins, corresponding to two consecutive integers i, i + 1 such that π(i + 1) = π(i) ± 1. Merging the twins in the graph amounts to merging the two elements i, i + 1. We know that polynomial interchanges are characterized by the fact the iteration of this merging procedure eventually leads to the trivial permutation with n = 1. Definition. — A finite graph is called a cograph if it can be reduced to a trivial 1-vertex graph by merging twins successively. The terminology “cograph” comes from the fact that the complement of a cograph is also a cograph. A graph G and its complement G have the same vertices and two vertices are adjacent in G if and only if they are not adjacent in G. Proposition. — A permutation is a polynomial interchange if and only if its permutation graph is a cograph. We just explained why the permutation graph of a polynomial interchange is a cograph. To prove the converse, it suﬃces to show that if G(π) is a cograph, there are two consecutive integers with consecutive images. The proof is by induction on n. If i < j are false (resp. true) twins, the image by π of the interval [[i, i + 1, . . . , j]] (2)

We are not responsible for the terminology which is classical and more convenient than dizygotic (or fraternal) twins. It is also closer to the French “faux jumeaux”.

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Figure 25. A cograph

is [[π(i), π(i + 1), . . . , π(j)]] (resp. [[π(j), π(j + 1), . . . , π(i)]]). If j ≥ i + 2, the image π([[i, . . . , j − 1]]) is also an interval and we apply the induction hypothesis to the restriction of π to [[i, . . . , j − 1]] so that one finds two consecutive integers with consecutive images. Cographs have been introduced in the 1970’s under diﬀerent names (D⋆ -graphs, hereditary Dacey graphs, and 2-parity graphs; see [2] for references). They are not very diﬃcult to describe. We list some of their properties whose (elementary) proofs are left to the reader. In what follows, all graphs are finite, with no loops and no multiple edges. A connected graph defines a metric space on its set of vertices. The distance between two vertices is, by definition, the length of the shortest path connecting them. A subgraph H of a graph G is induced if any edge of G connecting two vertices of H is also an edge of H. Theorem. — The following properties of a finite graph G are equivalent. 1. G is a cograph. 2. G is the permutation graph of some polynomial interchange. 3. Any connected induced subgraph of G has a diameter at most 2. 4. There is no induced subgraph isomorphic to

(denoted P4 ).

Note that the permutation graphs associated to the forbidden Kontsevich permutations (2, 4, 1, 3) and (3, 1, 4, 2) are both isomorphic to P4 . Conversely a permutation whose graph is isomorphic to P4 is one of these two permuations. The previous theorem can therefore be translated as the following. Theorem. — Let n ≥ 2 be some integer. A permutation π of {1, 2, . . . , n} is a polynomial interchange if and only if it does not “contain” one of the two forbidden permutations, i.e., if there do not exist four indices 1 ≤ i1 < i2 < i3 < i4 ≤ n such that π(i2 ) < π(i4 ) < π(i1 ) < π(i3 ) or π(i3 ) < π(i1 ) < π(i4 ) < π(i2 ). 2.2. Collapsible graphs. — Starting from a tree, one can strip oﬀ its leaves one by one until it has been stripped completely naked. Let us say that a vertex in a graph is pendant if it is adjacent to a unique vertex. Any tree can be constructed by successive additions of pendant vertices, starting with the tree with only one vertex.

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Figure 26. Pendant vertex

Definition. — A finite graph is collapsible if it can be reduced to a 1-vertex graph by applying two kinds of elementary operations: deleting a pendant vertex and merging twins. One can express the same thing in the opposite way. Start with the trivial graph with one vertex and apply two kinds of operations: adding a pendant vertex or creating a pair of twins. The second operation simply consists in duplicating a vertex and connecting the newly born twin to the rest of the graph as the original vertex was. Then, decide if you want true or false twins. The key point is the following. Proposition. — A chord diagram is analytic if and only if its interlace graph is collapsible. This will follow from the algorithmic description of analytic diagrams given in the previous section. Before the proof, let us make an elementary remark, as an appetizer. Let w be a diagram and A be a subset of its 2n letters on the circle. We will say that A is stable under w if any chord with one end in A has its other end in A. Said diﬀerently A is a sub-chord diagram wA of w. Suppose that there is an interval A which is stable under w, and let B be its complement. Clearly the interlace graph G(w) of w is the disjoint union of the graphs G(wA ) and G(wB ) of wA and wB . It follows that G(w) is collapsible if and only if G(wA ) and G(wB ) are collapsible. Our algorithm shows that if wA , wB are analytic so is w. Conversely, if w is analytic, so are their sub-diagrams wA and wB . Let us prove now the proposition. Start with an analytic diagram w. If two chords of w are parallel or antiparallel, the associated vertices in the interlace graph are twins and our algorithm merges them. A pitchfork gives a pendant vertex in the graph and the algorithm deletes the short chord and keeps the handle. A solitary chord defines an isolated vertex, which is removed by the algorithm. It follows that the interlace graph associated to an analytic diagram is collapsible.

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A

21

B

Figure 27. G(w) = G(wA ) ⊔ G(wB )

For the converse, we show that every diagram w whose interlace graph G(w) is collapsible contains a solitary chord, or a pitchfork, or a pair of parallel or antiparallel chords. A solitary chord in a diagram corresponds to two consecutive identical letters · · · aa · · · in the cyclic word. A pitchfork corresponds to a subword of the form · · · aba · · · . A pair of parallel (resp. antiparallel) chords corresponds to · · · ab · · · ba · · · (resp. · · · ab · · · ab · · · ). Our proof will be by contradiction. We use the symbol to mean “contradiction”. Consider a possible counterexample w to the previous assertion with a minimal number of chords. So G(w) is collapsible and w contains no solitary chord, no pair of parallel or antiparallel chords, and no pitchfork. Since G(w) is collapsible, there is a vertex α which is either isolated, or pendant, or is part of a pair of twins. Let w be the diagram obtained by deleting α from w. Of course G(w) is collapsible so that, by minimality, w contains a subword · · · aa · · · or · · · aba · · · , or · · · ab · · · ba · · · , or · · · ab · · · ab · · · . The problem is that these are subwords of w and not of w, which also contains two copies of the letter α, which could sneak into the above subwords. Note that by minimality any interval which is stable under w is either empty or everything. A priori: – 0, 1 or 2 letters α could sneak in the subword, i.e., in non dotted intervals – the subword of w could correspond to a solitary chord, or a pitchfork, or to a pair of parallel or antiparallel chords, – α could be isolated, pendant or twin, true or false, in G(w). That makes 3 × 4 × 4 cases to examine! Fortunately, many cases can be studied simultaneously.

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1/ If no letter α sneaks into the above subwords, there is no problem: our solitary chord or pitchfork, or pair of parallel or antiparallel chords in w have the same property for w . 2/ If α is isolated in G(w), this means that no chord intersects α. Therefore α decomposes the circle in two stable intervals, of which one has to be empty . 3/ If two letters α sneak in, they cannot occur as consecutive letters since that would force the chord α to be solitary in w . Still in this case, we have to look at · · · aαbαa · · · , or · · · aαb · · · bαa · · · , or · · · aαb · · · aαb · · · This produces respectively a pitchfork (α, b), a pair of parallel chords (α, b) or antiparallel chords (α, b) in w . 4/ Inserting one α in a solitary chord yields · · · aαa · · · which produces a pitchfork in w with handle α . So far, we did not use the fact that α is pendant or twin. This will be used in the remaining cases, when a single α enters in a pitchfork or a pair of parallel or antiparallel chords of w. If α is pendant, let β be the only chord in w intersecting α. If α has twin siblings, we denote β one of them. The two chords α, β determine four intervals in the circle, excluding α, β, that will be called sectors. If α is pendant, the union of two sectors which are on the same side of α is stable. If α, β are twins, the union of opposite sectors is stable.

α

α β

β

β

α β

α Figure 28. Linked or unlinked

5/ Suppose now that a single letter α enters · · · aba · · · , or · · · ab · · · ba · · · , or · · · ab · · · ab · · · and that one of the two letters a, b is equal to β. Then, the two letters α and β are consecutive in w. This implies that one of the sectors is empty. 5-1 In the pendant case, this implies that the other sector, on the same side of α, is a stable interval and therefore also empty. So, α, β is a pitchfork in w with handle β . 5-2 In the twin case, this implies that the opposite sector is a stable interval, and therefore empty. So α, β is a pair of parallel or antiparallel chords in w . 6/ Finally, suppose that a single letter α enters · · · aba · · · , or · · · ab · · · ba · · · , or · · · ab · · · ab · · · and that none of the two letters a, b is equal to β.

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6-1 Assume that (a, b) are parallel or antiparallel chords in w. Inserting one α in ab yields · · · aαb · · · ba · · · or · · · aαb · · · ab · · · . Since the letters ba (and ab) are consecutive, these two occurrences are on the same side of the chord α. And since the letters aαb are consecutive in w, it follows that these other two occurrences of a and b are on diﬀerent sides of α. Hence the chord α intersects only one of the chords a, b. 6-1-1 If α is pendant, this forces a or b to be equal to β . 6-1-2 If α, β are twins, this is not possible . 6-2 Assume that (a, b) is a pitchfork in w. Inserting a letter α in · · · aba · · · yields · · · aαba or · · · abαa so that the chord α should intersect the chord a. 6-2-1 If α is pendant, this forces a = β . 6-2-2 If (α, β) are twins, then β (which is not a) also intersects a. Since the letters aαba are consecutive, we must have β = b . This finishes the proof. Ouf! Now, we have to understand the nature of collapsible graphs. 2.3. Distance hereditary and treelike graphs. — Collapsible graphs have been defined by several authors forty years ago, under diﬀerent names, with very diﬀerent motivations. We will see that these graphs are very close to being trees. Howorka [12] defined distance hereditary graphs in 1977. Definition. — A finite graph G is distance hereditary if for every connected induced subgraph H ⊂ G, the distance between two vertices of H in H is equal to the distance between the same vertices in G.

Figure 29. A cycle of length ≥ 5 is not distance hereditary

For instance, a tree is distance hereditary and a cycle of length at least 5 is not. It suﬃces to choose H as the induced subgraph defined by a path inside the cycle whose length is greater than one half of the length of the cycle. Let us introduce metric graphs along with a property characterizing the subspaces of metric trees. A weakened version of this property will give an alternative description of our distance hereditary graphs. Consider a finite graph and choose some length for each edge, which could be any positive real number. Define the length of a path as the sum of the lengths of its edges and the distance between two vertices as the smallest length of a path connecting them. One speaks of a metric graph. We are looking for a characterization of metric spaces (usually called metric trees) arising in this way from trees. Here is the answer. Let (V, d) be a finite metric space.

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Choose four points x1 , x2 , x3 , x4 in V and compute the sums of the lengths of the three pairs of diagonals: d(x1 , x2 ) + d(x3 , x4 ) ; d(x1 , x3 ) + d(x2 , x4 ) ; d(x1 , x4 ) + d(x2 , x3 ). Let s (resp. m, l) be the smallest (resp. medium, largest) of these three numbers: s ≤ m ≤ l. It turns out that a finite metric space is isometric to a subset of a metric tree if and only if m = l for every quadruple of points. This is not diﬃcult to prove but the lazy reader might find the proof in this short paper [3].

x1

x2

x3

x4

Figure 30. A quadrangle in a tree

We should be careful. A graph, where all edges have length 1, can be isometric to a subset of a metric tree without being itself a tree. Look at the example in Figure 31. In graph theory, those graphs are called block graphs.

Figure 31. A block graph

In order to construct them, start with a tree, delete some of its vertices and replace them by cliques, i.e., finite graphs where all pairs of vertices are adjacent, as in the figure. This is indeed a characterization of block graphs (see [11]). A metric space (E, d) is geodesic if for every pair of points (x, y) there exists an isometric embedding i : [0, d(x, y)] → E such that i(0)) = x and i(d(x, y)) = y. In the 1980’s, Gromov developed a geometric theory for hyperbolic spaces which had a very strong inﬂuence on combinatorial and geometric group theory. The definition is the following. A metric space (E, d) is called hyperbolic if there exists some δ ≥ 0 such that for every quadruple of points as above, m and l are “almost equal”, i.e., l −m ≤ δ. Note that any finite metric space is trivially hyperbolic (for δ suﬃciently big) so that this concept is only relevant for geometry in the large. There are many equivalent formulations of this property, the most popular (for geodesic metric spaces) being that all geodesic triangles are slim. Consider three points x, y, z and choose three

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25

geodesics [x, y], [x, z], [y, z] connecting them. Every point in [x, y] should be at some uniformly bounded distance from the union [x, z] ∪ [y, z], independently of the choice of x, y, z (Figure 32). z

y x

Figure 32. A slim triangle

This concept is remarkably robust. For instance, the universal cover of a negatively curved compact Riemannian manifold is hyperbolic. These metric spaces are well approximated by trees, in a quantitative way (see for instance [6]). In 1986, Bandelt and Mulder published a paper [1] proposing purely metrical characterizations of distance hereditary graphs, close to Gromov’s hyperbolicity conditions. Definition. — A finite graph G is treelike if for every 4-tuple of vertices x1 , x2 , x3 , x4 two of the following three numbers are equal: d(x1 , x2 ) + d(x3 , x4 );

d(x1 , x3 ) + d(x2 , x4 );

d(x1 , x4 ) + d(x2 , x3 ).

For instance, one can show easily that a treelike graph is hyperbolic in the sense of Gromov with δ=2.

Figure 33. A cycle of length 4 is treelike but not a tree

All these definitions turn out to be equivalent, and correspond to Theorem. — Let G be a finite graph. The following properties are equivalent. 1. G is the interlace graph of some analytic chord diagram, 2. G is collapsible, 3. G is distance hereditary, 4. G is treelike,

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Figure 34. The house, the gem and the domino

5. G does not contain a cycle of length at least five, or a house, a gem, or a domino (see Figure 34), as an induced subgraph. All the equivalences in the previous theorem (except of course the first item) are proved in the above mentioned papers. However, we will soon propose some elementary proofs. It is now time to harvest the fruits of our labor and to get a very simple description of analytic chord diagrams.

It should not be surprising that the interlace graphs of , , are the house, , , are the only the gem, and the domino. In fact, one can easily show that chord diagrams whose interlace graphs are the house, the domino and the gem. In the same way, we have already described the non-analytic chord diagram Cn defined by Z/2nZ (n ≥ 5) where there is a chord connecting 2k and 2k + 3 (for k = 1, . . . , n). Its interlace graph is a cycle of length n. It also clear that Cn is the only chord diagram whose interlace graph is a cycle of length n. Finally, note that a sub-chord diagram defines an induced subgraph in the interlace graph. Therefore, we get a very satisfactory description of analytic chord diagrams, which is the main result of this paper. Theorem. — A chord diagram is analytic if and only if it does not contain or Cn (n ≥ 5) as a sub-chord diagram.

,

,

Note the complete analogy with our characterization of polynomial interchanges as those permutations which do not contain Kontsevich’s examples (2, 4, 1, 3) and (3, 1, 4, 2).

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2.4. Some proofs. — We present now the proofs of the equivalences of the definitions in the previous subsection. They are mostly elementary. In this specific case, it was probably more challenging to find the significant definitions than to prove their equivalence. No

≥5

as an induced subgraph =⇒ Distance hereditary.

Let H be an induced subgraph of a graph G. Connect two vertices p, q of H at distance n in H by a path c = (x0 , x1 , . . . , xn ) (with p = x0 and q = xn ) in H. Two vertices xi , xj are adjacent if and only if i, j are consecutive since otherwise there would be a shortcut. In other words, the path c is induced in G. It follows that in order to show that a graph is distance hereditary, we should prove that the distance between the endpoints of any induced path in G is equal to the length of the path. ≥5 Suppose that no is induced in G. Choose an induced path c1 = (x0 , x1 , . . . , xn ) and let us show, by induction on n, that the distance in G between x0 and xn is exactly n. Connect x0 to xn by a shortest path c2 = (y0 , y1 , . . . , yl ) in G (with y0 = x0 and yl = xn ). Of course, c2 is also induced and d(y0 , yi ) = j for 0 ≤ i ≤ l. By the induction hypothesis, d(x0 , xi ) = i for 0 ≤ i ≤ n − 1 so l is equal to n − 2, n − 1 or n. We must show that the first two cases are not possible. Suppose that l = n − 2 or n − 1. By induction hypothesis, we can assume that the two paths c1 , c2 only intersect at their endpoints: any other intersection point could be used as the starting point of shorter paths c′1 and c′2 .

n-1

xn-1

n-2

xn-2

n-3

xn-3

xn-1

xn=yn-2

n-2

xn-2

yn-2

yn-3

n-3

xn-3

yn-3

x1

y1

c2

c1 1

xn=yn-1

n-1

x1

y1

0

1 0

x0=y0

x0=y0

Figure 35. Checking heredity. . .

Draw a picture in the plane in such a way that the height of a vertex of c1 or c2 is the distance from x0 . The cases l = n − 2 and l = n − 1 are pictured in Figure 35. Vertices of c1 are red and vertices of c2 are blue. The union of c1 and c2 defines a cycle c in G. The length of c is at least 5. The cycle c cannot be induced since there is no induced cycle of length ≥ 5. Therefore there must exist diagonals connecting vertices of c1 with vertices of c2 .

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By the triangle inequality, the height diﬀerence of the two endpoints of a diagonal can only be −1, 0, 1. Moreover, diagonals connect points of diﬀerent colors. Let us order the diagonals (xi , yj ) from top to bottom, i.e., (xi , yj ) is before (xi′ , yj ′ ) if j > j ′ or j = j ′ and i > i′ .

δ1

δ1

δ1

δ2

≥5

Figure 36. No

=⇒ distance hereditary

Now, try to construct the ladder, one diagonal at a time. The rule of the game is the following. You have to draw an ordered sequence of diagonals δ1 , δ2 , . . . respecting ≥5 . Note that the the conditions above and without creating any induced diagonal δk together with the part of c which is above it defines a cycle. Any chord in this cycle has to be one of the previously chosen chords δ1 , . . . , δk−1 . In the case l = n − 1, there are only two possibilities for the first diagonal δ1 . It could be (xn−1 , yn−2 ) or (xn−2 , yn−2 ). In the case l = n−2 there is only one possibility ≥5 for δ1 . Then, try to select the second diagonal δ2 , avoiding . Only one of the three choices of δ1 allows you to do so. Finally, try to draw the third diagonal, in the only case where you could draw δ1 , δ2 . It is not possible to continue without creating one of the forbidden graphs. Distance hereditary =⇒ Collapsible. We only need to show that G has a pendant vertex or a pair of twins. Indeed deleting the former or merging the latters preserves the distance hereditary property so that we can repeatedly collapse the graph to a single vertex. y x

Sk z

y' Sk-1

Figure 37. Hereditary =⇒ Collapsible

ASTÉRISQUE 415

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Choose some vertex x in a connected distance hereditary graph G and look at the largest k such that the sphere Sk in G of radius k and centered on x is non-empty. Let C be a connected component of Sk . If C contains only one element, then it is a pendant vertex in G. Otherwise, choose two vertices y, y ′ in C which are adjacent in C. Choose a vertex z adjacent to y and at distance k − 1 from x. Choose a chain c of length k − 1 from x to z and call c′ the chain of length k + 1 obtained by adding the edge between z and y and from y to y ′ . Since the distance between x and y ′ is exactly k, this chain cannot be induced and y ′ has to be adjacent to z. This implies that two points in C are simultaneously adjacent or not to any point z at distance k − 1 from x. It follows that C cannot contain an induced path P4 of length 3, since together with z, it would produce a gem in G, which is not distance hereditary. Hence C is a cograph and in particular contains a pair of twins. By the above observation, two twins in C are twins in G. Collapsible =⇒ Treelike. Easy by induction. Take four points in a graph, delete a pendant vertex or merge two twins. One of the four points might be the vertex which has been removed. If this is the case replace it by the other end of the removed edge. Look at the corresponding points in the stripped graph (taking into account for instance the fact that two of our four points could be the two twins which have been merged). Apply the induction hypothesis. Treelike =⇒ No induced

≥5

.

Obvious since one checks easily that none of these examples of graphs are treelike.

2.5. Appendix: completely decomposable graphs. — This subsection is some kind of bonus but it can shed some light on the structure of collapsible graphs. Graphs that can be stripped to a point by deleting only pendant vertices are trees. Graphs that can be stripped to a point by merging only pairs of twins are cographs. Collapsible graphs should not be far from being trees. This is indeed true as explained now. B1

B2 x1 x2

x1 x2

A2 A1

G1

G G2

Figure 38. Splitting a graph

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Let G be a finite connected graph. Suppose that its vertices have been partitioned in two parts A1 and A2 . Let B1 ⊂ A1 (resp. B2 ⊂ A2 ) the set of vertices of A1 (resp. A2 ) which are adjacent to some vertex in A2 (resp. A1 ). Suppose that every element of B1 is adjacent to every element of B2 and that those are the only edges between A1 and A2 . This condition is trivially satisfied if A1 or A2 contains less than one element, so we assume that A1 and A2 contain at least two elements each. In this situation, the graph G is called decomposable and the partition A1 , A2 is a split. In order to keep track of this decomposition, let us create two graphs G1 , G2 in the following way. The set of vertices of G1 (resp. G2 ) is A1 plus one extra vertex x1 (resp. x2 ) called the control vertex. As for the edges of G1 (resp. G2 ), choose the edges of G plus extra edges connecting x1 (resp. x2 ) to all elements of B1 (resp. B2 ). The graph G can be reconstructed from (G1 , x1 ) and (G2 , x2 ) by an elementary join construction. Notice that the control points x1 , x2 are not vertices of G: they are only useful to define the edges connecting the two parts. Note that when A2 contains two elements, as in Figure 39, the graph G has a pendant vertex or a pair of twins.

Figure 39. A2 contains 2 elements

Hammer and Maﬀray [10] introduced in 1987 completely decomposable graphs, analog to trees. A finite connected graph is a tree if and only if every induced connected subgraph contains a cut edge, i.e., an edge that disconnects it. Definition. — A finite graph is completely decomposable if every induced connected subgraph with at least four vertices is decomposable. It is not hard to prove that completely decomposable (connected) graphs are precisely the collapsible (connected) graphs. Indeed in the join construction, if G1 and G2 are collapsible, the same is true for G, so that completely decomposable graphs are collapsible, by induction. Conversely, we have seen that a pendant vertex or a pair of twins gives rise to a decomposition. Therefore collapsible graphs are decomposable and even completely decomposable, since an induced subgraph of a collapsible graph is collapsible (with the distance hereditary definition). In order to give a precise description of completely decomposable graphs, let us state first the important split decomposition theorem for general connected finite graphs. If a finite connected graph G is decomposable, consider it as the join of G1 and G2 as before. Then, try to decompose G1 and G2 , and so on, until the resulting graphs

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become non-decomposable. The final result of this decomposition into “elementary pieces” can be conveniently described by a graph-labeled tree, as explained below. It consists of a tree T where each internal node x is equipped with a connected finite graph Gx which is either a clique, a star or indecomposable (no splits). Moreover some bijection has been chosen between the vertices of Gx and the edges getting out from the node x in T . Assume that the valency of each node is at least 3. Given such a structure, we construct a graph G(T ) which is a “composition of the Gx ’s controlled by T ”. The definition is the following. The vertices of G(T ) are the leaves of T . In order to understand the edges of G(T ), let us just draw a picture, inspired by the paper of Gioan and Paul [9] who introduced this concept of graph-labeled tree. We see a tree with 16 leaves and 6 internal nodes, in pink. The associated graph, with 16 vertices, is drawn on the right. 6

7

6 8

7

5

8

5 4

9

3

9

10

34

11 10 16 2

11 12

2 1

1 13

15

16 15 14

12 13

14

Figure 40. Graph-labeled tree and its composition

Choose two leaves of T and connect them by the shortest path in the tree. For each node x which is visited by this path, there is an entrance edge and an exit edge. In turn, these two edges define two vertices of Gx . Two vertices of G(T ), that is two leaves of T , are adjacent in G(T ) if, for every node x visited by this path, the two corresponding vertices of Gx are adjacent in Gx . The vertices of the Gx ’s generalize the two control vertices, as in the simple case where T contains only one edge. The main result, proved by Cunningham and Edmonds [5] in 1980 (and reformulated by Giona and Paul), is that any finite connected graph is obtained by such a construction in which the Gx ’s are elementary, in an essentially unique way. The existence of this splitting is easy. The hard part is the “essential uniqueness” that we don’t define since we will not need it. Let us come back to completely decomposable graphs. In this special case the Gx ’s must have at most 3 vertices. Indeed, they are indecomposable and induced subgraphs of G, so that the claim follows from the definition of complete decomposability. This gives a fairly precise geometric description of completely decomposable graphs. Take a tree such that every node has valency 3. For each node, choose a connected graph

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with 3 vertices (not that many choices!), and construct a graph-labeled tree as in Figure 41. All completely decomposable graphs are produced in this way. This is absolutely no surprise. Indeed look at the third picture, showing three small graphs with three leaves. When you hook one of them to some (blue) leaf of a graph-labeled tree T , you get another graph-labeled tree T ′ with one more leaf. If you examine the eﬀect on the associated graph G(T ), you see that you have split a vertex in a pair of twins (true or false) or you have created a pendant vertex, depending on the cases. We are back to the original definition of collapsible graphs as graphs that can be constructed from a point by successive introductions of twins or of pendant vertices.

4

5

4

5

3

6

3

6

2

7

2

7

1

8

1

8

Figure 41. A collapsible graph

References [1] H.-J. Bandelt & H. M. Mulder – “Distance-hereditary graphs”, J. Combin. Theory Ser. B 41 (1986), p. 182–208. [2] A. Brandstädt, V. B. Le & J. P. Spinrad – Graph classes: a survey, SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), 1999. [3] P. Buneman – “A note on the metric properties of trees”, J. Combin. Theory Ser. B 17 (1974), p. 48–50. [4] S. Chmutov, S. Duzhin & J. Mostovoy – Introduction to Vassiliev knot invariants, Cambridge Univ. Press, 2012. [5] W. H. Cunningham & J. Edmonds – “A combinatorial decomposition theory”, Canad. J. Math. 32 (1980), p. 734–765. [6] É. Ghys & P. de la Harpe (eds.) – Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Math., vol. 83, Birkhäuser, 1990.

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[7] É. Ghys – “Intersecting curves (variation on an observation of Maxim Kontsevich)”, Amer. Math. Monthly 120 (2013), p. 232–242. [8]

, A singular mathematical promenade, ENS Editions, 2017.

[9] E. Gioan & C. Paul – “Split decomposition and graph-labelled trees: characterizations and fully dynamic algorithms for totally decomposable graphs”, Discrete Appl. Math. 160 (2012), p. 708–733. [10] P. L. Hammer & F. Maffray – “Completely separable graphs”, Discrete Appl. Math. 27 (1990), p. 85–99, Computational algorithms, operations research and computer science (Burnaby, BC, 1987). [11] F. Harary – “A characterization of block-graphs”, Canad. Math. Bull. 6 (1963), p. 1–6. [12] E. Howorka – “A characterization of distance-hereditary graphs”, Quart. J. Math. Oxford Ser. 28 (1977), p. 417–420. [13] A. Stoimenow – “On the number of chord diagrams”, Discrete Math. 218 (2000), p. 209–233.

É. Ghys, Unité de Mathématiques Pures et Appliquées de l’École normale supérieure de Lyon, U.M.R. 5669 du CNRS, 46, allée d’Italie, 69364 Lyon Cedex 07, France E-mail : [email protected] C.-L. Simon, Unité de Mathématiques Pures et Appliquées de l’École normale supérieure de Lyon, U.M.R. 5669 du CNRS, 46, allée d’Italie, 69364 Lyon Cedex 07, France E-mail : [email protected]

SOCIÉTÉ MATHÉMATIQUE DE FRANCE 2020

Astérisque 415, 2020, p. 35–43 doi:10.24033/ast.1098

A CLOSING LEMMA FOR POLYNOMIAL AUTOMORPHISMS OF C2 by Romain Dujardin

Abstract. — We prove that for a dissipative polynomial diﬀeomorphism of C2 , the support of any invariant measure is, apart from a few well-understood cases, contained in the closure of the set of saddle periodic points. Résumé (Un lemme de fermeture pour les automorphismes polynomiaux de C2 ) Nous montrons que pour un automorphisme polynomial dissipatif de C2 , le support de toute mesure invariante est contenu dans l’adhérence de l’ensemble des points selles, à l’exception de quelques cas bien compris.

1. Introduction and results Let f be a polynomial diﬀeomorphism of C2 with non-trivial dynamics. This nontriviality can be expressed in a variety of ways, for instance it is equivalent to the exponential growth of the algebraic degrees of the iterates f n or to the positivity of topological entropy. The dynamics of such transformations has attracted a lot of attention in the past few decades (the reader can consult e.g., [1] for basic facts and references). In this paper we make the standing assumption that f is dissipative, i.e., that the (constant) Jacobian of f satisfies |Jac(f )| < 1. We denote by J + the forward Julia set, which can be classically characterized in terms of normal families, or by saying that J + = ∂K + , where K + is the set of points with bounded forward orbits. Reasoning analogously for backward iteration gives the backward Julia set J − = ∂K − . Thus the 2-sided Julia set is naturally defined by J = J + ∩ J − . Another interesting dynamically defined subset is the closure J ∗ of 2010 Mathematics Subject Classification. — 37F45, 37F10, 37F15. Key words and phrases. — Ergodic closing lemma, polynomial automorphisms of C2 . Research partially supported by ANR project LAMBDA, ANR-13-BS01-0002 and a grant from the Institut Universitaire de France.

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the set of saddle periodic points (which is also the support of the unique entropy maximizing measure by [2]). The inclusion J ∗ ⊂ J is obvious. It is a major open question in this area of research whether the converse inclusion holds. Partial answers have been given in [3, 5, 7, 13, 11]. Let ν be an ergodic f -invariant probability measure. If ν is hyperbolic, that is, its two Lyapunov exponents (1) are non-zero and of opposite sign, then the so-called Katok closing lemma [12] implies that Supp(ν) ⊂ J ∗ . It may also be the case that ν is supported in the Fatou set: then from the classification of recurrent Fatou components in [4], this happens if and only if ν is supported on an attracting or semi-Siegel periodic orbit, or is the Haar measure on a cycle of k circles along which f k is conjugate to an irrational rotation (recall that f is assumed dissipative). Here by semi-Siegel periodic orbit, we mean a linearizable periodic orbit with one attracting and one irrationally indiﬀerent multipliers. The following “ergodic closing lemma” is the main result of this note: Theorem 1.1. — Let f be a dissipative polynomial diffeomorphism of C2 with nontrivial dynamics, and ν be any invariant measure supported on J. Then Supp(ν) is contained in J ∗ . A consequence is that if J \J ∗ happens to be non-empty, then the dynamics on J \J ∗ is “transient” in a measure-theoretic sense. Indeed, if x ∈ J, we can form an invariant Pn probability measure by taking a cluster limit of n1 k=0 δf k (x) and the theorem says that any such invariant measure will be concentrated on J ∗ . More generally the same argument implies: Corollary 1.2. — Under the assumptions of the theorem, if x ∈ J + , then ω(x)∩J ∗ 6= ∅. Here as usual ω(x) denotes the ω-limit set of x. Note that for x ∈ J + it is obvious that ω(x) ⊂ J. It would be interesting to know whether the conclusion of the corollary can be replaced by the sharper one: ω(x) ⊂ J ∗ . Theorem 1.1 can be formulated slightly more precisely as follows. Theorem 1.3. — Let f be a dissipative polynomial diffeomorphism of C2 with nontrivial dynamics, and ν be any ergodic invariant probability measure. Then one of the following situations holds: (i) either ν is atomic and supported on an attracting or semi-Siegel cycle; (ii) or ν is the Haar measure on an invariant cycle of circles contained in a periodic rotation domain; (iii) or Supp(ν) ⊂ J ∗ . (1)

Recall that in holomorphic dynamics, Lyapunov exponents always have even multiplicity.

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Note that the additional ergodicity assumption on ν is harmless since any invariant measure is an integral of ergodic ones. The only new ingredient with respect to Theorem 1.1 is the fact that measures supported on periodic orbits that do not fall in case (i), that is, are either semi-parabolic or semi-Cremer, are supported on J ∗ . For semi-parabolic points this is certainly known to the experts although apparently not available in print. For semi-Cremer points this follows from the hedgehog construction of Firsova, Lyubich, Radu and Tanase (see [14]). For completeness we give complete proofs below. A final comment on the dissipativity assumption. Of course Theorem 1.1 also holds if |Jac(f )| > 1 by simply replacing f by f −1 . On the other hand our methods break down completely when f is conservative (|Jac(f )| = 1), since they are based on the analysis of strong stable manifolds. Acknowledgments. — Thanks to Sylvain Crovisier and Misha Lyubich for inspiring conversations. This work was motivated by the work of Crovisier and Pujals on strongly dissipative diﬀeomorphisms (see [6, Thm 4]) and by the work of Firsova, Lyubich, Radu and Tanase [10, 14] on hedgehogs in higher dimensions (and the question whether hedgehogs for Hénon maps are contained in J ∗ ).

2. Proofs In this section we prove Theorem 1.3 by dealing separately with the atomic and the non-atomic case. Theorem 1.1 follows immediately. Recall that f denotes a dissipative polynomial diﬀeomorphism with non trivial dynamics and ν an f -invariant ergodic probability measure. 2.1. Preliminaries. — Using the theory of laminar currents, it was shown in [2] that any saddle periodic point belongs to J ∗ . More generally, if p and q are saddle points, then J ∗ = W u (p) ∩ W u (q) (see Theorems 9.6 and 9.9 in [2]). This result was generalized in [8] as follows. If p is any saddle point and X ⊂ W u (p), we respectively denote by Inti X, cli X, ∂i X the interior, closure and boundary of X relative to the intrinsic topology of W u (p), that is the topology induced by the biholomorphism W u (p) ≃ C. Lemma 2.1 ([8, Lemma 5.1]). — Let p be a saddle periodic point. Relative to the intrinsic topology in W u (p), ∂i (W u (p) ∩ K + ) is contained in the closure of the set of transverse homoclinic intersections. In particular ∂i (W u (p) ∩ K + ) ⊂ J ∗ . Here is another statement along the same lines, which can easily be extracted from [2]. Lemma 2.2. — Let ψ : C → C2 be an entire curve such that ψ(C) ⊂ K + . Then for any saddle point p, ψ(C) admits transverse intersections with W u (p). Proof. — This is identical to the first half of the proof of [8, Lemma 5.4].

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We will repeatedly use the following alternative which follows from the combination of the two previous lemmas. Recall that a Fatou disk is a holomorphic disk along which the iterates (f n )n≥0 form a normal family. Lemma 2.3. — Let E be an entire curve contained in K + , p be any saddle point, and t be a transverse intersection point between E and W u (p). Then either t ∈ J ∗ or there is a Fatou disk ∆ ⊂ W u (p) containing t. Proof. — Indeed, either t is in ∂i (W u (p) ∩ K + ) so by Lemma 2.1, t belongs to J ∗ , or t is in Inti (W u (p) ∩ K + ). In the latter case, pick any open disk ∆ ⊂ Inti (W u (p)∩K + ) containing t. Since ∆ is contained in K + , its forward iterates remain bounded so it is a Fatou disk. 2.2. The atomic case. — Here we prove Theorem 1.3 when ν is atomic. By ergodicity, this implies that ν is concentrated on a single periodic orbit. Replacing f by an iterate we may assume that it is concentrated on a fixed point. Since f is dissipative there must be an attracting eigenvalue. A first possibility is that this fixed point is attracting or semi-Siegel. Then we are in case (i) and there is nothing to say. Otherwise p is of saddle, semi-parabolic or semi-Cremer type and we must show that p ∈ J ∗ . The case of saddles was treated in [2, Thm 9.2]. In both remaining cases, p admits a strong stable manifold W ss (p) associated to the contracting eigenvalue, which is biholomorphic to C by a theorem of Poincaré. Let q be a saddle periodic point and t be a point of transverse intersection between W ss (p) and W u (q). If t ∈ J ∗ , then since f n (t) converges to p as n → ∞ we are done. Otherwise there is a non-trivial Fatou disk ∆ transverse to W ss (p) at t. Let us show that this is contradictory. In the semi-parabolic case, this is classical. A short argument goes as follows (compare [18, Prop. 7.2]). Replace f by an iterate so that the neutral eigenvalue is equal to 1. Since f has no curve of fixed points there are local coordinates (x, y) near p in ss which p = (0, 0), Wloc (p) is the y-axis {x = 0} and f takes the form (x, y) 7−→ (x + xk+1 + h.o.t., by + h.o.t.) , with |b| < 1 (see [18, §6]). Then f n is of the form (x, y) 7−→ (x + nxk+1 + h.o.t., bn y + h.o.t.) , from which it follows that f n cannot be normal along any disk transverse to the y axis, so we are done. In the semi-Cremer case we rely on the hedgehog theory of [10, 14]. Let φ : D → ∆ be any parameterization, and fix local coordinates (x, y) as before in which p = (0, 0), ss (p) is the y-axis and f takes the form Wloc (x, y) 7−→ (ei2πθ x, by) + h.o.t. Let B be a small neighborhood of the origin in which the hedgehog H is well-defined. Reducing ∆ and iterating a few times if necessary, we can assume that for all k ≥ 0, f k (∆) ⊂ B and φ is of the form s 7→ (s, φ2 (s)). Then the first coordinate of f n ◦ φ is of the form s 7→ ei2nπθ s + h.o.t. If (nj )j≥0 is a subsequence such that f nj ◦ φ

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converges to some ψ = (ψ1 , ψ2 ), we get that ψ1 (s) = αs + h.o.t., where |α| = 1. Thus ψ(D) = lim f nj (∆) is a non-trivial holomorphic disk Γ through 0 that is smooth at the origin. For every k ∈ Z we have that f k (Γ) = lim f nj +k (∆) ⊂ B. Therefore by the local uniqueness of hedgehogs (see [14, Thm 2.2]), Γ is contained in H . It follows that H has non-empty relative interior in any local center manifold of p and from [14, Cor. D.1] we infer that p is semi-Siegel, which is the desired contradiction.

2.3. The non-atomic case. — Assume now that ν is non-atomic. If ν gives positive mass to the Fatou set, then by ergodicity it must give full mass to a cycle of recurrent Fatou components. Such components were classified in [4, §5]: they are either attracting basins or rotation domains. Since ν is non-atomic we must be in the second situation. Replacing f by f k we may assume that ν is supported in a fixed Fatou component Ω. Then Ω retracts onto some Riemann surface S which is biholomorphic to a disk or an annulus and on which the dynamics is that of an irrational rotation. Furthermore all orbits in Ω converge to S. Thus ν must give full mass to S, and since S is foliated by invariant circles, by ergodicity ν gives full mass to a single circle. Finally the unique ergodicity of irrational rotations implies that ν is the Haar measure. Therefore we are left with the case where Supp(ν) ⊂ J, that is, we must prove Theorem 1.1. Let us start by recalling some facts on the Oseledets-Pesin theory of our mappings. Since ν is ergodic by the Oseledets theorem there exists 1 ≤ k ≤ 2, a set R of Lk full measure and for x ∈ R a measurable splitting of Tx C2 , Tx C2 = i=1 Ei (x) such P that for v ∈ Ei (x), limn→∞ n1 log kdfxn (v)k = χi . Moreover, χi = log |Jac(f )| < 0, and since ν is non-atomic both χi cannot be both negative (this is already part of Pesin’s theory, see [2, Prop. 2.3]). Thus k = 2 and the exponents satisfy χ1 < 0 and χ2 ≥ 0 (up to relabelling). Without loss of generality, we may further assume that points in R satisfy the conclusion of the Birkhoﬀ ergodic theorem for ν. As observed in the introduction, the ergodic closing lemma is well-known when χ2 > 0 so we can consider only the case χ2 = 0 (our proof actually treats both cases simultaneously). To ease notation, let us denote by E s (x) the stable Oseledets subspace and by χs the corresponding Lyapunov exponent (χs < 0). The Pesin stable manifold theorem (see e.g., [9] for details) asserts that there exists a measurable set R ′ ⊂ R of full s measure, and a family of holomorphic disks Wloc (x), tangent to E s (x) at x for x ∈ R ′ , s s and such that f (Wloc (x)) ⊂ Wloc (f (x)). In addition for every ε > 0 there exists a set Rε′ of measure ν(Rε′ ) ≥ 1 − ε and constants rε and Cε such that for x ∈ Rε′ , s Wloc (x) contains a graph of slope at most 1 over a ball of radius rε in E s (x) and for s y ∈ Wloc (x), d(f n (y), f n (x)) ≤ Cε exp((χs + ε)n) for every n ≥ 0. Furthermore, local stable manifolds vary continuously on Rε′ .

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From this we can form global stable manifolds by declaring (2) that W s (x) is the s increasing union of f −n (Wloc (f n (x))). Then it is a well-known fact that W s (x) is a.s. biholomorphically equivalent to C (see e.g., [2, Prop 2.6]). Indeed, almost every point visits Rε′ infinitely many times, and from this we can view W s (x) as an increasing union of disks Dj such that the modulus of the annuli Dj+1 \Dj is uniformly bounded from below. Discarding a set of zero measure if necessary, it is no loss of generality to S assume that ε>0 Rε′ = R ′ and that for every x ∈ R ′ , W s (x) ≃ C. To prove the theorem we show that for every ε > 0, Rε′ ⊂ J ∗ . Fix x ∈ Rε′ and a saddle point p. By Lemma 2.2 there is a transverse intersection t between W s (x) and W u (p). Since x is recurrent and d(f n (x), f n (t)) → 0, to prove that x ∈ J ∗ it is enough to show that t ∈ J ∗ . We argue by contradiction so assume that this is not the case. Then by Lemma 2.3 there is a Fatou disk ∆ through t inside W u (p). Reducing ∆ a little if necessary we may assume that f n is a normal family in some neighborhood of ∆ in W u (p). Since ν is non-atomic and stable manifolds vary continuously for the C 1 topology on Rε′ , there is a set A of positive measure such that if y ∈ A, W s (y) admits a transverse intersection with ∆. The iterates f n (∆) form a normal family and f n (∆) is exponentially close to f n (A). Let (nj ) be some subsequence such that f nj |∆ converges. Then the limit map has either generic rank 0 or 1, that is if φ : D → ∆ is a parameterization, f nj ◦ φ converges uniformly on D to some limit map ψ, which is either constant or has generic rank 1. Set Γ = ψ(D). Let ν ′ be a cluster value of the sequence of measures (f nj )∗ (ν | ). Then ν ′ is a measure of mass ν(A), supported on A Γ and ν ′ ≤ ν. Since ν gives no mass to points, the rank 0 case is excluded so Γ is a (possibly singular) curve. Notice also that if z is an interior point of ∆ (i.e., z = φ(ζ) for some ζ ∈ D), then lim f nj (z) = ψ(ζ) is an interior point of Γ. This shows that ν ′ gives full mass to Γ (i.e., it is not concentrated on its boundary). Then the proof of Theorem 1.1 is concluded by the following result of independent interest. Proposition 2.4. — Let f be a dissipative polynomial diffeomorphism of C2 with nontrivial dynamics, and ν be an ergodic non-atomic invariant measure, giving positive measure to a subvariety. Then ν is the Haar measure on an invariant cycle of circles contained in a periodic rotation domain. In particular a non-atomic invariant measure supported on J gives no mass to subvarieties. Proof. — Let f and ν be as in the statement of the proposition, and Γ0 be a subvariety such that ν(Γ0 ) > 0. Since ν gives no mass to the singular points of Γ0 , by reducing Γ0 a bit we may assume that Γ0 is smooth. If M is an integer such that 1/M < ν(Γ0 ), by the pigeonhole principle there exists 0 ≤ k < l ≤ M such that ν(f k (Γ0 ) ∩ f l (Γ0 )) > 0, so f k (Γ0 ) and f l (Γ0 ) intersect along a relatively open set. Thus replacing f by some (2) If ν has a zero exponent, this may not be the stable manifold of x in the usual sense, that is, there might exists points outside W s (s) whose orbit approach that of x.

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iterate f N (which does not change the Julia set)Swe can assume that Γ0 ∩ f (Γ0 ) is relatively open in Γ0 and f (Γ0 ). Let now Γ = k∈Z f k (Γ0 ). This is an invariant, injectively immersed Riemann surface with ν(Γ) > 0. Notice that replacing f by f N may corrupt the ergodicity of ν so if needed we replace ν by a component of its ergodic decomposition (under f N ) giving positive (hence full) mass to Γ. We claim that Γ is biholomorphic to a domain of the form {z ∈ C, r < |z| < R} for some 0 ≤ r < R ≤ ∞, that f | is conjugate to an irrational rotation, and ν is Γ0 the Haar measure on an invariant circle. This is a priori not enough to conclude the proof since at this stage nothing prevents such an invariant “annulus” to be contained in J. To prove the claim, note first that since Γ is non-compact, it is either biholomorphic to C or C∗ , or it is a hyperbolic Riemann surface (3). In addition Γ possesses an automorphism f with a non-atomic ergodic invariant measure. In the case of C and C∗ all automorphisms are aﬃne and the only possibility is that f is an irrational rotation. In the case of a hyperbolic Riemann surface, the list of possible dynamical systems is also well-known (see e.g., [15, Thm 5.2]) and again the only possibility is that f is conjugate to an irrational rotation on a disk or an annulus. The fact that ν is a Haar measure follows as before. e ⊂ Γ be a relatively compact invariant Let γ be the circle supporting ν, and Γ annulus containing γ in its interior. To conclude the proof we must show that γ is contained in the Fatou set. This will result from the following lemma, which will be proven afterwards. e Lemma 2.5. — The mapping f admits a dominated splitting along Γ. See [17] for generalities on the notion of dominated splitting. In our setting, since Γ is an invariant complex submanifold and f is dissipative, the dominated splitting actually implies a normal hyperbolicity property. Indeed, observe first that f | is Γ an isometry for the Poincaré metric PoinΓ of Γ, which to the induced

is equivalent

e In particular C −1 ≤ Riemannian metric along Γ.

df n |T Γ˜ ≤ C for some C > 0 independent of n. Therefore a dominated splitting for f |e means that there is a Γ e Tx C2 = Tx Γ ⊕ Vx , and for every x ∈ Γ e and continuous splitting of T C2 along Γ,

n ′ n ′ n ≥ 0 we have dfx | ≤ C λ for some C > 0 and λ < 1. In other words, f is Vx e Thus in a neighborhood of γ, all orbits converge to Γ. normally contracting along Γ.

This completes the proof of Proposition 2.4.

Proof of Lemma 2.5. — By the cone criterion for dominated splitting (see [16, Thm 1.2] or [17, Prop. 2.2]) it is enough to prove that for every x ∈ Γ there exists a cone Cx about Tx Γ in Tx C2 such that the field of cones (Cx )x∈Γe is strictly contracted by the dynamics. For x ∈ Γ, choose a vector ex ∈ Tx Γ of unit norm (3)

In the situation of Theorem 1.1 we further know that Γ ⊂ K so the first two cases are excluded.

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relative to the Poincaré metric PoinΓ and pick fx orthogonal to ex in Tx C2 and such that det(ex , fx ) = 1. Since PoinΓ |e is equivalent to the metric induced by Γ e the ambient Riemannian metric, there exists a constant C such that for all x ∈ Γ,

C −1 ≤ kex k ≤ C. Thus, the basis (ex , fx ) diﬀers from an orthonormal basis by bounded multiplicative constants, i.e., there exists C −1 ≤ α(x) ≤ C such that (α(x)ex , α−1 (x)fx ) is orthonormal. Let us work in the frame {(ex , fx ), x ∈ Γ}. Since df | is an isometry for the Poincaré Γ metric and f (Γ) = Γ, the matrix expression of dfx in this frame is of the form ! eiθ(x) a(x) , 0 e−iθ(x) J

where J is the (constant) Jacobian. Fix λ such that |J| < λ < 1, and for ε > 0, let Cxε ⊂ Tx C2 be the cone defined by ε

Cx

= {uex + vfx , |v| ≤ ε |u|} .

Let also A = supx∈Γe |a(x)|. Working in coordinates, if (u, v) ∈ Cxε then dfx (u, v) =: (u1 , v1 ) = (eiθ(x) u + a(x)v, e−iθ(x) Jv), e we get hence for x ∈ Γ |u1 | ≥ |u| − A |v| ≥ |u| (1 − Aε) and |v1 | = |Jv| ≤ ε |J| |u| . e we have that We see that if ε is so small that |J| < λ(1 − Aε), then for every x ∈ Γ λε ε |v1 | ≤ λε |u1 |, that is, dfx (Cx ) ⊂ Cf (x) . The proof is complete. References [1] E. Bedford – “Dynamics of polynomial diﬀeomorphisms in C2 : foliations and laminations”, ICCM Not. 3 (2015), p. 58–63. [2] E. Bedford, M. Lyubich & J. Smillie – “Polynomial diﬀeomorphisms of C2 . IV. The measure of maximal entropy and laminar currents”, Invent. math. 112 (1993), p. 77–125. [3] E. Bedford & J. Smillie – “Polynomial diﬀeomorphisms of C2 : currents, equilibrium measure and hyperbolicity”, Invent. math. 103 (1991), p. 69–99. [4]

, “Polynomial diﬀeomorphisms of C2 . II. Stable manifolds and recurrence”, J. Amer. Math. Soc. 4 (1991), p. 657–679.

[5]

, “Polynomial diﬀeomorphisms of C2 . III. Ergodicity, exponents and entropy of the equilibrium measure”, Math. Ann. 294 (1992), p. 395–420.

[6] S. Crovisier & E. Pujals – “Strongly dissipative surface diﬀeomorphisms”, Comment. Math. Helv. 93 (2018), p. 377–400. [7] R. Dujardin – “Some remarks on the connectivity of Julia sets for 2-dimensional diﬀeomorphisms”, in Complex dynamics, Contemp. Math., vol. 396, Amer. Math. Soc., 2006, p. 63–84. [8] R. Dujardin & M. Lyubich – “Stability and bifurcations for dissipative polynomial automorphisms of C2 ”, Invent. math. 200 (2015), p. 439–511.

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[9] A. Fathi, M.-R. Herman & J.-C. Yoccoz – “A proof of Pesin’s stable manifold theorem”, in Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., vol. 1007, Springer, 1983, p. 177–215. [10] T. Firsova, M. Lyubich, R. Radu & R. Tanase – “Hedgehogs for neutral dissipative germs of holomorphic diﬀeomorphisms of (C, 0)”, Astérisque 416 (2020), p. 193–211. [11] L. Guerini & H. Peters – “Julia sets of complex Hénon maps”, Internat. J. Math. 29 (2018), 1850047. [12] A. Katok – “Lyapunov exponents, entropy and periodic orbits for diﬀeomorphisms”, Inst. Hautes Études Sci. Publ. Math. 51 (1980), p. 137–173. [13] M. Lyubich & H. Peters – “Structure of partially hyperbolic Hénon maps”, preprint arXiv:1712.05823. [14] M. Lyubich, R. Radu & R. Tanase – “Hedgehogs in higher dimension and their applications”, Astérisque 416 (2020), p. 213–251. [15] J. Milnor – Dynamics in one complex variable, third ed., Annals of Math. Studies, vol. 160, Princeton Univ. Press, 2006. [16] S. Newhouse – “Cone-fields, domination, and hyperbolicity”, in Modern dynamical systems and applications, Cambridge Univ. Press, 2004, p. 419–432. [17] M. Sambarino – “A (short) survey on dominated splittings”, in Mathematical Congress of the Americas, Contemp. Math., vol. 656, Amer. Math. Soc., 2016, p. 149–183. [18] T. Ueda – “Local structure of analytic transformations of two complex variables. I”, J. Math. Kyoto Univ. 26 (1986), p. 233–261.

Romain Dujardin, Sorbonne Université, CNRS, Laboratoire de Probabilités, Statistiques et Modélisations (LPSM), F-75005 Paris, France • E-mail : [email protected]

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Astérisque 415, 2020, p. 45–57 doi:10.24033/ast.1099

ON THE MINIMA OF MARKOV AND LAGRANGE DYNAMICAL SPECTRA by Carlos Gustavo T. de A. Moreira

Dedicated to Jean-Christophe Yoccoz

Abstract. — We consider typical Lagrange and Markov dynamical spectra associated to horseshoes on surfaces. We show that for a large set of real functions on the surface, the minima of the corresponding Lagrange and Markov dynamical spectra coincide, are isolated, and are given by the image of a periodic point of the dynamics by the real function. This solves a question by Jean-Christophe Yoccoz. Résumé (Sur les minima des Spectres Dynamiques de Markov et Lagrange). — Nous considérons des spectres dynamiques typiques de Lagrange et Markov associés aux fers à cheval des surfaces. Pour un grand ensemble de fonctions définies sur la surface à valeurs réelles, nous montrons que les minima des spectres dynamiques de Markov et Lagrange coïncident, sont isolés et sont l’image par la fonction d’un point périodique de la dynamique. Cela répond à une question de Jean-Christophe Yoccoz.

1. Introduction The classical Lagrange spectrum (cf. [2]) is defined as follows: Given an irrational number α, according to Dirichlet’s theorem the inequality α − pq < q12 has infinitely many rational solutions pq . Markov and Hurwitz improved this result (cf. [2]), proving 1 that, for all irrational α, the inequality α − pq < √5q has infinitely many rational 2 p solutions q . 2010 Mathematics Subject Classification. — 11J06, 37D20, 28A78. Key words and phrases. — Markov and Lagrange dynamical spectra, regular Cantor sets, fractal dimensions.

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On the other hand, for a fixed irrational α, better results can be expected. We associate, to each α, its best constant of approximation (Lagrange value of α), given by 1 p p k(α) = sup k > 0 : α − < 2 has infinitely many rational solutions q kq q =

−1

lim sup |q(qα − p)|

∈ R ∪ {+∞}.

|p|,q→∞

p∈Z,q∈N

Then, we always have k(α) ≥

√

5. The Lagrange spectrum is the set

L = {k(α) : α ∈ R \ Q and k(α) < ∞}. Let α be an irrational number expressed in continued fractions by α = [a0 , a1 , . . . ]. Define, for each n ∈ N, αn = [an , an+1 , . . . ] and βn = [0, an−1 , an−2 , . . . ]. Using elementary continued fractions techniques it can be proved that k(α) = lim sup(αn + βn ). n→∞

The study of the geometric structure of L is a classical subject, which began with Markov, proving in 1879 ([3]) that √ √ √ 221 L ∩ (−∞, 3) = {k1 = 5 < k2 = 2 2 < k3 = < · · ·} 5 where kn is a sequence (of irrational numbers whose squares are rational) converging to q 3 - more precisely, the elements kn of L ∩ (−∞, 3) are the numbers the form 9−

4 z2 ,

where z is a positive integer such that there are other positive integers x, y

with x ≤ y ≤ z and x2 + y 2 + z 2 = 3xyz. Another interesting set is the classical Markov spectrum defined by (cf. [2]) M= inf2 |f (x, y)|−1 : f (x, y) = ax2 + bxy + cy 2 with b2 − 4ac = 1 . (x,y)∈Z \(0,0)

It is possible to prove (cf. [2]) that L and M are closed subsets of the real line with L ⊂ M and that L ∩ (−∞, 3) = M ∩ (−∞, 3). Both the Lagrange and Markov spectrum have a dynamical interpretation. This fact is an important motivation for our work. Let Σ = (N∗ )Z and σ : Σ → Σ the shift defined by σ((an )n∈Z ) = (an+1 )n∈Z . If f : Σ → R is defined by f ((an )n∈Z ) = α0 + β0 = [a0 , a1 , . . . ] + [0, a−1 , a−2 , . . . ], then L = lim sup f (σ n (θ)) : θ ∈ Σ n→∞

and M= √

sup f (σ n (θ)) : θ ∈ Σ . n∈Z

Notice that 5, which is the common minimum of L and M , is the image by f of the fixed point (. . . , 1, 1, 1, . . . ) of the shift map σ.

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This last interpretation, in terms of a shift, admits a natural generalization of Lagrange and Markov spectrum in the context of hyperbolic dynamics (at least in dimension 2, which is the focus of this work). We will define, as in [4], the Markov and Lagrange dynamical spectra associated to a hyperbolic set as follows. Let M 2 be a surface and ϕ : M 2 → M 2 be a diﬀeomorphism with Λ ⊂ M 2 a hyperbolic set for ϕ (which means that ϕ(Λ) = Λ and there is a decomposition TΛ M 2 = E s ⊕ E u such that Dϕ|E s is uniformly contracting and Dϕ |E u is uniformly expanding). In this paper we will asume that Λ is a horseshoe: it is a compact, locally maximal, hyperbolic invariant set of saddle type (and thus Λ is not an attractor nor a repellor, and is topologically a Cantor set). Let f : M 2 → R be a continuous real function, then the Lagrange Dynamical Spectrum associated to (f, Λ) is defined by L(f, Λ) = lim sup f (ϕn (x)) : x ∈ Λ , n→∞

and the Markov Dynamical Spectrum associate to (f, Λ) is defined by M (f, Λ) = sup f (ϕn (x)) : x ∈ Λ . n∈Z

Here we prove the following theorem, which solves a question posed by JeanChristophe Yoccoz to the author in 1998: Main Theorem. — Let Λ be a horseshoe associated to a C 2 -diffeomorphism ϕ. Then there is a dense set H ⊂ C ∞ (M, R), which is is C 0 -open, such that for all f ∈ H, we have min L(f, Λ) = min M (f, Λ) = f (p), where p = p(f ) ∈ Λ is a periodic point of ϕ. Moreover, f (p) is an isolated point both in L(f, Λ) and in M (f, Λ). Remark. — In the previous statement, a horseshoe means a compact, locally maximal (which means that it is the maximal invariant set in some neighborhood of it), transitive hyperbolic invariant set of saddle type (and so it contains a dense subset of periodic orbits). Acknowledgements. — I would like to thank Carlos Matheus, Davi Lima, Sergio Romaña and Sandoel Vieira for helpful discussions on the subject of this paper. I would also like to thank the anonymous referee for his very valuable comments and suggestions, which helped to substantially improve this work. 2. Preliminaries from dynamical systems If Λ is a hyperbolic set associated to a C 2 -diﬀeomorphism, then the stable and unstable foliations F s (Λ) and F u (Λ) are C 1+ε for some ε > 0. Moreover, these foliations can be extended to C 1+ε foliations defined on a full neighborhood of Λ(cf. the comments at the end of Section 4.1 of [6], pp. 60).

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We will consider the following setting. Let Λ be a horseshoe of ϕ. Let us fix a geometrical Markov partition {Ra }a∈A with suﬃciently small diameter consisting of rectangles Ra ≃ Ias × Iau delimited by compact pieces Ias , resp. Iau , of stable, resp. unstable, manifolds of certain points of Λ (cf. [7, pp 129] or [6] for more details). The set B ⊂ A2 of admissible transitions consist of pairs (a0 , a1 ) such that ϕ(Ra0 ) ∩ Ra1 6= ∅. So, we can define the following transition matrix B which induces the same transitions than B ⊂ A2 bai aj = 1 if ϕ(Rai ) ∩ Raj 6= ∅,

bai aj = 0 otherwise, for (ai , aj ) ∈ A2 .

Let ΣA = {a = (an )n∈Z : an ∈ A for all n ∈ Z}. We can define the homeomorphism of ΣA , the shift, σ : ΣA → ΣA defined by σ((a n )n∈Z ) = (an+1 )n∈Z . Let ΣB = a ∈ ΣA : ban an+1 = 1, ∀n ∈ Z , this set is a closed and σ-invariant subspace of ΣA . Still denote by σ the restriction of σ to ΣB . The pair (ΣB , σ) is called a subshift of finite type of (ΣA , σ). Given x, y ∈ A, we denote by Nn (x, y, B) the number of admissible strings for B of length n + 1, beginning at x and ending with y. Then the following holds Nn (x, y, B) = bnxy . In particular, since ϕ| is transitive, there is N0 ∈ N∗ such that for all x, y ∈ A, Λ NN0 (x, y, B) > 0. Subshifts of finite type also have a sort of local product structure. First we define the local stable and unstable sets: (cf. [7, chap 10]) s W1/3 (a) = {b ∈ ΣB : ∀n ≥ 0, d(σ n (a), σ n (b)) ≤ 1/3}

= {b ∈ ΣB : ∀n ≥ 0, an = bn } , u W1/3 (a)

= {b ∈ ΣB : ∀n ≤ 0, d(σ n (a), σ n (b)) ≤ 1/3} = {b ∈ ΣB : ∀n ≤ 0, an = bn } ,

P∞

−(2|n|+1) where d(a, b) = δn (a, b) and δn (a, b) is 0 when an = bn and 1 n=−∞ 2 otherwise. u u So, if a, b ∈ ΣB and d(a, b) < 1/2, then a0 = b0 and W1/3 (a) ∩ W1/3 (b) is a unique point, denoted by the bracket [a, b] = (. . . , b−n , . . . , b−1 , b0 , a1 , . . . , an , . . .). If ϕ is a diﬀeomorphism of a surface (2-manifold), then the dynamics of ϕ on Λ is topologically conjugate to a subshift ΣB defined by B, namely, there is a homeomorphism Π : ΣB → Λ such that, the following diagram commutes

ΣB

σ

// ΣB

ϕ

// Λ

Π

Λ

Π

i.e.,

ϕ ◦ Π = Π ◦ σ.

Moreover, Π is a morphism of the local product structure, that is, Π[a, b] = [Π(a), Π(b)], (cf. [7, chap 10]). If p = Π(θ) ∈ Λ, we say that θ is the kneading sequence of p.

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Next, we recall that the stable and unstable manifolds of Λ can be extended to locally invariant C 1+ε -foliations in a neighborhood of Λ for some ε > 0. Therefore, we can use these foliations to define projections πau : Ra → Ias × {iua } and πas : Ra → {isa } × Iau of the rectangles into the connected components Ias × {iua } and {isa } × Iau of the stable and unstable boundaries of Ra where iua ∈ ∂Iau and isa ∈ ∂Ias are fixed arbitrarily. Using these projections, we have the stable and unstable Cantor sets [ [ Ks = πau (Λ ∩ Ra ) and K u = πas (Λ ∩ Ra ) a∈A

a∈A

associated to Λ. The stable and unstable Cantor sets K s and K u are C 1+ε -dynamically defined / 1+ε C -regular Cantor sets, i.e., the C 1+ε -maps gs (πau1 (y)) = πau0 (ϕ−1 (y)) for y ∈ Ra1 ∩ ϕ(Ra0 ) and gu (πas0 (z)) = πas1 (ϕ(z)) for z ∈ Ra0 ∩ ϕ−1 (Ra1 ) are expanding of type ΣB defining K s and K u in the sense that F — the domains of gs and gu are disjoint unions I s (a1 , a0 ) and (a0 ,a1 )∈B F I u (a0 , a1 ) where I s (a1 , a0 ), resp. I u (a0 , a1 ), are compact subinter(a0 ,a1 )∈B

vals of Ias1 , resp. Iau0 ; — for each (a0 , a1 ) ∈ B, the restrictions gs | s and gu | u are C 1+ε difI (a0 ,a1 ) I (a0 ,a1 ) feomorphisms onto Ias0 and Iau0 with |Dgs (t)| > 1, resp. |Dgu (t)| > 1, for all t ∈ I s (a0 , a1 ), resp. I u (a0 , a1 ) (for appropriate choices of the parametrization of Ias and Iau ); — K s , resp. K u , are the maximal invariant sets associated to gs , resp. gu , that is, \ [ \ [ Ks = gs−n I s (a1 , a0 ) and K u = gu−n I u (a0 , a1 ) n∈N

(a0 ,a1 )∈B

n∈N

(a0 ,a1 )∈B

(see Section 1 of Chapter 4 of [6] and Chapter 1 of [5] for more informations on regular Cantor sets associated to horseshoes in surfaces). Moreover, we will think the intervals Iau , resp. Ias , a ∈ A inside an abstract line so that it makes sense to say that the interval Iau , resp. Ias , is located to the left or to the right of the interval Ibu , resp. Ibs , for a, b ∈ A. The stable and unstable Cantor sets K s and K u are closely related to the geometry of the horseshoe Λ: for instance, the horseshoe Λ is locally diﬀeomorphic to the Cartesian product of the two regular Cantor sets K s and K u (since the stable and unstable foliations of Λ are of class C 1 ). Moreover, the Hausdorﬀ dimension of any regular Cantor set coincides with its box dimension. It follows that HD(Λ) = HD(K s ) + HD(K u ) =: ds + du ,

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where HD stands for the Hausdorﬀ dimension (cf. Proposition 4 in [6, chap. 4] and the comments before it). In the paper [1], we were interested in the fractal geometry (Hausdorﬀ dimension) of the sets M (f, Λ) ∩ (−∞, t) and L(f, Λ) ∩ (−∞, t) as t ∈ R varies. For this reason, we will also study the fractal geometry of \ Λt := ϕ−n ({y ∈ Λ : f (y) ≤ t}) = {x ∈ Λ : mϕ,f (x) = sup f (ϕn (x)) ≤ t} n∈Z

n∈Z

for t ∈ R.

3. Proofs We may study the subsets Λt introduced above through its projections [ [ πau (Λt ∩ Ra ) and Ktu = πas (Λt ∩ Ra ) Kts = a∈A

a∈A

on the stable and unstable Cantor sets of Λ. It follows from the proof of Theorem 1.2 of [1] (see Remarks 1.3, 1.4 and 2.10 of [1]) that (even in the non-conservative case) the box dimensions Ds (t) of Kts and Du (t) of Ktu depend continuously on t. In particular, if t0 = min M (f, Λ), then Ds (t0 ) = Du (t0 ) = 0, since, for any t < t0 , the sets Λt , Kts and Ktu are empty, so Ds (t) = Du (t) = 0. Since the stable and unstable foliations of Λ are of class C 1 , the box dimension of Λt0 is at most the sum of the box dimensions of Kts0 and Ktu0 , and so is equal to 0. It follows that, for any ǫ > 0, there is a locally maximal subhorseshoe ˜ and HD(Λ) ˜ < ǫ (we may fix a large positive ˜ ⊂ Λ with Λt ⊂ Λ (of finite type) Λ 0 ˜ as the set of points of Λ in whose kneading sequences all factors integer m and take Λ of size m are factors of the kneading sequence of some element of Λt0 —the number of such factors grow subexponentially in m since the box dimension of Λt0 is 0, so ˜ is small when m is large by the estimates on fractal the Hausdorﬀ dimension of Λ dimensions of [6, chap. 4]). ˜ < 1/2k, then, for any Proposition 1. — Let k > 1 be an integer. If HD(Λ) r r ∈ N ∪ {∞}, there is a dense set (in the C topology) of C r real functions f such ˜ (and in particular that, for some c > 0, |f (p) − f (q)| ≥ c · |p − q|k/(k−1) , ∀p, q ∈ Λ f | ˜ is injective; moreover, its inverse function is (1 − 1/k)-Hölder). Λ Proof. — Given a smooth function f , there are, as in Proposition 2.7 of [1], arbitrarily small perturbations of it whose derivative does not vanish at the stable and unstable ˜ so we will assume that f satisfies this property. directions in points of Λ, Given a Markov partition {Ra }a∈A with suﬃciently small diameter as before, we may perturb f by adding, for each a, independently, a small constant ta to f in a ˜ ∩ Ra (notice that the compact sets (Λ ˜ ∩ Ra ) are mutually small neighborhood of Λ

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˜ ∩ Ra ) and f (Λ ˜ ∩ Rb ) have box dimensions disjoint). Since, for a 6= b, the images f (Λ smaller than 1/2k, their arithmetic diﬀerence ˜ ∩ Ra ) − f (Λ ˜ ∩ Rb ) = {x − y, x ∈ f (Λ ˜ ∩ Ra ), y ∈ f (Λ ˜ ∩ Rb )} f (Λ ˜ ∩ Ra ) ∩ (f (Λ ˜ ∩ Rb ) + t) 6= ∅ = {t ∈ R|f (Λ has box dimension smaller than 1/k < 1, and so, for almost all ta , tb , the perturbed ˜ ∩ Ra ) + ta and f (Λ ˜ ∩ Rb ) + tb are disjoint. images f (Λ Consider now parametrizations of small neighborhoods of the pieces Ra according ˜ are C 1 close to be horizontal and the unstable leaves to which the stable leaves of Λ 1 ˜ are C close to be vertical, and such that, in the coordinates given by these of Λ paramerizations, f (x, y) is C 1 close to an aﬃne map f (x, y) = ax + by + c, with a and b far from 0 (these parametrizations exist since the pieces Ra are chosen very small, so f is close to be aﬃne in Ra ). Then we may consider, in each coordinate system as above, perturbations of f of the type fλ (x, y) = f (x, λy), where λ is a parameter close to 1. ˜ ∩ Ra is a decomposition of it in a union Given ρ > 0 small, a ρ-decomposition of Λ s u ˜ of rectangles Ij ×Ij intersected with Λ such that both intervals Ijs and Iju have length of the order of ρ. Let r be a large positive integer, and consider 2−kr and 2−(k−1)r -de˜ ∩ Ra . Given two rectangles of the 2−kr -decomposition which belong compositions of Λ to diﬀerent rectangles of the 2−(k−1)r -decomposition (and so have distance at least of the order of 2−(k−1)r ), the measure of the interval of values of λ such that the images by fλ of the two rectangles of the 2−kr -decomposition have distance smaller than 2−kr is at most of the order of 2−kr /2−(k−1)r = 2−r . Since the box dimension ˜ is d < 1/2k, the number of pairs of rectangles in the 2−kr -decomposition is of the of Λ order of (2−kr )−2d = 22dkr , and so the the measure of the set of values of λ such that the images of some pair as before of two rectangles of the 2−kr -decomposition have non-empty intersection is at most of the order of 22dkr · 2−r = 2−(1−2dk)r ≪ 1 (notice that 2dk < 1). The sum of these measures for all r ≥ r0 is O(2−(1−2dk)r0 ) ≪ 1, and so ˜ ∩ Ra are such that |p − q| ≥ ǫ there is λ close to 1 such that, if ǫ is small and p, q ∈ Λ k/(k−1) then |f (p) − f (q)| is at least of the order of ǫ (consider r in the above discussion such that 2−(k−1)r is of the order of ǫ). This implies the result: for some c > 0, ˜ It follows that f | is injective and its inverse |f (p)−f (q)| ≥ c·|p−q|k/(k−1) , ∀p, q ∈ Λ. ˜ Λ function g is (1 − 1/k)-Hölder: indeed, it satisfies |g(x) − g(y)| ≤ (c−1 |x − y|)(1−1/k) , ˜ for any x, y ∈ f (Λ). Let ϕ : M → M be a diﬀeomorphism of a compact 2-manifold M and let Λ be a horseshoe for ϕ. We recall the following remark from [4]: Remark. — We have L(f, Λ) ⊂ M (f, Λ) for any f ∈ C 0 (M, R). In fact: Let a ∈ L(f, Λ), then there is x0 ∈ Λ such that a = lim sup f (ϕn (x0 )). Since n→+∞

Λ is a compact set, then there is a subsequence (ϕnk (x0 )) of (ϕn (x0 )) such that

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lim ϕnk (x0 ) = y0 and

k→+∞

a = lim sup f (ϕn (x0 )) = lim f (ϕnk (x0 )) = f (y0 ). n→+∞

k→+∞

Claim. — We have f (y0 ) ≥ f (ϕn (y0 )) for all n ∈ Z. Otherwise, suppose there is n0 ∈ Z such that f (y0 ) < f (ϕn0 (y0 )). Put ǫ = f (ϕn0 (y0 )) − f (y0 ), then, since f is a continuous function, there is a neighborhood U of y0 such that ǫ f (y0 ) + < f (ϕn0 (z)) for all z ∈ U. 2 Thus, since ϕnk (x0 ) → y0 , then there is k0 ∈ N such that ϕnk (x0 ) ∈ U for k ≥ k0 , therefore, ǫ f (y0 ) + < f (ϕn0 +nk (x0 )) for all k ≥ k0 . 2 This contradicts the definition of a = f (y0 ). Proposition 2. — Assume that f | ˜ is injective. Then min L(f, Λ) = min M (f, Λ) = Λ ˜ such that the restriction of ϕ to the closure of the f (p) for only one value of p ∈ Λ orbit of p is minimal. Proof. — Let p ∈ Λ be such that f (p) = min M (f, Λ), which is unique since f | ˜ is Λ injective. We have f (ϕj (p)) < f (p) for all integer j such that ϕj (p) 6= p. If some subsequence ϕnk (p) converges to a point q such that p does not belong to the closure of the orbit of q, f (p) does not belong to the image by f of the closure of the orbit of q and so the Markov value of the orbit of q is strictly smaller than f (p), a contradiction. This implies that the restriction of ϕ to the closure of the orbit of p is minimal and, in particular, f (p) is the Lagrange value of its orbit, so we also have f (p) = min L(f, Λ). Let Λ be a horseshoe associated to a C 2 -diﬀeomorphism ϕ. We define X ⊂ C 0 (M, R) as the set of real functions f for which min L(f, Λ) = min M (f, Λ) = f (p), where p = p(f ) ∈ Λ is a periodic point of ϕ and there is ε > 0 such that, for every q ∈ Λ which does not belong to the orbit of p, supn∈Z f (ϕn (q)) > f (p) + ε. Proposition 3. — The set X is open in C 0 (M, R). Proof. — Suppose that f ∈ X. If ε > 0 is as in the definition of X, let g ∈ C 0 (M, R) such that |g(x) − f (x)| < ε/3, ∀x ∈ M . Then we have supn∈Z g(ϕn (p)) = g(˜ p) < f (˜ p) + ε/3 ≤ f (p) + ε/3, for some point p˜ in the (finite) orbit of p. Moreover, for every q ∈ Λ which does not belong to the orbit of p, since g(ϕn (q)) > f (ϕn (q))−ε/3, we have supn∈Z g(ϕn (q)) ≥ supn∈Z f (ϕn (q)) − ε/3 > f (p) + ε − ε/3 = f (p) + 2ε/3 > g(˜ p) + ε/3. So we have min L(g, Λ) = min M (g, Λ) = g(˜ p) and g ∈ X.

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Proof of the Main Theorem. — Let t0 = min M (f, Λ). Fix a large positive inte˜ ⊂ Λ with Λt ⊂ Λ ˜ and ger K and consider a locally maximal subhorseshoe Λ 0 ˜ HD(Λ) < 1/2K. We take symbolic representations of points of Λ associated to a Markov partition of Λ. We will assume that f satisfies the conclusions of Proposition 1 (replacing k by K). We will prove that, under these conditions, we have f ∈ X, which will conclude the proof, since, if f ∈ X, then clearly min M (f, Λ) = min L(f, Λ) is isolated in M (f, Λ), and thus also in L(f, Λ) (and X is C 0 -open, by Proposition 3). ˜ is injective, by Proposition 2 there is a unique Since the restriction of f to Λ p ∈ Λ such that f (p) = t0 , and the restriction of ϕ to the closure of the orbit of p is minimal. Let θ = (. . . , a−2 , a−1 , a0 , a1 , a2 , . . . ) be the kneading sequence of p. Assume by contradiction that p is not a periodic point. We will consider the following regular Cantor sets, which we may assume, using s ˜ the set of parametrizations, to be contained in the real line: K s = Wloc (p) ∩ Λ, ˜ points of Λ whose kneading sequences are of the type (. . . , b−2 , b−1 , a0 , a1 , a2 , . . . ), for some b−1 , b−2 , . . . (the point corresponding to this sequence will be denoted u ˜ the set of points of Λ ˜ whose kneadby π s (b−1 , b−2 , . . . )) and K u = Wloc (p) ∩ Λ, ing sequences are of the type (. . . , a−2 , a−1 , a0 , b1 , b2 , . . . ), for some b1 , b2 , . . . (the point corresponding to this sequence will be denoted by π u (b1 , b2 , . . . )). Given a finite sequence (c−1 , c−2 , . . . , c−r ) such that (c−r , . . . , c−2 , c−1 , a0 ) is admissible, we define the interval I s (c−1 , c−2 , . . . , c−r ) to be the convex hull of {(. . . , b−2 , b−1 , a0 , a1 , a2 , . . . ) ∈ K s |b−j = c−j , 1 ≤ j ≤ r}. Analogously, given a finite sequence (d1 , d2 , . . . , ds ) such that (a0 , d1 , d2 , . . . , ds ) is admissible, we define the interval I u (d1 , d2 , . . . , ds ) to be the convex hull of {(. . . , a−2 , a−1 , a0 , b1 , b2 , . . . ) ∈ K u |bj = dj , 1 ≤ j ≤ s}. If a and b are the values of the derivative of f at p applied to the unit tangent s u vectors of Wloc (p) and Wloc (p), respectively, we have that a and b are non-zero and, s u considering local isometric parametrizations of Wloc (p) and Wloc (p) which send p s s u to 0, we have that, locally, for x ∈ Wloc (p) ⊃ K and y ∈ Wloc (p) ⊃ K u , f (x, y) = ax + by + O(x2 + y 2 ), in coordinates given by extended local stable and unstable ˜ (here, the point p has coordinates (x, y) = (0, 0); we will foliations of the horseshoe Λ use this local form in small neighborhoods of p, i.e, for |x| and y small). Let k 6= 0 such that ak = a0 . We define dk = max{d(π s (ak−1 , ak−2 , . . . ), p), d(π u (ak+1 , ak+2 , . . . ), p)}. There are 0 < λ1 < λ2 < 1 such that the norm of the derivative of ϕ restricted to a stable direction and the inverse of the norm of the derivative of ϕ restricted to a unstable direction always belong to (λ1 , λ2 ). We say that k > 0 is a weak record if dk < dj for all j with 1 ≤ j < k such that aj = a0 . We will construct the sequence 0 < k1 < k2 < . . . of the records. Let k1 be the smallest k > 0 with ak = a0 and, given a record kn , kn+1 will be the smallest weak record k > kn with dk < min{|a/b|, |b/a|} · λ31 · dkn . We say that k > 0 such that ak = a0 is left-good if

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f (π s (ak−1 , ak−2 , . . . )) < f (p) and that k is right-good if f (π u (ak+1 , ak+2 , . . . )) < f (p). We say that k is left-happy if k is left-good but not right-good or if k is left-good and right-good and d(π s (ak−1 , ak−2 , . . . ), p) ≥ d(π u (ak+1 , ak+2 , . . . ), p). We say that k is right-happy if k is right-good but not left-good or if k is left-good and rightgood and d(π s (ak−1 , ak−2 , . . . ), p) < d(π u (ak+1 , ak+2 , . . . ), p). Notice that k cannot be simultaneously left-happy and right-happy. We say that an index k > 0 is cool if it is left-happy or right-happy and ak+j = aj for every j with |j| ≤ K. By the minimality, we have lim dkn = 0, and, since f (ϕj (p)) < f (p) for all j 6= 0, for all large values of n, kn is cool. If k is a cool index then we define its basic cell as follows: if k is lefthappy, we take rk to be the positive integer r such that ak−j = a−j for 0 ≤ j < r and ak−r 6= a−r , and sk to be the smallest positive integer s such that |I u (ak+1 , ak+2 , . . . , ak+s )| ≤ |I s (ak−1 , ak−2 , . . . , ak−rj )|. Analogously, if k is right-happy, we take sk to be the positive integer s such that ak+j = aj for 0 ≤ j < s and ak+s 6= as , and rk to be the smallest positive integer r such that |I s (ak−1 , ak−2 , . . . , ak−r )| ≤ |I u (ak+1 , ak+2 , . . . , ak+sk )|. The basic cell of k is the finite sequence (ak−rk , . . . , ak−1 , ak , ak+1 , . . . , ak+sk ) indexed by the interval [−rk , sk ] of integers (so that the index 0 in this interval corresponds to ak ). Notice that rkn+1 > rkn and skn+1 > skn for every n large. We define the extended cell of k as the finite sequence (ak−˜rk , . . . , ak−1 , ak , ak+1 , . . . , ak+˜sk ) indexed by the interval [−˜ rk , s˜k ] of integers, where r˜k := ⌊(1 + 2ˆ c/K)rk ⌋, s˜k := ⌊(1 + 2ˆ c/K)sk ⌋ and cˆ = log λ1 / log λ2 > 1. A crucial remark is that, since f satisfies the conclusions of Proposition 1 and f (ϕj (p)) < f (p) for all j 6= 0, • There is a positive integer r0 such that for every m > 0 which is not cool (in par˜ whose kneading sequence (. . . , b−1 , b0 , b1 , . . . ) ticular if am 6= a0 ), and any point q ∈ Λ satisfies bj = am+j for −r0 ≤ j ≤ r0 , we have f (q) < f (p). Indeed, there is a constant r˜ such that, if m > 0 is not cool, then (am−˜r , . . . , am−1 , am , am+1 , . . . , am+˜r ) 6= (a−˜r , . . . , a−1 , a0 , a1 , . . . , ar˜). We have f (ϕm (p)) < f (p), and, if r0 is much larger ˜ whose kneading sequence (. . . , b−1 , b0 , b1 , . . . ) satisfies than r˜, if q is a point in Λ bj = am+j for −r0 ≤ j ≤ r0 , we have f (q) much closer to f (ϕm (p)) than to f (p), and ˜ so f (q) < f (p) (recall that f is injective in Λ). • For any cool index k > 0, if (ak−˜rk , . . . , ak−1 , ak , ak+1 , . . . , ak+˜sk ) is the extended ˜ is any point whose kneading sequence (. . . , b−1 , b0 , b1 , . . . ) cell of k, and q ∈ Λ satisfies bj = ak+j for −˜ rk ≤ j ≤ s˜k , we have f (q) < f (p). Indeed, since K/(K−1) ˜ by definition of the extended cells, |f (p) − f (q)| ≥ c · |p − q| , ∀p, q ∈ Λ, f (p) does not belong tho the convex hull of the image by f of the image by Π of the cylinder {(. . . , b−1 , b0 , b1 , . . . )|bj = ak+j , −˜ rk ≤ j ≤ s˜k }, which contains the point f (ϕk (p)) < f (p). We now show the following Claim. — There is a positive integer m such that we never have (a−mt , . . . , a−1 , a0 , a1 , . . . , amt−1 ) = γ 2m = γγ · · · γ

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(2m times), for any finite sequence γ, where t = |γ|. Indeed, take a positive integer k0 such that λk20 < λ1 , a large positive integer m0 (with m0 ≥ max{r0 , K}) and m = m0 k0 . Suppose by contradiction that (a−mt , . . . , a−1 , a0 , a1 , . . . , amt−1 ) = γ 2m for some γ = (c1 , c2 , . . . , ct ). We may assume that γ is not of the form αn for a smaller sequence α (otherwise we may replace γ by α). The Markov value t˜0 of Π(γ) = Π(. . . γγγ . . . ), which is larger than t0 , is attained at c1 . Indeed, if 2 ≤ j ≤ t, either j − 1 is not cool or the extended cell of j − 1 (centered in cj ) is contained in γ 3 , so f (ϕj−1 (Π(γ))) < f (p) = t0 . Take the maximum values of m1 , m2 ≥ m for which (a−m1 t , . . . , a−1 , a0 , a1 , . . . , am2 t−1 ) = γ m1 +m2 . Since f (ϕ−(m1 −m0 )t (p)), f (ϕ−(m1 −m0 −1)t (p)), f (ϕ(m2 −m0 −1)t (p)) and f (ϕ(m2 −m0 )t (p)) are smaller than f (p) = t0 , it follows that, for every j ≥ 1, f (π u (γ j am2 t , am2 t+1 , am2 t+2 , . . . )) < f (π u (γ j+1 am2 t , am2 t+1 , am2 t+2 , . . . )) (the suﬃx (am2 t , am2 t+1 , am2 t+2 , . . . ) helps diminishing the value of f ) and f (π s ((γ t )j a−m1 t−1 , a−m1 t−2 , . . . )) < f (π s ((γ t )j+1 a−m1 t−1 , a−m1 t−2 , . . . )), where γ t = (ct , . . . , c2 , c1 ) (the prefix (a−m1 t−1 , a−m1 t−2 , . . . ) also helps diminishing the value of f ). Thus, by comparison, and using the previous remark on extended cells, if we delete from each factor of (. . . , a−1 , a0 , a1 , . . . ) equal to (a−(m1 +1)t , . . . , a−1 , a0 , a1 , . . . , a(m2 +1)t−1 ) the factor (a0 , a1 , . . . , at−1 ) = γ, we will reduce the Markov value of the sequence, a contradiction (notice that the number of consecutive copies of γ in a factor of (. . . , a−1 , a0 , a1 , . . . ) is bounded, since t˜0 > t0 ). This concludes the proof of the Claim. Notice that we can assume that K is much larger than m2 (K > 5 cˆm2 is enough ˜ if necessary. for our purposes), by reducing Λ, In order to conclude the proof we will have two cases: (i) There are arbitrarily large values of n for which kn is left-happy. ˜ In this case we will show that, for such a large value of n, the periodic point q of Λ whose kneading sequence (. . . , b−1 , b0 , b1 , . . . ) has period (a0 , a1 , . . . , akn −1 ) (and so ˜ is locally maximal and p ∈ Λ, ˜ satisfies bm = am (mod kn ) , for all m; notice that, since Λ ˜ we have q ∈ Λ for n large) has Markov value smaller than t0 , a contradiction. In order to do this, notice that if rkn > 2m2 kn , then we get a contradiction by the previous Claim. If there is 0 < k < kn which is cool and satisfies rk > 2m2 k or sk > 2m2 (kn −k), we also get a contradiction by the previous Claim. Otherwise, except perhaps for the terms whose indices are multiple of kn , any term equal to a0 in the periodic sequence with period (a0 , a1 , . . . , akn −1 ) has a neighborhood in this periodic sequence which coincides with the extended cell of a corresponding element of the original sequence (. . . , a−1 , a0 , a1 , . . . ), and so the Markov value of the point corresponding to this periodic sequence is smaller than t0 . For the terms whose indices are multiple of kn (which correspond to the point q), if sˆ is the positive integer such that akn +j = aj for 0 ≤ j < sˆ and akn +ˆs 6= asˆ, then bj = aj for all 0 ≤ j < kn + sˆ, and so d(π u (b1 , b2 , . . . ), p) = o(d(π u (akn +1 , akn +2 , . . . ), p)) = o(d(π s (akn −1 , akn −2 , . . . ), p)), which implies f (q) < f (p) = t0 (since f (x, y) = ax + by + O(x2 + y 2 )).

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(ii) For all n large, kn is right-happy. ˜ whose kneading In this case we will show that, for n large, the periodic point q ′ of Λ ′ ′ ′ sequence (. . . , b−1 , b0 , b1 , . . . ) has period (akn , akn +1 , . . . , akn+1 −1 ) (and so satisfies ˜ is locally maximal and bm = am (mod kn+1 −kn )+kn , for all m; notice that, since Λ ′ ˜ we have q ∈ Λ ˜ for n large) has Markov value smaller than t0 , a contradiction. In p ∈ Λ, order to do this, notice that if rk > 2m2 (k − kn ) for some k which is cool and satisfies kn < k ≤ kn+1 or sk > 2m2 (kn+1 − k) for some k which is cool and satisfies and kn ≤ k < kn+1 , then we get a contradiction by the previous Claim. Otherwise, except perhaps for the terms whose indices are multiple of kn+1 − kn , any term equal to a0 in the periodic sequence with period (akn , akn +1 , . . . , akn+1 −1 ) has a neighborhood in this periodic sequence which coincides with the extended cell of a corresponding element of the original sequence (. . . , a−1 , a0 , a1 , . . . ), and so the Markov value of the point corresponding to this periodic sequence is smaller than t0 . For the terms whose indices are multiple of kn+1 − kn (which correspond to the point q ′ ), if rˆ is the positive integer such that akn+1 −j = a−j for 0 ≤ j < rˆ and akn+1 −ˆr 6= a−ˆr , then bj = akn+1 +j for all − min{ˆ r, kn+1 − kn + rkn } < j ≤ 0 and bj = akn +j for 0 ≤ j ≤ s˜kn (notice that bj = aj for 0 ≤ j < skn , but bskn 6= askn ), and so, by the definition of record, d(π s (akn+1 −1 , akn+1 −2 , . . . ), p) ≤ (1 + o(1))|b/a|λ31 d(π u (akn +1 , akn +2 , . . . ), p) and thus d(π s (b−1 , b−2 , . . . ), p) ≤ (1 + o(1))|b/a|λ1 d(π u (b1 , b2 , . . . ), p), which implies f (q) < f (p) = t0 (since f (x, y) = ax + by + O(x2 + y 2 )). Now we concluded that p and θ = (. . . , a−2 , a−1 , a0 , a1 , a2 , . . . ) are periodic. Let α = (a0 , a1 , . . . , as−1 ) be a minimal period of θ. If f ∈ / X then, for each positive integer n, there is a point qn ∈ Λ which does not belong to the orbit of p such that supr∈Z f (ϕr (qn )) ≤ f (p)+1/n. Given a kneading sequence (. . . , c−2 , c−1 , c0 , c1 , c2 , . . . ) of some point of Λ, we say that k ∈ Z is a regular position of it if there is an integer j such that ck+i = aj+i (mod s) for 1 − s ≤ i ≤ 1, and that k is a strange position otherwise. Since qn does not belong to the orbit of p, there is kn ∈ Z which is a strange position of the kneading sequence of qn . Let q˜n := ϕkn (qn ). Then 0 is a strange position of its kneading sequence. Take a subsequence of (˜ qn ) converging to a point q˜ ∈ Λ. Then 0 is a strange position of the kneading sequence of q˜ (and thus q˜ does not belong to the orbit of p) and supr∈Z f (ϕr (˜ q )) ≤ f (p). This implies (since f is ˜ injective in Λ and f (p) is the smallest element of L(f, Λ)) that f (ϕr (˜ q )) < f (p), ∀r ∈ Z and lim supr→+∞ f (ϕr (˜ q )) = lim supr→−∞ f (ϕr (˜ q )) = f (p). Let θ˜ = (. . . , b−2 , b−1 , b0 , b1 , b2 , . . . ) be the kneading sequence of q˜, and let m and m0 be as in the Claim. We should have factors of θ˜ with (arbitrarily large) positive indices equal to α2m , and, analogously, we should have factors of θ˜ with negative indices equal to α2m . This implies that there is a factor of θ˜ of the form α2m βα2m , where β is a finite sequence which does not belong or end by α (and thus is not of the form αr for any positive integer r), and which me may assume not to contain any factor of the form α2m (otherwise we find a smaller factor with these properties). We claim that if z ∈ Λ is a point whose kneading sequence is periodic with period α2m β then f (ϕr (z)) < f (p), ∀r ∈ Z, and so, since z is periodic, supr∈Z f (ϕr (z)) < f (p),

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a contradiction. In order to do this, we use an argument somewhat analogous to the proof of the Claim: let j such that (bj , bj+1 , . . . , bj+M ) = α2m βα2m , where M = 4ms + b − 1, with b = |β|. Since f (ϕj+(2m−m0 )s−i (˜ q )) < f (p) for 1 ≤ i ≤ 2s and f (ϕj+(2m+m0 )s+b+i (˜ q )) < f (p) for 1 ≤ i ≤ 2s, we have that f (π u (αj βα)) < f (p) (so the suﬃx β helps diminishing the value of f ) and f (π s ((αt )j β t αt )) < f (p) for every j ≥ 0 (so the prefix β helps diminishing the value of f ). So, for the positions corresponding to a0 in α2m , we use the fact that β helps diminishing the value of f both as a suﬃx and as a prefix in order to show that the corresponding values of f are smaller than f (p). For the other positions, including the positions inside β, we use the extended cell argument in order to show that the corresponding values of f are also smaller than f (p). This concludes the proof. References [1] C. M. A. G. Cerqueira & C. G. Moreira – “Continuity of Hausdorﬀ dimension across generic dynamical Lagrange and Markov spectra”, Journal of Modern Dynamics 12 (2018), p. 151–174. [2] T. W. Cusick & M. E. Flahive – The Markoﬀ and Lagrange spectra, Math surveys and Monographs, vol. 30, A.M.S., 1989. [3] A. Markov – “Sur les formes quadratiques binaires indéfinies”, Math. Ann. 15 (1879), p. 381–406. [4] C. G. Moreira & S. A. R. Ibarra – “On the Lagrange and Markov dynamical spectra”, Ergodic Theory and Dynamical Systems 37 (2016), p. 1570–1591. [5] C. G. Moreira & J.-C. Yoccoz – “Tangencies homoclines stables pour des ensembles hyperboliques de grande dimension fractale”, Annales Scientifiques de l’École Normale Supérieure 43 (2010), p. 1–68. [6] J. Palis & F. Takens – Hyperbolicity & sensitive chaotic dynamiscs at homoclinic bifurcations, Cambridge studies in abvanced mathematics, vol. 35, 1993. [7] M. Shub – Global stability of dynamical systems, Springer, 1986.

C.G.T. de A. Moreira, IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil. E-mail : [email protected]

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Astérisque 415, 2020, p. 59–85 doi:10.24033/ast.1100

QUENCHED AND ANNEALED TEMPORAL LIMIT THEOREMS FOR CIRCLE ROTATIONS by Dmitry Dolgopyat & Omri Sarig

In memory of Jean-Christophe Yoccoz

P Abstract. — Let h(x) = {x} − 12 . We study the distribution of n−1 k=0 h(x + kα) when x is fixed, and n is sampled randomly uniformly in {1, . . . , N }, as N → ∞. Beck proved in [2, 3] that if x = 0 and α is a quadratic irrational, then these distributions converge, after proper scaling, to the Gaussian distribution. We show that the set of α where a distributional scaling limit exists has Lebesgue measure zero, but that the following annealed limit theorem holds: Let (α, n) be chosen randomly uniformly Pn−1 in R/Z × {1, . . . , N }, then the distribution of k=0 h(kα) converges after proper scaling as N → ∞ to the Cauchy distribution. Résumé (Théorèmes limites temporels modifiés pour les rotations du cercle). — Soit h(x) = Pn−1 {x} − 21 . On étudie la distribution de k=0 h(x + kα) pour x fixé et n tiré au hasard uniformément dans {1, . . . , N }, quand N → ∞. Beck a montré dans [2, 3] que pour x = 0 et α irrationnel quadratique, ces distributions convergent, après un changement d’échelle approprié, vers une distribution gaussienne. Nous montrons que l’ensemble des α pour lesquels la distribution limite après changement d’échelle existe est de mesure de Lebesgue nulle, mais qu’on a le théorème limite modifié suivant: soit (α, n) choisi au hasard uniformément dans R/Z × {1, . . . , N }, alors la distribution de Pn−1 k=0 h(kα) converge après un changement d’échelle approprié quand N → ∞ vers la distribution de Cauchy.

2010 Mathematics Subject Classification. — 37D25; 37D35. Key words and phrases. — Uniform distribution, temporal distributional limit theorems, circle rotations, annealed limit theorems, Cauchy distribution. D. D. acknowledges the support of the NSF grant DMS1665046. O.S. acknowledges the support of the ISF grants 199/14 and 1149/18.

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1. Introduction We study the centered ergodic sums of functions h : T → R for the rotation by a an irrational angle α ! Z n X (1.1) Sn (α, x) = h(x + kα) − n h(z)dz. T

k=1

Weyl’s equidistribution theorem says that for every α ∈ R \ Q, and for every h Riemann integrable, n1 Sn (α, x) −−−−→ 0 uniformly in x. We are interested in highern→∞ order asymptotics. We aim at results which hold for a set of full Lebesgue measure of α. If h is suﬃciently smooth, then Sn (α, x) is bounded for almost every α and all x (see [12] or Appendix A). The situation for piecewise smooth h is more complicated, and not completely undertstood even for functions with a single singularity. Setup. — Here we study (1.1), for the simplest example of a piecewise smooth function with one discontinuity on T = R/Z: 1 h(x) = {x} − . 2 The fractional part {x} is the unique t ∈ [0, 1) s.t. x ∈ t + Z. Case (1.2) is suﬃcient for understanding the behavior for typical α for all functions f (t) on T which are diﬀerentiable everywhere except one point x0 , and whose derivative on T\{x0 } extends to a function of bounded variation on T. This is because of the following result proven in Appendix A: (1.2)

Proposition 1.1. — If f (t) is differentiable on T \ {x1 , . . . , xν } and f ′ extends to a function with bounded variation on T, then there are A1 , . . . , Aν ∈ R s.t. for a.e. α there is ϕα ∈ C(T) s.t. for all x 6= xi , Z ν X Ai h(x + xi ) + f (t)dt + ϕα (x) − ϕα (x + α). f (x) = i=1

T

Of course there are many functions h for which Proposition 1.1 holds. The choice (1.2) is convenient, because of its nice Fourier series. Methodology. — Sn (α, x) is very oscillatory. Therefore, instead of looking for simple asymptotic formulas for Sn (α, x), which is hopeless, we will look for simple scaling limits for the distribution of Sn (α, x) when x, or α, or n (or some of their combinations) are randomized. There are several natural ways to carry out the randomization: (1) Spatial vs temporal limit theorems: In a spatial limit theorem, the initial condition x chosen randomly from the space T. In a temporal limit theorem, the initial condition x is fixed, and the “time” n is chosen randomly uniformly in {1, . . . , N } as N → ∞. Neither limit theorem implies the other, see [10].

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(2) Quenched vs annealed limit theorems: In a quenched limit theorem, α is fixed. In an annealed limit theorem α is randomized. The terminology is motivated by the theory of random walks in random environment; the parameter α is the “environment parameter.” We indicate what is known and what is still open in our case. Known results on spatial limit theorems:— The quenched spatial limit theorem fails; the annealed spatial limit theorem holds. The failure of the quenched spatial limit theorem is very general. It follows from the Denjoy-Koksma inequality that there are no quenched spatial distributional limit theorems for any rotation by α ∈ R \ Q, and every function of bounded variation which is not a coboundary (e.g., h(x) = {x} − 12 ). In the coboundary case, the spatial limit theorem is trivial. Many people have looked for weaker quenched versions of spatial distributional limit theorem (e.g., along special subsequences of “times”). See [7, 10] for references and further discussion. The annealed spatial limit theorem is a famous result of Kesten. Theorem 1.2 ([14]). — If (x, α) is uniformly distributed on T × T then the distribution converges as n → ∞ to a symmetric Cauchy distribution: ∃ρ1 6= 0 s.t. for of Snln(α,x) n all t ∈ R, Sn (α, x) 1 arctan(t/ρ1 ) lim P ≤t = + . n→∞ ln n 2 π See [14] for the value of constant ρ1 . The same result holds for h(x) = 1[0,β) ({x}) − β with β ∈ R, with diﬀerent ρ1 = ρ1 (β) [14, 15]. Known results on temporal limit theorems:— Quenched temporal limit theorems are known for special α; There were no results on the annealed temporal limit theorem until this work. The first temporal limit theorem for an irrational rotation (indeed for any dynamical system) is due to J. Beck [2, 3]. Let MN (α, x) :=

N 1 X Sn (α, x) , Sn (α, x) := Sn (α, x) − MN (α, x). N n=1

Theorem 1.3 (Beck). — Let α be an irrational root of a quadratic polynomial with n (α,x) integer coefficients. Fix x = 0. If n is uniformly distributed on {1 . . . N } then S√ ln N converges to a normal distribution as N → ∞. A similar result holds for the same x and α with h(x) = {x} − 12 replaced by 1[0,β) ({x}) − β, β ∈ Q [2, 3]. [1, 10] extended this to all x ∈ [0, 1). A remarkable recent paper by Bromberg & Ulcigrai [5] gives a further extension to all x, all irrational α of bounded type, and for an uncountable collection of β (which depends on α). Recall that the set of α of bounded type is a set of full Hausdorﬀ dimension [13], but zero Lebesgue measure [16].

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This paper. — We show that for h(x) = {x}− 21 , the quenched temporal limit theorem fails for a.e. α, but that the annealed temporal limit theorem holds. See §2 for precise statements. Heuristic overview of the proof. — When we expand the ergodic sums of h into Fourier series, we find that the resulting trigonometric series can be split into the contribution of “resonant” and “non-resonant” harmonics. The non-resonant harmonics are many in number, √ but small in size. They tend to cancel out, and their total contribution is of order ln N . It is natural to expect that this contribution has Gaussian statistics. If α has bounded type, all harmonics are non-resonant, and as Bromberg and Ulcigrai show in the case 1[0,β) − β the limiting distribution is indeed Gaussian. The resonant harmonics are small in number, but much larger in size: individual resonant harmonics have contribution of order ln N . For typical α, the number, strength, and location of the resonant harmonics changes erratically with N in a nonuniversal way. This leads to the failure of temporal distributional limit theorems for typical α. We remark that a similar obstruction to quenched limit theorems have been observed before in the theory of random walks in random environment [9, 18, 6]. To justify this heuristic we fix N and compute the distribution of resonances when α is uniformly distributed. Since the distribution of resonances is non-trivial, changing a scale typically leads to a diﬀerent temporal distribution proving that there is no limit as N → ∞. As a by-product of our analysis we obtain some insight on the frequency with which a given limit distribution occurs. Functions with more than one discontinuity. — In a separate paper [11] we use a diﬀerent method to show that given a piecewise smooth discontinuous function with arbitrary finite number of discontinuities, the quenched temporal limit theorems fails for Lebesgue almost all α. But this method does not provide an annealed result, and it does not give us as detailed information as we get here on the scaling limits which appear along subsequences for typical α.

2. Statement of results

Fix x ∈ T arbitrary. Let Sn (α, x) :=

n X

h(x + kα), and

k=1

MN (α, x) = Sn (α, x)

ASTÉRISQUE 415

N 1 X Sn (α, x), N n=1

= Sn (α, x) − MN (α, x).

63

QUENCHED AND ANNEALED TLTS FOR CIRCLE ROTATIONS.

: Consider the cumulative distribution function of Snln(α,x) N 1 Sn (α, x) x FN (α)(z) = FN (α)(z) = Card 1 ≤ n ≤ N : ≤z . N ln N When α is random FN (α) becomes a random element in the space X of distribution functions endowed with Prokhorov topology. We begin with the annealed temporal distributional limit theorem: Theorem 2.1. — Fix x ∈ T arbitrary. Let (α, n) be uniformly distributed on T × {1, . . . , N }. Then Snln(α,x) converges in law as N → ∞ to the symmetric N Cauchy distribution with scale parameter 3π1√3 : 1 arctan(t/ρ2 ) Sn (α, x) (2.1) ≤t = + (t ∈ R) lim P N →∞ ln N 2 π 1 √ . (2.2) ρ2 = 3π 3 Next we turn to the quenched result, beginning with some preparations. Recall (see e.g., [8, Section 2.1]) that the Cauchy random variable can be represented up to scaling as ∞ X Θm , (2.3) C= ξ m=1 m where Ξ = {ξm } is a Poisson process on R and Θm are i.i.d bounded random variables with zero mean independent of Ξ. To make our exposition more self-contained we recall the derivation of (2.3) in Appendix B. In our case Θm will be distributed like the following random variable Θ. Let θ be uniformly distributed on [0, 1] and define ∞ X cos(2πkθ) (2.4) Θ(θ) = , θ ∼ U [0, 1]. 2π 2 k 2 k=1

Notice that (2.5)

Θ(θ) =

as can be verified by expanding that for every θ ∈ [0, 1], Θ(θ) = (2.6)

θ2 − θ 1 + on [0, 1], 2 12

1 θ 2 −θ + 12 on [0, 1] 2 2 ζ 1 2 − 24 , where ζ =

0q P(Θ < t) = 2 2t + 1

Next, given a sequence Ξ = {ξm } s.t. (2.7)

CΞ =

1 θ − 12 . Thus − 24 ≤Θ≤

1 12 ,

and

1 , t ≤ − 24 1 12

1 1 t ∈ (− 24 , 12 ),

t>

P

−2 m ξm

X Θm m

into a Fourier series. Notice also

ξm

1 12 .

< ∞ we can define ,

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where Θm are i.i.d. random variables with distribution given by (2.6). (In the proof of Theorem 2.1, {ξm } would describe the small denominators of Sn and {Θm } would describe the corresponding numerators, see Formula (3.2) in Section 3.) The sum in (2.7) converges almost surely due to Kolmogorov’s Three Series Theorem (note that (2.4) easily implies that E(Θ) = 0). Let FΞ be the cumulative distribution function of CΞ . If Ξ is a Poisson process on R, then FΞ is a random element of Prokhorov’s space X. Theorem 2.2. — Fix x ∈ T arbitrary. If α has absolutely continuous distribution on T with bounded density then FN (α) converges in law as N → ∞ to FΞ where Ξ is the Poisson process on R with intensity 6 (2.8) c = 2. π A similar result has been proven for sub-diﬀusive random walks in random environment in [9] with the following distinctions: (1) for random walks Θm + 1 have exponential distribution rather than the distribution given by (2.4); (2) for random walk the Poisson process in the denominator of (2.3) is supported on R+ and can have intensity e cx−s with s 6= 0.

We now explain how to use Theorem 2.2 to show that for a.e. α, there is no nontrivial temporal distributional limit theorem for Sn (α, x). It is enough to show that for a.e. α, one can find several sequences Nk with diﬀerent scaling limits for Sn (α, x) as n ∼ U {1, . . . , Nk }. Let D(Θ) := {finite linear combinations of i.i.d. with distribution Θ} (closure in X). Corollary 2.3. — Fix x ∈ T arbitrary. For a.e. α, for every Y ∈ D(Θ), there are Nk → ∞, Bk → ∞, Ak ∈ R s.t. Sn (α, x) − Ak dist −−−−→ Y, as n ∼ U {1, . . . , Nk }. (2.9) k→∞ Bk In particular, for a.e. α, the distribution of Θ (2.6), and the normal distribution are distributional limit points of properly rescaled ergodic sums Sn (α, x) as n ∼ U {1, . . . , N }. Proof. — Put on X the probability measure µ induced by the FΞ , when Ξ is the Poisson point process with intensity c as in Theorem 2.2. Observe that for every Y ∈ D(Θ), (2.10)

∃ decreasing seq. of open Un ⊂ X s.t. µ(Un ) > 0 and Un ↓ {Y }.

Indeed, fix a countable dense set {fn }n≥1 ⊂ Cc (R), and let Un :=

n \ i=1

ASTÉRISQUE 415

{X : |EX (fi ) − EY (fi )|

0 s.t. if X1 , X2 are random variables s.t. there is a coupling with P(|X1 − X2 | ≥ δ) ≤ δ then for each i ∈ {1, . . . n} we have |EX1 (fi ) − EX2 (fi )| < n1 . Note that given a sequence {bi } ∈ ℓ2 ! ! X X X b2i . E bi Θi = Var(Θ) bi Θi = 0, Var i

i

i

Accordingly, the Chebyshev inequality shows that if Ξ = {ξi } is a sequence s.t. 2 k ∞ X X 1 1 δ2 1 (2.11) − ≤ , + ai ξi ξi2 Var(Θ) i=1 i=k+1

then FΞ ∈ Un . It remains to note that if Ξ = {ξi } is a Poisson point process then for each δ > 0, (2.11) holds with positive probability. Indeed let Aδ,R be the event that of Ξ ∩ [−R, R] consists of there being exactly k points ξ1 , . . . , ξk and 2 Pk 1 δ2 1 i=1 ai − ξi ≤ 2Var(Θ) . Then for each δ, R the probability of Aδ,R is positive while P 8cVar(Θ) 1 E then = 2c |ξi |≥R ξi2 R (see Formula (B.3) in Appendix B). Thus if R = δ2 P 2 δ 1 ≤ 21 . Since ξ ∩ [−R, R] the Markov inequality shows that P |ξ|≥R ξi2 ≥ 2Var(Θ) and ξ ∩ (R \ [−R, R]) are independent, (2.11) has positive probability proving (2.10). We claim that for any n, for almost every α, every sequence has a subsequence ′ ′ (α) ∈ Un for all m. A diagonal argument then produces a {Nm } such that FNm subsequence {Nk } along which we have (2.9) with Ak := MNk (α) and Bk := ln Nk . ′ Fix n and set U := Un . To produce {Nm } it is enough to show that for each N , (2.12)

mes(α ∈ T : ∃N ≥ N such that FN (α) ∈ U) = 1.

Let µ(U) = 2ε. Let α be uniformly distributed. By Theorem 2.2 there exists n1 ≥ N and a set A1 ⊂ T such that mes(A1 ) ≥ ε so that for every α ∈ A1 , Fn1 (α) ∈ U. If mes(A1 ) = 1 we are done; otherwise we apply Theorem 2.2 with α uniformly distributed on T \ A1 and find n2 ≥ n1 and a set A2 ⊂ T \ A1 such that mes(A2 ) ≥ εmes(A2 ) so that for each α ∈ A2 , Fn2 (α) ∈ U. Continuing in this way, we obtain nm ↑ ∞ such that for α ∈ Aj , Fnj (α) ∈ U and mes(T \ ∪kj=1 Aj ) ≤ (1 − ε)k . Letting k to infinity we obtain (2.12). Corollary 2.3 shows that for every x, for a.e. α, there is no non-trivial temporal distributional limit theorem for Sn (α, x).

3. The main steps in the proofs of Theorems 2.1, 2.2 We state the main steps in the proofs of Theorems 2.1, 2.2. The technical work needed to carry out these steps is in the next section.

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Step 1: Identifying the resonant harmonics. — It is a classical fact that the Fourier series of h(x) = {x} − 21 converges to h(x) everywhere: (3.1)

h(x) = −

∞ X sin(2πjx)

πj

j=1

for all x ∈ R.

We will use this identity to represent Sn (α, x)

:=

n X

h(x + kα) −

N 1 X h(x + kα) N k=1

k=1

as a trigonometric sum, and then work to separate the “resonant frequencies,” which contribute to the asymptotic distributional behavior of Sn (α), from those which do not. We need the following definitions: T := N ln2 N, gj,n := Sn,T (α)

:=

cos((2n + 1)πjα + 2πjx) , 2πj sin(πjα) T X

gj,n .

j=1

Proposition 3.1. — Let (α, n) be uniformly distributed on T × {1, . . . , N }. Then dist = Sn,T (α) + εˆn where lnεbnN −−−−→ 0.

Sn (α, x)

N →∞

The proof is given in §4.2. We follow the analysis of [2, 3], but we obtain weaker estimates since we consider a larger set of rotation numbers than in [2, 3]. In what follows, indices j in gj,n are called “harmonics.” We will separate the harmonics into diﬀerent classes, according to their contribution to Sn,T (α). We begin with some standard definitions: Given x ∈ R there is a unique pair y ∈ (− 12 , 21 ], m ∈ Z such that x = m + y. We will call y the signed distance from x to the nearest integer and denote it by ((x)). We let kxk = |((x))|, hhxii = (−1)m ((x)) where m is as above. Fix N . An integer 1 ≤ j ≤ T = N ln2 N is called a prime harmonic, if jα = ((jα)) + m where gcd(j, m) = 1. If j and m are not co-prime, that is r := gcd(j, m) 6= 1, then we call j/r the prime harmonic associated to j. (If j is prime then the prime harmonic associated with j is j itself.) Definition 3.2. — Fix δ > 0, and N ≫ 1. (1) p ∈ N is called a prime resonant harmonic, if p ≤ N , p is a prime harmonic, and kpαk ≤ (δp ln N )−1 . (2) j ∈ N is called a resonant harmonic, if j ≤ N ln2 N , and the prime harmonic associated to j is a prime resonant harmonic.

ASTÉRISQUE 415

67

QUENCHED AND ANNEALED TLTS FOR CIRCLE ROTATIONS.

Let R = R (δ, N ) denote the set of resonant harmonics, P = P (δ, N ) the set of prime resonant harmonics, and O = O (δ, N ) be the set of non resonant harmonics which are less than T (N ) = N ln2 N. Split X J R O Sn,T (α) = Sn,T (α) + Sn,T (α) where Sn,T = gj,n . j∈J O O Let VN (α) := En [Sn,T (α)2 ] ≡

1 N

PN

O 2 n=1 Sn,T (α) .

Proposition 3.3. — Suppose α ∈ T is distributed according to an absolutely continuous measure with bounded density. For every ε > 0 there are δ0 > 0 and EN (ε) ⊂ T Borel with the following properties: (1) mes(EN (ε)) > 1 − ε for all N large enough; ! O (δ,N )

(2) for all 0 < δ < δ0 , lim

N →∞

VN

sup

α∈EN (ε)

(α) ln2 N

≤ ε.

The proof is given in §4.2. Here is a corollary. Corollary 3.4. — For ε >0 there is a δ0 > 0 s.t. for all N large enough and every O (δ,N ) |Sn,T | > ε < ε, where 1 ≤ n ≤ N is distributed uniformly, 0 < δ < δ0 , Pα,n ln N α ∈ T is sampled from an absolutely continuous measure with bounded density, and α, n are independent. Proof. — Without loss of generality 3ε2 < ε. Fubini’s theorem gives ! ! Z O (δ,N ) O (δ,N ) |Sn,T | |Sn,T | 4 Pα,n Pn >ε ≤ε + > ε dα. ln N ln N EN (ε4 ) O (δ,N )

By Chebyshev’s Inequality, the integrand is less than ε12 × ln21 N VN this is less than 2ε2 for all N large enough (uniformly in α). O (δ,N ) So Pα,n ( ln1N |Sn,T | > ε) ≤ ε4 + 2ε2 < ε.

. On EN (ε4 ),

Proposition 3.1 and Corollary 3.4 say that the asymptotic distributional behavior R of Sn (α) is determined by the behavior of the sum of the resonant terms Sn,T (α). Step 2: An identity for the sum of resonant terms. — Let mod 2 (3.2) ξj := jhhjαii ln N and Θj (n) := Θ [(2n+1)jα+2jx] , 2 P∞ where Θ(t) = k=1 cos(2πkt) 2π 2 k2 , see (2.4). Proposition 3.5. — For all δ small enough, R (δ,N )

(3.3)

Sn,T

ln N

=

X j∈P (δ,N )

Θj (n) + O ξj

1 ln N

.

The big Oh is uniform in j but not in δ.

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For the proof, see §4.3. Step 3: Limit theorems for resonant harmonics. — We will use (3.3) to study of the R distributional behavior of Sn,T . First we will describe the distribution of the set of denominators {ξj = jhhjαii ln N }j∈P (δ,N ) , and then we will describe the conditional joint distribution of set of numerators, given {ξj }. Notice that we need information on the point process (“random set”) {ξj : j ∈ P (δ, N )}, not just on individual terms. Proposition 3.6. — Suppose that α is distributed according to a bounded density on T. For each δ the point process ln j , hhjαiij ln N ln N j∈P (δ,N ) converges in distribution as N → ∞ to the Poisson Point Process on [0, 1] × [− 1δ , 1δ ], with constant intensity c = π62 . The second coordinate contains the information we need on {ξj }. The information contained in the first coordinate is needed in the proof of Proposition 3.7 below. Proposition 3.6 is proven in [8, Theorem 5] with hhxii replaced by ((x)). The proof given in [8] relies on the Poisson limit theorem for the sum along the orbit of the diagonal ﬂow on SL2 (R)/SL2 (Z) of the Siegel transform of functions of the form F (x, y) = 1IN (x)1JN (y) where {IN } and {JN } are sequences of shrinking intervals. To obtain Proposition 3.6 one needs to change slightly the definition of IN but all the estimates used in [8] remain valid in the present context. Proposition 3.7. — Suppose that α is distributed according to a bounded density on T. For every r > 1 and ε > 0 there are δ, N0 and A(N, δ, r) ⊂ T such that mes(A(N, δ, r)) > 1 − ε and (1) If α ∈ A(N, δ, r) then |P (δ, N )| > r for all N > N0 . (2) For each neighborhood V of the uniform distribution on [0, 2]r there exists NV such that for N ≥ NV the following holds. Let α ∈ A(N, δ, r) and jk be an enumeration of the prime resonant harmonics in P (δ, N ) which orders kjk αkjk in decreasing order, then the distribution of the random vector (j1 (α(2n + 1) + 2x), . . . , jr (α(2n + 1) + 2x)) mod 2 where n ∼ Uniform{1, . . . , N } belongs to V. Proposition 3.7 is proved in §4.4. Proof of Theorem 2.1. — This theorem describes the distributional behavior of ln1N Sn (α) as (α, n) ∼ U (T × {1, . . . , N }), when N → ∞. Step 1 says that for every ε there are δ,N0 such that for all N > N0 , 1 1 R (δ) Sn (α, x) = S + ∆n (α) ln N ln N n,T

ASTÉRISQUE 415

QUENCHED AND ANNEALED TLTS FOR CIRCLE ROTATIONS.

69

where P(|∆n (α)| ≥ ε) ≤ ε, as (α, n) ∼ U (T × {1, . . . , N }). To see this take ∆n := 1 O εn + Sn,T ), and use Proposition 3.1 and Corollary 3.4. ln N (b dist R (δ,N ) 1 (α) −−−−→ Cδ as (α, n) ln N Sn N →∞ dist variables such that Cδ −−−→Cauchy. δ→0

We will prove Theorem 2.1 by showing that U (T × {1, . . . , N }), where Cδ are random

∼

Let jk be an enumeration of P (δ, N ) which orders kjk αkjk in decreasing order. By step 2, R (δ,N )

Sn,T

ln N

=

X Θjk (n) + O(

1 ln N )

ξjk

.

Proposition 3.6 says that the point process {ξjk } converges in law to the Poisson Point Process on [− 1δ , 1δ ]. Proposition 3.7 says that given {ξjk }, (Θj1 (n)) + dist

O( ln1N ), . . . , Θj|P | (n) + O( ln1N )) −−−−→ (Θ1 , . . . , Θ|P | ) where Θi are are independent N →∞

identically distributed random variables with distribution (2.6). R (δ,N ) P Θm S dist It follows that n,T −−−−→ ln N ξm , where Θi are independent, distributed like N →∞

(2.6), and {ξm } is a Poisson Point Process Cδ on [− 1δ , 1δ ] with density c = 6/π 2 . In the limit Cδ −−−→ Cauchy random variable, see Appendix B. This completes the proof δ→0

of Theorem 2.1 except for the Formula (2.2) which is proven in the appendix.

Proof of Theorem 2.2. — Theorem 2.2 describes the convergence in distribution of the (X-valued) random variable FN (α)(·)

:=

1 Card 1 ≤ n ≤ N : N

Sn (α,x)

ln N

≤·

to FΞ as N → ∞, when α is sampled from an absolutely continuous distribution on T with bounded density. We will assume for simplicity that the density is constant, the changes needed to treat the general case are routine and are left to the reader. R Again we claim that it is enough to prove the result with Sn,T replacing Sn . Let ∆n (α) be as above. By step 1, for every ε there are δ and N0 s.t. for all N > N0 , P(|∆n (α)| ≥ ε) ≤ ε as (α, n) ∼ U (T × {1, . . . , N }). By Fubini’s theorem for such N , Card(1 ≤ n ≤ N : |∆n (α)| ≥ ε) √ mes α : ≥ ε N

≤

√

ε.

R (α) It follows that the set of α where the asymptotic distributional behavior of ln1N Sn,T 1 is diﬀerent from that of ln N Sn (α) in the limit N → ∞, δ → 0 has measure zero. Thus to prove Theorem 2.2, it is enough to show that

δ FN (α)(·)

1 := Card 1 ≤ n ≤ N : N

R

Sn,T (α)

ln N

≤·

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converges in law, as N → ∞, to an X-valued random variable FδΞ such that dist

FδΞ −−−→ FΞ , where FΞ is the cumulative distribution function of the random variδ→0

able CΞ defined in (2.7). This is done as before, using Propositions 3.6 and 3.7. 4. Proofs of the Key Steps. 4.1. Preliminaries. — The following facts are elementary: Lemma 4.1. — (a) If the harmonic j is not prime and is associated to the prime harmonic p, j = rp, then ((jα)) = r((pα)). (b) For every x ∈ R,

2 π

≤

| sin(πx)| πkxk

≤

π 2,

and

| sin(πx)| = πkxk + O(kxk3 ) as x → 0. (c) For every x ∈ R and m, N ∈ N, m X

(4.1)

sin(y + jx) =

j=1 m X

(4.2)

cos(y + 2jx) =

j=1

cos(y + x/2) − cos(y + (2m + 1)x/2) , 2 sin(x/2) sin(mx) cos((m + 1)x + y) . sin x

(d) If α is uniformly distributed on T then for each j 6= 0, jα is also uniformly distributed on T. (e) For every 0 < a < 21 , mes(α ∈ T : kjαk < a) = 2a. (mes =Lebesgue). R R dα 1 dα (f) T∩[kjαk>a] kjαk , = 2 ln 2a = a2 − 1. T∩[kjαk>a] kjαk2 Part (e) of Lemma 4.1 implies the following estimates: (4.3) lim mes α ∈ T : jkjαk > ln−1.1 j for all j ≥ N/ ln10 N = 1, N →∞

(4.4)

lim mes(α ∈ T : jkjαk > ln−2 N for all j ≤ 2T ) = 1, 6 lim mes α ∈ T : # j : kjαk < lnNN , j < 2T ≤ ln9 N } = 1. N →∞

(4.5)

N →∞

We prove (4.5), and leave the proofs of (4.3),(4.4) (which are easier) to the reader. Let FeN denote the set of α in (4.5), then FeNc = {α ∈ T :

2T X

1[kjαk< ln6 N ] (α) > ln9 N }.

j=1

By Lemma 4.1(e), the sum has expectation P2N ln2 N 2 ln6 N −−−−→ 0. mes(Fec ) ≤ 91 N

ln N

ASTÉRISQUE 415

j=1

N

N →∞

N

P2N ln2 N j=1

2 ln6 N . N

By Markov’s inequality,

QUENCHED AND ANNEALED TLTS FOR CIRCLE ROTATIONS.

71

We also observe the following consequnce of (4.2) X m π ≤ min , m . cos(y + 2πmx) (4.6) 2||x|| j=1 π To see that (4.6) is less than 2||x|| we estimate the numerator of (4.2) by 1 and the denominator by Lemma 4.1(b). To see that (4.6) is less than m, note that each term in the LHS is less than 1 in absolute value. We note that (4.3) is a very special case of Khinchine’s Theorem on Diophantine approximations (seePe.g., [4, Thm 2.3]). This theorem says that if ϕ : N → R+ is a function such that q ϕ(q) < ∞, then for almost every α the inequality

(4.7)

kqαk < ϕ(q) P has only finitely many solutions, while if q ϕ(q) = ∞ and ϕ is non increasing then (4.7) has infinitely many solutions. Next we list some tightness estimates. Recall that a family of real-valued functions {fn } on a probability space (Ω, F , P) is called tight, if for every ε > 0 there is an a > 0 s.t. P(|fn | > a) < ε for all n. n o PN 1 1 Lemma 4.2. — Let α ∼ U (T) then N ln is tight. j=1 kjαk N N ∈N

Proof. — For every ε > 0, N N X X 1 ε 1 (4.8) mes α : 6= 1[ε/(4N ),1/2] (kjαk) ≤ , kjαk kjαk 2 j=1 j=1 SN ε because the event in the brackets equals j=1 {α : kjαk < 4N } up to measure zero, ε ε and mes{α : kjαk < 4N } = 2N by Lemma 4.2(e). On the other hand, one can check using Lemma 4.2(f) that N X 2 1 1[ε/(4N ),1/2] (kjαk) = 2N ln N + ln ≤ 3N ln N E kjαk ε j=1 if N is suﬃciently large. Hence by Markov’s inequality N X 1 6N ln N ≤ ε. (4.9) P 1[ε/(4N ),1/2] (kjαk) ≥ kjαk ε 2 j=1 P N 1 6N ln N ≤ ε. Combining (4.8) and (4.9) we see that P j=1 kjαk ≥ ε

Lemma 4.3. — Let α ∼ U (T), then the following families of functions are tight as N → ∞ (recall that T := N ln2 N ): PT 1 . (a) ln21 N j=1 jkjαk PT lnβ2 T 1 (b) (ln ln1 T )2 j=T ln−β1 T jkjαk 1[ln−β3 T,lnβ4 T ] (jkjαk), for every β1 , β2 , β3 , β4 .

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(c)

1 ln T ln ln T

PT

1 j=1 jkjαk 1[ln−β3 T,lnβ4 T ] (jkjαk),

for every β3 , β4 .

We omit the proof, because it is similar to the proof of Lemma 4.2. 4.2. Step 1 (Propositions 3.1 and 3.3). — The proofs of these propositions follow [3] closely, but we decided to give all the details, since our assumptions are diﬀerent. Proof of Proposition 3.1. — The starting point is the Fourier series expansion PT of h(x) = {x} − 12 given in (3.1). Let hT (x) = − j=1 sin(2πjx) . Summation by parts πj (see [3, formula (8.6)]) gives us that n n N n X X 1 X 1 1 X 1 hT (x + kα) ≤ h(x + kα) − ≤ . T kkαk T kkαk k=1

k=1

k=1

k=1

The last expression converges to 0 in distribution as α ∼ U (T) and N → ∞, by Lemma 4.2 (recall that T = N ln2 N ). Thus it suﬃces to study the distribution of ! n N n X 1 X X (4.10) hT (x + kα) hT (x + kα) − N n=1 k=1

k=1

as α ∼ U (T) and N → ∞. A direct calculation using Lemma 4.1(c) shows that (4.10) = Sn,T (α) +

T X

fj ,

j=1

where fj =

N 1 X sin(2πjx + 2πjα) − sin(2πjx + 2π(N + 1)jα) gj,n = N n=1 4πN j sin2 (πjα)

(cf. [3, Lemma 8.2]). Observe that by (4.6) there is a universal constant C such that 1 1 . , (4.11) |fj | ≤ C min jN kjαk2 jkjαk Let T := N/ ln10 N. By (4.4), with probability close to 1 we have 1/(jkjαk) < ln2 N for all j ≤ T , so with probability close to 1, all j ≤ T satisfy kjαk ∈ [ T ln12 N , 12 ]. We split T ln12 N , 12 = T ln12 N , N1 ∪ N1 , 12 and apply the bounds in (4.11) to each piece. Thus on (4.12)

FN := {α : ∀j ≤ T, jkjαk > ln−2 N },

we have T X j=T +1

ASTÉRISQUE 415

|fj | ≤ C

T X j=T +1

|fej |,

QUENCHED AND ANNEALED TLTS FOR CIRCLE ROTATIONS.

73

where fej =

1[ N1 , 12 ] (kjαk)

[ T ln12 N , 12 ] (kjαk)

1

−

1[ N1 , 12 ] (kjαk)

. N jkjαk2 jkjαk jkjαk P T 2 e Since by Lemma 4.1(f) Eα −−−→ 1, j=T +1 fj ≤ C(ln ln N ) , and by (4.4) mes(FN ) − N →∞ P dist T we conclude that ln1N j=T +1 |fj | −−−−→ 0 as (α, n) ∼ U (T × {1, . . . , N }). N →∞ PT PT dist 1 Next, we show that ln T j=1 fj −−−−→ 0. By (4.11), j=1 |fj | ≤ CBN with N →∞ PT dist BN 1 BN = j=1 N jkjαk −−−→ 0. 2 , so it is enough to show that ln N − +

N →∞

− + Here is the proof. Split BN = BN + BN where the first term contains the j s.t. + δ −1 δ −1 jkjαk ≥ ln N , and BN contains the j s.t. jkjαk < ln N. By Lemma 4.1(f) − BN δT ln N 1 ≤C =O . Eα ln N N ln N ln10 N

Hence by Markov’s inequality

dist − 1 −−−→ ln N BN − N →∞

0.

Fix ε > 0 and let FN be as in (4.12) above, then (4.4) says that mes(FN ) > 1 − ε for all N large enough. Let R (N ) = j ≤ T : δjkjαk ln N ≤ 1 . By Lemma 4.1(a) [ (4.13) R (N ) ⊂ {kp : k ∈ Z, p ∈ P (δ, N ), kp ≤ T }. p∈P (δ,N )

Given p ∈ P , let Rp (δ)

:= {resonant harmonics in R , associated to p}.

(4.13) implies that for some constants C, C X X X X C C (Lemma 4.1(a)) + ≤ BN = N jkjαk2 N k 3 pkpαk2 p∈P (δ,N ) j∈Rp (δ) p∈P (δ,N ) kp∈Rp (δ) ! ∞ X X 1 X C C ≤ ≤ 2 3 N pkpαk k N pkpαk2 k=1

p∈P (δ,N )

≤

X p∈P (δ,N )

CT N (pkpαk)2

(α∈FN )

≤

p∈P (δ,N )

C T (ln2 N )2 CCard(P (δ, N )) . ≤ N ln6 N p∈P (δ,N ) X

By Proposition 3.6, if α is distributed according to a bounded density on T, then Card(P (δ, N )) converges in law as N → ∞ to a Poissonian random variable. Therefore there is K(ε) so that mes{α ∈ T : Card(P (δ, N )) ≤ K(ε)} > 1 − ε. + It follows that mes{α ∈ T : BN ≤ K(ε)C ln−6 N } > 1 − 2ε, whence

dist + 1 −−−→ ln N BN − N →∞

0.

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To summarize, εbN := Sn (α, x) − Sn,T (α) = and

1 ln N

PT

j=1 |fj | =

1 ln N

T X

fj + error which tends to 0 in distribution,

j=1

P

T j=T +1

+ − + BN |fj | + BN

above.

dist

−−−−→ 0 by the arguments N →∞

Proof of Proposition 3.3. — We give the proof assuming that α ∼ U (T). The modifications needed to deal with absolutely continuous measures with bounded densities are routine, and are left ot the reader. N P P P O 2 Simple algebra gives VN = N1 + N1 gj,n gj1 ,n gj2 ,n , where gj,n = j∈O n=1

cos((2n+1)πjα+2πjx) . To 2πj sin(πjα) 1 |gj,n | < jkjαk , whence

j1 ,j2 ∈O j1 6=j2

bound the sum of diagonal terms, we use that by (4.6)

N N X 1 X 1 1 XX 2 gj,n ≤ =: Diag(N ). 2 N N n=1 j kjαk2 n=1

(4.14)

j∈O

j∈O

To bound the sum of oﬀ-diagonal terms, we first collect terms to get X 1 X 1 e j ,j ,N , gj1 ,n gj2 ,n = Γ N 4π 2 j1 j2 sin(πj1 α) sin(πj2 α) 1 2 j1 ,j2 ∈O j1 6=j2

j1 ,j2 ∈O j1 6=j2

where N 1 X e Γj1 ,j2 ,N := cos((2n + 1)πj1 α + 2πjx) cos((2n + 1)πj2 α + 2πjx). N n=1 A−B Now the identity cos A+B = 12 [cos A + cos B] and (4.6) give 2 cos 2 i h 1 1 e j ,j ,N ≤ C min . Γ 1 2 N k(j1 −j2 )αk , 1 + min N k(j1 +j2 )αk , 1

This and the estimate j| sin(πjα)| ≥ 2jkjαk (Lemma 4.1(b)) implies that for some universal constant C 1 X gj1 ,n gj2 ,n ≤ C OﬀDiag− (N ) + OﬀDiag+ (N ) , N j1 ,j2 ∈O j1 6=j2

where (4.15)

OﬀDiag± :=

X min j1 ,j2 ∈O j1 6=j2

1 N k(j1 ±j2 )αk , N k(j1

± j2 )αk

j1 kj1 αk · j2 kj2 αk

Since min(x, x−1 ) ≤ 1 for all x, the numerator is bounded by one.

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O ≤ Diag(N ) + C[OﬀDiag− + OﬀDiag+ ]. We need the following additional Thus VN decomposition:

(4.16)

± ± ± OﬀDiag± := OﬀDiag± i + OﬀDiagii + OﬀDiagiii + OﬀDiagiv ,

where (i) OﬀDiag− i : sum over the terms s.t. k(j1 − j2 )αk ≥

ln6 N N ;

(ii) OﬀDiag− ii : k(j1 − j2 )αk ≤

ln6 N N ,

and j1 kj1 αk ≥ ln10 N ;

(iii) OﬀDiag− iii : k(j1 − j2 )αk ≤

ln6 N N ,

and

j1 kj1 αk ≤ ln10 N, (iv) OﬀDiag− iv : k(j1 − j2 )αk ≤

j2 kj2 αk ≥ ln10 N ;

ln6 N N ,

j1 kj1 αk ≤ ln10 N,

j2 kj2 αk ≤ ln10 N.

Similarly for OﬀDiag+ with k(j1 + j2 )αk instead of k(j1 − j2 )αk. We now have the following upper bound (4.17)

O

VN

≤ Diag + C

iv X

+ OﬀDiag− + OﬀDiag k k .

k=i

For every ε > 0, and for each of the nine summands D1 , . . . , D9 above, we will construct δ0 > 0 and Borel sets A1 (ε, N ), . . . , A9 (ε, N ) ⊂ T s.t. mes[Ai ] > 1 − 9ε , and with the following property: Di ε sup 2 (4.18) ∀0 < δ < δ0 , lim ≤ . N →∞ α∈Ai ln N 9 T9 This will prove the proposition, with EN (ε) := i=1 Ai . We begin by recalling some facts on the typical behavior of kjαk for α ∼ U (T). Recall that T := N ln2 N , and let N (A) ∀j > ln10 , jkjαk > ln−1.1 j N ∗ −2 EN := α ∈ T : (B) ∀j ≤ 2T, jkjαk > ln N . 9 ln6 N (C) #{1 ≤ j ≤ 2T : kjαk < N } ≤ ln N ∗ Then mes(EN ) −−−−→ 1, by (4.3),(4.4), and (4.5). Most of our sets Ai will be subsets ∗ of EN .

N →∞

The summand Diag =

1 j∈O N

P

PN

1 n=1 j 2 kjαk2 .

∗ Suppose α ∈ EN . By the definition of resonant harmonics, for every j ∈ O either δjkjαk ln N ≥ 1, or N ≤ j ≤ T.

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∗ In the first case, jkjαk ≥ (δ ln N )−1 . In the second case, by property (A) of EN , −1.1 −1.1 jkjαk > ln j ≥ ln T . Accordingly, N Z X 1 −1 dα Eα (Diag · 1EN∗ ) ≤ 1 2 kjαk2 [jkjαk>(δ ln N ) ] j j=1 T Z T X 1 −1.1 T ] dα 1 + 2 kjαk2 [jkjαk>ln j j=N +1 T N T 1.1 X X 2δj ln N 2j ln T ≤ + (by Lemma 4.1(f)) 2 j j2 j=1 j=N +1

2

1.1

≤ 2δ ln N + 4 ln

N ln ln N.

∗ Let δ0 := ε2 /1000, and choose N0 so large that for all N > N0 , mes(EN ) > 1 − ε/9 4 ln1.1 N ln ln N 2 < ε /1000. By Markov’s inequality, for all N > N and δ < δ0 , and 0 ln2 N ∗ mes{α ∈ EN : Diag >

ε 9

ln2 N } ≤

2δ ln2 N + 4 ln ln N ε < . 9 (ε/9) ln2 N

∗ We obtain (4.18) with Di = Diag, Ai := {α ∈ EN : Diag ≤ 2 ε /1000. Notice that this Ai depends on ε.

ε 9

ln2 N }, and δ0 :=

The summand OﬀDiag− i : This is the part of (4.15) with j1 , j2 s.t. k(j1 −j2 )αk ≥

ln6 N N .

1/ ln6 N j1 kj1 αk · j2 kj2 αk j1 ,j2 ∈O 2 2 T T X X 1 1 1 1 1 ≤ 6 = 2 2 . ln N j=1 jkjαk ln N ln N j=1 jkjαk

OﬀDiag− i ≤

X

By Lemma 4.3(a), the term in the brackets is tight: there is a constant K = K(ε) s.t. PT 1 for all N , A := {α ∈ T : ln21 N j=1 jkjαk ≤ K} has measure more than 1 − 9ε . q 2 − 1 K2 ε For all N > exp 4 9K ε , for every α ∈ A, ln2 N OﬀDiagi ≤ ln4 N < 9 , and we get q 2 (4.18) with Ai = A, N0 = exp 4 9K ε . ∗ The summand OﬀDiag− ii : Suppose α ∈ EN , and let j2 ∈ O be an index which appears − in OﬀDiagii . Let I(j2 ) be the set of j1 s.t. (j1 , j2 ) appear in OﬀDiag− ii , namely

I(j2 ) := {j1 ∈ O : k(j1 − j2 )αk ≤

ln6 N N ,

j1 kj1 αk ≥ ln10 N }.

∗ By property (C) in the definition of EN (applied to |j1 −j2 |), the cardinality of I(j2 ) is 9 bounded by 2 ln N .

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77

Since the numerator in (4.15) is bounded by one, and since in case (ii) j1 kj1 αk ≥ ln10 N , we have T X X 1 1 OﬀDiag− ii 1 ≤ 2 ln2 N ln N j2 =1 j2 kj2 αk j ∈I(j ) j1 kj1 αk 1

T X

2

|I(j2 )| ln2 N j2 ln10 N T T 2 X 1 2 1 X 1 . ≤ 3 = 2 ln N ln N j=1 jkjαk ln N j=1 jkjαk

≤

1

1 j kj 2 2 αk =1

The term in the brackets is tight by Lemma 4.3(a), and we can continue to obtain (4.18) as we did in case (i). − The summand OﬀDiag− iii : This is similar to OﬀDiagii . − The summand OﬀDiag− iv : Suppose the pair of indices (j1 , j2 ) appears in OﬀDiagiv : 10 10 ln6 N k(j1 − j2 )αk ≤ N , j1 kj1 αk ≤ ln N , j2 kj2 αk ≤ ln N . Let jmax := max(j1 , j2 ) and jmin := min(j1 , j2 ). ∗ We claim that if α ∈ EN , then

(4.19)

ln−2 N ≤ jmin kjmin αk ≤ ln10 N ;

(4.20)

ln−1.1 T ≤ jmax kjmax αk ≤ ln10 N ;

(4.21)

jmax ≥

N . ln8 N

Here is the proof. The left side of (4.19) is because jmin ≤ T := N ln2 N by definition ∗ of O and by property (B) in the definition of EN . The right side is because we are in case (iv). Let ∆j := jmax − jmin , then ∆j ≤ jmax ≤ T , whence by property (B), 6 ∆jk(∆j)αk > ln−2 N . So ∆j > (k(∆j)αk ln2 N )−1 . In case (iv), k(∆j)αk ≤ lnNN , so ∆j ≥ lnN 8 N . Since jmax ≥ ∆j, (4.21) follows. The right side of (4.20) is by the definition of case (iv). The left side is because of (4.21), property (A) in the definition ∗ of EN , and because jmax ≤ T . ∗ We can now see that for every α ∈ EN , T − X 2 1 OﬀDiagiv (N ) 1[ln−1.1 T ≤jkjαk≤ln10 N ] ≤ 2 jkjαk ln2 N ln N N j= ln8 N

×

T X j=1

1 10 1 −2 jkjαk [ln T ≤jkjαk≤ln N ]

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3

1 2(ln ln T ) ln T × 2 (ln ln T )2 ln N

≤

×

1 ln T ln ln T

T X j=1

T X T j= ln10 T

1 1 −1.1 T ≤jkjαk≤ln10 N ] jkjαk [ln

1 . 10 1 −2 jkjαk [ln T ≤jkjαk≤ln N ]

The first term tends to zero (because T = N ln2 N ), and the second and third terms are tight by Lemma 4.3(b),(c). We can now proceed as before to obtain (4.18) for Di = OﬀDiag− iv . The summands OffDiag+ k , k = i, . . . , iv: These can be handled in the same way as OﬀDiag− , except that in cases (ii),(iii) and (iv) we need to apply properties (B) k and (C) to j1 + j2 instead of |j1 − j2 |. This is legitimate since j1 + j2 ≤ 2T . 4.3. Step 2 (Proposition 3.5) Proof of Proposition 3.5. — Fix δ > 0 small, α ∈ T \ Q, and N ≫ 1, and set P = P (δ, N ), R = R (δ, N ). Recall that Rp (δ)

:= {resonant harmonics in R , associated to p}.

R / ln N Then we have the following decomposition for Sn,T R

(4.22)

Sn,T

ln N

≡

1 X 1 X gj,n = ln N ln N j∈R

X

p∈P kp∈Rp (δ)

cos((kp)tn ) , 2πkp sin(πkpα)

where (4.23)

tn = (2n + 1)πα + 2πx.

To continue, we need the following observations on Rp (δ): Claim 1: Suppose p ∈ P , and Lp := min(⌊ 12 δp ln N ⌋, ⌊N ln2 N/p⌋). For every 1 ≤ k ≤ Lp (a) kp ∈ Rp (δ) and ((kpα)) = k((pα)); (b)

1 2πkp sin(πkpα)

=

1 phhpαii

1 2π 2 k2

+ O kpαk2 .

Proof. — Write pα = m + ((pα)) with m ∈ Z. If j = kp with 1 ≤ k ≤ Lp , then jα = km + k((pα)) and, since p is prime resonant, 1 . 2 Since α is irrational, this is a strict inequality, whence ((jα)) = k((pα)). This also shows that jα = km + ((jα)) whence r := gcd(j, km) = k, and j is associated to p ≡ j/r. To complete the proof that j ∈ Rp (δ) we just need to check that j ≤ N ln2 N . This is immediate from the definition of Lp . This proves (a). |k((pα))| = kkpαk ≤ Lp (δp ln N )−1 ≤

ASTÉRISQUE 415

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QUENCHED AND ANNEALED TLTS FOR CIRCLE ROTATIONS.

For (b), we write again pα = m + ((pα)) with m ∈ Z and note that sin(πkpα) = sin(πkm + π((kpα))) = (−1)m sin π((kpα)). We saw above that if k ≤ Lp , then ((kpα)) = k((pα)). So sin(πkpα) = (−1)m sin πk((pα)) = (−1)m πk((pα)) + O(k 3 kpαk3 ). Note that (−1)m ((pα)) = hhpαii because of the decomposition pα = m + ((pα)) above. So sin(πkpα) = πkhhpαii + O(k 3 kpαk3 ). Thus 1 + O k 2 ||pα||3 1 1 = = 2πkp sin(πkpα) 2π 2 k 2 phhpαii (1 + O (k 2 ||pα||3 )) 2π 2 k 2 phhpαii proving (b).

Claim 2. — If p ∈ P and kp ∈ Rp (δ), then

1 2πkp sin(πkpα)

=O

1

k2 pkpαk

.

Proof. — Write kpα = ℓ + ((kpα)) with ℓ ∈ Z. Since p is associated to kp, gcd(kp, ℓ) = k. We have kp ℓ kkpαk = |kpα − ℓ| = gcd(kp, ℓ) α− gcd(kp, ℓ) gcd(kp, ℓ) ℓ ≥ kkpαk. = k pα − gcd(kp, ℓ) In particular, kkpαk ≤ kkpαk ≤ πkkpαk = πkkpαk, whence

1 2.

Therefore kkpαk = kkpαk, and | sin(πkpα)| ≍ 1 = O k2 pkpαk .

1 2πkp sin(πkpα)

Claim 3. — Suppose p ∈ P , then {kp : k = 1, . . . , Lp } ⊂ Rp (δ) ⊂

kp : k = 1, . . . ,

N ln2 N p

.

Proof. — The first inclusion is by Claim 1(a), the second is because by definition, every j ∈ R is less than T = N ln2 N . P We now return to (4.22). Claims 2 and 3 say that the inner sum is kp∈Rp (δ) gkp,n = PN ln2 N/p 1 PLp 2 1 k=Lp +1 k2 . If Lp = ⌊N ln N/p⌋, then the error term k=1 gkp,n + O pkpαk 1 term can be found by sumvanishes. If not, then Lp = the error ⌊ 2 δp ln N ⌋ and PLp P 1 1 mation to be equal to O δpkpαk · p ln N . Thus kp∈Rp (δ) gkp,n = k=1 gkp,n + 1 1 O pkpαk · p ln N (where the implied constant is not uniform in δ).

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Applying Claim 1 to

PLp

k=1 gkp,n

we obtain (recall (4.23))

Lp X 1 1 X cos((kp)t ) 1 n = + O(kpαk2 ) + O ln N ln N phhpαii 2π 2 k 2 p ln N k=1 p∈P Lp X 1 X cos((kp)t ) 1 1 n . = + O(kpαk2 Lp ) + O ln N phhpαii 2π 2 k 2 p ln N

R

Sn,T

k=1

p∈P

1 δp ln N so kpαk2 Lp ≤ δ22p2 ln2 N = O p ln1 N . Thus Lp R X X Sn,T 1 cos((kp)tn ) 1 . = +O ln N phhpαii ln N 2π 2 k 2 p ln N

For p ∈ P , kpαk

1 and a small real number ε > 0. Claim 1. — There exist δ > 0, N0 ≥ 1 s.t. for all N > N0 , Ω(N, δ, r) := {α ∈ T : |P (δ, N )| ≥ r} has measure bigger than 1 − ε. Proof. — Proposition 3.6 says that {jhhjαii ln N : j ∈ P (δ, N )} converges as a point dist process to a Poisson point process with density π62 on [− 1δ , 1δ ]. So #P (δ, N ) −−−−→ N →∞

Poisson distribution with expectation π12 2 δ . If we choose δ small enough, the probability that this random variable is less than 2r is smaller than 2ε . So the claim holds with some N0 . PN Let µα,r,N := N1 n=1 δ(j1 tn ,...,jr tn ) where tn = (2n+1)πα+2πx (see (4.23)) and fix some weak-star open neighborhood V of the normalized uniform Lebesgue on [0, 2]r . We will construct NV s.t. for all N ≥ NV , mes (α ∈ Ω(N, δ, r) : µα,r,N ∈ V) > 1 − 2ε, where jk := jk (α, δ, N ) enumerate P (δ, N ) as in Proposition 3.7.

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Without loss of generality, V = and fν (x1 , . . . , xr ) = eiπ l

(ν)

Pr

:=

(ν) k=1 lk xk

Tν0

ν=1 {µ

:|

R

[0,2]r

fν dµ| < ε} where ε > 0, ν0 ∈ N,

, where

(ν) (l1 , . . . , lr(ν) )

∈ Zr \ {0} (1 ≤ ν ≤ ν0 ).

This is because any weak star open neighborhood of Lebesgue’s measure on [0, 2]r contains a neighborhood of this form. Let Lν := Lν (α, δ, N ) :=

r X

(ν)

lk jk .

k=1

Claim 2. — We have mes (α ∈ Ω(N, δ, r) : Lν = 0 for some 1 ≤ ν ≤ ν0 ) −−−−→ 0. N →∞

Proof. — Proposition 3.6 says that the sequence jk is superlacunary in the sense that for each R max(jk′ , jk′′ ) ′ ′′ ≥ R −−−−→ 1. (4.24) mes α : ∀1 ≤ k < k ≤ r : N →∞ min(jk′ , jk′′ ) To see this recall that the gaps between neighboring points of a Poisson process have an exponential distribution, therefore for any ε there exists δ such that the gaps between ln jk′ and ln jk′′ are, with probability 1 − ε, bounded below by δ ln N. (ν) Let kν∗ = arg max(jk : lk 6= 0). Applying (4.24) with X (ν) |lk |, R = Rν := 2(r − 1) k

P (ν) we see that for all N large enough, with almost full probability, jkν∗ ≥ 2 k6=k∗ |lk jk |, whence Lν 6= 0. This proves the claim. Notice that this argument also shows that with almost full probability, (ν) |Lν | ≤ 2lk∗ jkν∗ . Claim 3. — We have mes α ∈ Ω(N, δ, r) : ∃ν ≤ ν0 s.t. kLν α/2k ≤ lnNN −−−−→ 0. N →∞

{ lnlnNj

Proof. — By Proposition 3.6, : j ∈ P (δ, N )} converges as a point process to a Poisson point process with intensity π12 2 δ on [0, 1], therefore mes α ∈ Ω(N, δ, r) : jkν∗ > lnN 4N ln N ≤ mes α ∈ T : ∃j ∈ P (δ, N ) s.t. lnlnNj > 1 − 4 ln −−−−→ 0. ln N N →∞

Therefore for all N large enough, with almost full probability in Ω(N, δ, r), jkν∗ ≤ N/ ln4 N , whence (for N large enough) also 1 ≤ |Lν | ≤ 2lkν∗ jkν∗

ln−2 N for all 1 ≤ j ≤ N ) −−−−→ 1. N →∞

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It follows that for all N large enough, with almost full probability in Ω(N, δ, r), Lν kLν α/2k > 1/ ln2 N , whence kLν α/2k >

1 > Lν ln2 N

N ln3 N

1 ln N = , N · ln2 N

proving the claim.

Fix NV s.t. for every N > NV , for each 1 ≤ ν ≤ ν0 , kLν α/2k > ln N/N with almost full probability in Ω(N, δ, r). Make NV so large that ln 1NV < ε. Then for every N > NV Z N X 1 2 1 πiL (α(2n+1)+2x) ν fν dµα,r,N = e ≤ N 2πiL ν α| N |1 − e n=1

=

1 1 1 = ≤ < ε, ln N N kLα/2k ln N N· N

whence µα,r,N ∈ V as required.

Appendix A

Proof of Proposition 1.1 Suppose f : T → R is diﬀerentiable on T\{x1 , . . . , xN }, and f ′ extends to a function of bounded variation on T. Since f ′ has bounded variation, f ′ is bounded. So f is + Lipschitz between its singularities. It follows that L− i := lim− f (t), Li := lim+ f (t) t→xi

t→xi

exist for each i. R PN − Let ϕ(t) := f (t) − i=1 (L+ i − Li )h(t + x1 ) − T f (s)ds. It is easy to see that ϕ| extends to a continuous function ψ on T s.t. T\{x1 ,...,xN } ψ(t) = f (t) −

N X i=1

(L+ i

−

L− i )h(t

+ x1 ) −

Z

f (s)ds

T

for every t ∈ T \ {x1 , . . . , xN }. (But maybe the identity breaks at xi .) By construction, ψ ∈ C(T) and ψ ′ |T\{x1 ,...,xN } = f ′ |T\{x1 ,...,xN } extends to a function with bounded variation on T. P Expand ψ to a Fourier series: ψ(x) = k∈Z−0 ak e2πikx . Our assumptions imply ak = O(k −2 ). A formal solution of ψ = κα − κα ◦ Rα gives X ak κα (x) ∼ bk,α e2πikx , where bk,α = . 1 − e2πikα k∈Z\{0}

P We claim that for a.e. func α, k |bk,α| < ∞ so that κα is a well-defined continuous P∞ |ak | 1 1 = O k2 ||kα|| so it suﬃces to check that k=1 k2 ||kα|| tion. Indeed bk,α = O ||kα|| converges for a.e. α.

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By Khinchine theorem for a.e. α we have ||kα|| > k −1.1 for all large k, so P∞ 1 k=1 k2 ||kα|| converges for a.e. α iﬀ ∞ X k=1

1

−1.1 ] 1 k 2 ||kα|| [||kα||>k

converges for a.e. α. This is indeed the case, because 1.1 Z X ∞ X 1 1 k −1.1 < ∞. 1 dα = · 2 ln ] 2 ||kα|| [||kα||>k 2 k k 2 T

k=1

Appendix B Cauchy and Poisson Let Ξ = {ξn } be a Poisson process on R with intensity c and let Θn be i.i.d bounded random variables with zero mean independent of Ξ. Note that the set of pairs {(ξm , Θm )} forms a marked Poisson process on R × R (see [17, Section 5.3] for background on marked Poisson processes). Accordingly one can apply the exponential formula for marked Poisson processes ([17, page 42]) which says that if u is a bounded continuous function on [L1 , L2 ] × R and t ∈ R then # " Z L2 X tu(y,Θ) EΘ e − 1 dy . (B.1) E exp tu(ξm , Θm ) = exp c L1

ξm ∈[L1 ,L2 ]

Note that computing the first and second derivative of (B.1) with respect to t at t = 0 we obtain Z L2 X (B.2) EΘ (u(y, Θ)) dy, u(ξm , Θm ) = c E L1

ξm ∈[L1 ,L2 ]

(B.3)

E

X ξm ∈[L1 ,L2 ]

u2 (ξm , Θm ) = c

Now consider YL =

Z

L2

EΘ u2 (y, Θ) dy.

L1

X Θn . ξn

|ξn |