Dynamical Systems of Algebraic Origin [1st ed.] 978-3-7643-5174-8;978-3-0348-9236-0

Table of contents :
Front Matter ....Pages i-xviii
Group actions by automorphisms of compact groups (Klaus Schmidt)....Pages 1-33
ℤd-actions on compact abelian groups (Klaus Schmidt)....Pages 35-75
Expansive automorphisms of compact groups (Klaus Schmidt)....Pages 77-92
Periodic points (Klaus Schmidt)....Pages 93-104
Entropy (Klaus Schmidt)....Pages 105-149
Positive entropy (Klaus Schmidt)....Pages 151-219
Zero entropy (Klaus Schmidt)....Pages 221-259
Mixing (Klaus Schmidt)....Pages 261-283
Rigidity (Klaus Schmidt)....Pages 285-300
Back Matter ....Pages 301-310

Citation preview

Progress in Mathematics Volume 128

Series Editors J. Oesterle A. Weinstein

Klaus Schmidt

Dynamical Systems of Algebraic Origin

Birkhauser Verlag Basel· Boston· Berlin

Author: Prof. Klaus Schmidt Mathematisches Institut Universitat Wien Strudlhofgasse 4 1090 Vienna Austria

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bibliothek Cataloging-in-Publication Data

Schmidt, Klaus: Dynamical systems of algebraic origin / Klaus Schmidt. Basel ; Boston ; Berlin : Birkhauser, 1995 (Progress in mathematics ; Vol. 128) ISBN-13: 978-3-0348-9957-4 DOT: 10.1007/978-3-0348-9236-0

e-ISBN-13: 978-3-0348-9236-0

NE:GT

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 1995 Birkhauser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Printed on acid-free paper produced of chlorine-free pulp Softcover reprint of the hardcover I st edition 1995

987654321

To Annelise

Contents

Introduction .........................................................

ix

Chapter I. Group actions by automorphisms of compact groups 1.

Ergodicity and mixing .............................................

1

2.

Expansiveness and Lie subshifts ...................................

9

3.

The descending chain condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

4.

Groups of Markov type ............................................

23

Chapter II. Zd-actions on compact abelian groups 5.

The dual module..................... .............................

35

6.

The dynamical system defined by a Noetherian module ............

43

7.

The dynamical system defined by a point ..........................

60

8.

The dynamical system defined by a prime ideal ....................

70

Chapter III. Expansive automorphisms of compact groups 9.

Expansive automorphisms of compact connected groups ............

77

10. The structure of expansive automorphisms .........................

83

Chapter IV. Periodic points

93

11. Periodic points of Zd-actions 12. Periodic points of ergodic group automorphisms vii

101

viii

CONTENTS

Chapter V. Entropy 13. Entropy of Zd-actions

105

14. Yuzvinskii's addition formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 114 15. Zd-actions on groups with zero-dimensional centres ................ 121 16. Mahler measure ................................................... 125 11'. Mahler measure and entropy of group automorphisms .............. 129 18. Mahler measure and entropy of Zd-actions

139

Chapter VI. Positive entropy 19. Positive entropy ................................................... 151 20. Completely positive entropy

162

21. Entropy and periodic points

174

22. The distribution of periodic points ................................. 193 23. Bernoullicity ...................................................... 196

Chapter VII. Zero entropy 24. Entropy and dimension ............................................ 221 25. Shift-invariant subgroups of (ZjpZ)Z2 .............................. 228 26. Relative entropies and residual sigma-algebras

241

Chapter VIII. Mixing 27. Multiple mixing and additive relations in fields

261

28. Masser's theorem and non-mixing sets

268

Chapter IX. Rigidity 29. Almost minimal Zd-actions and invariant measures

285

30. Cohomological rigidity ............................................ 293 31. Isomorphism rigidity .............. ,............................... 297

Bibliography ......................................................... 301 Index ................................................................. 306

Introduction

Although the study of dynamical systems is mainly concerned with single transformations and one-parameter flows (i.e. with actions of Z, N, JR, or JR+), ergodic theory inherits from statistical mechanics not only its name, but also an obligation to analyze spatially extended systems with multi-dimensional symmetry groups. However, the wealth of concrete and natural examples, which has contributed so much to the appeal and development of classical dynamics, is noticeably absent in this more general theory. A remarkable exception is provided by a class of geometric actions of (discrete subgroups of) semi-simple Lie groups, which have led to the discovery of one of the most striking new phenomena in multi-dimensional ergodic theory: under suitable circumstances orbit equivalence of such actions implies not only measurable conjugacy, but the conjugating map itself has to be extremely well behaved. Some of these rigidity properties are inherited by certain abelian subgroups of these groups, but the very special nature of the actions involved does not allow any general conjectures about actions of multi-dimensional abelian groups. Beyond commuting group rotations, commuting toral automorphisms and certain other algebraic examples (cf. [39]) it is quite difficult to find non-trivial smooth Zd-actions on finite-dimensional manifolds. In addition to scarcity, these examples give rise to actions with zero entropy, since smooth Zd-actions with positive entropy cannot exist on finite-dimensional, connected manifolds. Cellular automata (i.e. shift-commuting homeomorphisms of a sub shift of finite type) also generate zero-entropy Z2-actions, but any attempt to define a reasonably wide class of cellular automata, and to prove non-trivial results about that class, appears to run into logical quicksand (cf. e.g. [31]). The same applies more generally to higher-dimensional subshifts of finite type, where one can easily find examples with positive entropy, but where dynamical questions about even the simplest systems can lead to surprising difficulties (cf. e.g. [25],

[14], [15]).

ix

INTRODUCTION

x

The purpose of this book is to help remedy this lack of examples by introducing a class of continuous Zd-actions on compact, metric spaces which is diverse enough to exhibit many of the new phenomena encountered in the transition from Z to Zd, but which nevertheless lends itself to systematic study: the Zd-actions by automorphisms of compact, abelian groups. One aspect of these actions, which is a priori not surprising, but is quite striking in its extent and depth, is their connection with commutative algebra and arithmetical algebraic geometry. The algebraic framework resulting from this connection allows the construction of an unlimited supply of examples with specified dynamical properties, and by combining algebraic and dynamical tools one obtains a sufficiently detailed understanding of this class of Zd-actions to glimpse at least the beginnings of a general theory. Before describing the contents of this book in any detail I should mention a specific example, which-together with its ramifications-has provided much of the motivation for the work presented here. This example is due to Ledrappier [56], and consists of the shift-action (j of Z2 on the closed, shift-invariant subgroup

of the full two-dimensional two-shift. It is clear that (j has zero entropy, since the individual automorphisms (j m, m E Z2, have finite entropy. Ledrappier proved that (j is mixing (cf. Theorem 6.5 in this book), but observed that, for every x E X and k ;:::: 1, X(O,O) + X(2k,O) + X(O,2k) = 0 (mod 2). In particular, (j cannot be mixing of order 3. Much of Section 28 will be devoted to this kind of breakdown of higher order mixing for certain mixing Zd-actions. There is nothing special about the 'alphabet' Z/2Z of the group X in (0.1): one can replace it by any finite, abelian group G. Are the shift-actions (j = (j(G) on the resulting subgroups of X = X(G) C G7!,2 measurably conjugate for different choices of G? By considering the entropies of the individual shifts (j~), m E Z2, one sees immediately that the cardinality IGI of G is a measurable conjugacy invariant of (j(G): if G, G' are finite (abelian) groups such that (j(G) is measurably conjugate to (j(G'), then IGI = IG'I. Can anything more be said? We refer to the Sections 25-26 for further discussion. What happens if G is replaced by an uncountable group, like '][' = lR/Z? If we define X(1r) exactly as in (0.1) by X CIf) -_ { X -_ (Xm ) E

'JT'Z2. Jl

=0

•

+ X(ml +l,m2) + X(ml ,m2+l) (0.2) for every m = (ml' m2) E Z2},

X(ml,m2)

(mod 1)

what can be said about the shift-action (jell") of Z2 on XClI")? Is it mixing of all orders? Does it again have zero entropy? As it turns out, (jell") is not only mixing

INTRODUCTION

xi

(Theorem 6.5), but mixing of all orders (Theorem 27.3), and Proposition 19.7 shows that its entropy is positive and given by

where

L(s, X3)

=

f X3~n)

n=l

n

is Dirichlet's L-function associated with the character if n if n if n

== 0 (mod 3), == 1 (mod 3), == 2 (mod 3).

In fact, aCT) is Bernoulli (Theorem 23.1), and hence measurably conjugate to the shift-action a' of Z2 on the subgroup {X = X '=

(xm ) E']]' Z2 : X(ml,m2)

=0

+ X(ml-I,m2) + x(ml,m2+ 1)

(mod 1) for every m

=

(mI,m2) E Z2}.

(0.3)

However, its zero-dimensional cousin (0.1) does not allow any such change in its defining rule (Proposition 25.7 and Examples 25.8 (1)-(2)), and exhibits quite extraordinary rigidity properties (Chapter 9). Readers who get bored with abelian groups can carry the problem one step further and consider the shift-action a(SU(2)) of Z2 on the closed, shift-invariant subset X(SU(2))

=

{x

=

(xm)

E SU(2)Z2 : x(ml,m2) . x(ml +1,m2) . X(ml,m2+ 1 )

= 1 for every m = (mI, m2)

E Z2},

where 1 is the identity element in the group SU(2). What can be said about the dynamical properties of a(SU(2))? It must have positive entropy, but what is its value? Is a(SU(2)) again Bernoulli? Nothing appears to be known about this problem. This brief account of Ledrappier's example and the questions raised by it should convey the strong emphasis of this book on explicit examples, which serve not only as illustrations of definitions, results and techniques, but which provide motivation for much of the material presented here. The contents are organized as follows. Chapter I contains general background on actions of a countable group r by automorphisms of a compact (and always metrizable) group X. In Section 1 we review the connections between elementary spectral and dynamical properties of these actions and prove that such an action is ergodic if and only if it is topologically transitive (Theorem

xii

INTRODUCTION

1.1). Sections 2-4 introduce certain finiteness conditions for r-actions by automorphisms of compact groups. In Proposition 2.2 we establish that the first of these conditions, expansiveness, implies another finiteness condition, conjugacy to a Lie subshift: every expansive action a of an infinite, discrete group r by automorphisms of a compact group X is topologically and algebraically conjugate to the shift-action of r on a closed, shift-invariant subgroup Y c G r , where G is a compact Lie group. Furthermore, if X is connected, then G may be chosen to be a finite-dimensional torus (Theorem 2.4). For Z-actions, conjugacy to a Lie subshift is equivalent to a condition originally introduced in [84] (Proposition 2.15 and Corollary 2.16). The third finiteness condition, in Section 3, is the descending chain condition (d.c.c.): the action a of r on X satisfies the descending chain condition if every strictly decreasing sequence of closed, a-invariant subgroups of X is finite. The d.c.c. also implies conjugacy to a Lie subshift (Proposition 3.3), but not necessarily expansiveness, and makes life easier in a number of ways (Proposition 3.5 and Theorem 3.6). The most crucial aspect of the d.c.c. is, however, its connection with subshifts of finite type, which is explained in Definition 3.7 and Theorem 3.8. In Section 4 we discuss those countable groups r for which every action on a Lie subshift has the d.c.c., and prove that Zd has this property for every d ;::: 1 (Theorem 4.2 and Examples 4.8). In Chapter II we concentrate on Zd-actions by automorphisms of compact, abelian groups and their connection with commutative algebra. In Section 5 we show that every Zd-action a by automorphisms of a compact, abelian group X defines a dual module 9J1 = X over the ring of Laurent polynomials !Rd = Z[u~l, ... , ujl], and that this module is Noetherian whenever a is expansive or, more generally, whenever a satisfies the d.c.c. (Lemma 5.1 and Proposition 5.4). In Ledrappier's example (0.1) this module turns out to be given by 9J1 = !R2/(2,1 + Ul + U2), where (2,1 + Ul + U2) is the ideal 2!R2 + (1 + Ul + U2)!R2 C !R2, and in (0.2) 9J1 = !R2 /(1 + Ul + U2). Conversely, if 9J1 is an !Rd-module, then 9J1 defines a Zd-action a = aOOl byautomorphisms of the compact, abelian group X = Xool = 5Jt (Lemma 5.1). The extent of this correspondence between modules and Zd-actions is discussed in Theorem 5.9 and Corollary 5.10. Section 6 concentrates on Zd-actions corresponding to Noetherian modules and provides the first entries in a 'dictionary' which translates the dynamical properties of an action a on a compact, abelian group X into algebraic properties of the dual module 9J1 = X. Figure 1 illustrates the nature of this dictionary. In the second column of this table we assume that the !Rd-module 9J1 = X defining a is of the form !Rd/P, where P c !Rd is a prime ideal, and describe the algebraic condition on P equivalent to the dynamical condition on a = a!Jtd/p appearing in the first column. In the third column we consider a general (countable) !Rd-module 9J1, and state the algebraic property of 9J1 corresponding to the dynamical condition on a = aOOl in the first column. As can be seen from the entries in this column, these conditions are in all cases

xiii

INTRODUCTION

expressed in terms of the prime ideals associated with the module 9Jl, so that many of the dynamical properties of a!m are, in fact, determined by the dynamics of the Zd-actions a!R d / P , where p varies over the prime ideals associated with 9Jl. The last column gives reference to some of the relevant results in this text. Property of a a satisfies the descending chain condition a is expansive

a = a!Rd/P: property of p Always true

The set of a-periodic points is dense

Always true

Vc(p) nSd

=0

IFixA(a)1 < 00 for a subgroup A C Zd of finite index a is ergodic or topologically transitive a is strongly mixing

Vc(p) n {c E Cd: en = 1 for every n E A} = 0

a is mixing of every order

Either p is equal to p~ for some rational prime ~' or p n Z = {O} and a!Xd p is strongly mixing p is principal, and not generated by a generalized cyclotomic polynomial p f= {O}

h(a)

>0

h(a)

< 00

a has completely positive entropy a is Bernoulli a has a unique measure of (finite) maximal entropy

{u kn - 1: n E Zd} every k 2: 1

if-

p for

un - 1 f. p for every non-zero n E Zd

h(a!Xd/p)

>0

a!Xd/p has completely positive entropy o < h(a!Xd/p) < 00

a = a!m: property of 9J1 9J1 is Noetherian

Reference Propn.5.4

9J1 is Noetherian and a!Xd/p is expansive for every prime ideal p associated with 9J1 9J1 is Noetherian

Thm.6.5

9J1 is Noetherian and lFixA(a!Xd/p) I < 00 for every prime ideal p associated with 9J1 a!Xd/p is ergodic for every prime ideal p associated with 9J1 a!Xd/p is strongly mixing for every prime ideal p associated with 9J1 For every prime ideal p associated with 9J1, a!Xd/p is mixing of every order h(a!Xd/p) > 0 for at least one prime ideal p associated with 9J1 If 9J1 is Noetherian: p f= {O} for every prime ideal p associated with 9J1 h(a!Xd/p) > 0 for every prime ideal p associated with 9J1 a!m has completely positive entropy a!m has completely positive entropy and h(a!m) < 00

Thm.5.7

Thm.6.5

Thm.6.5, Lemma 6.6 Thm.6.5, Lemma 6.6 Thm.27.2

Propn. 19.4, Thm.19.5 Propn. 19.4

Thm.20.8

Thm.23.1 Thm.20.15

FIGURE 1

Apart from the results referred to in Figure 1, Section 6 contains a number of other consequences of this interplay between algebra and dynamics: if an ergodic Zd-action a on a compact, abelian group X satisfies the d.c.c., then there exists an n E Zd such that an is ergodic (Corollary 6.10); if a is an expansive

xiv

INTRODUCTION

zd-action by automorphisms of a compact, connected, abelian group X, then a is ergodic (Corollary 6.14); finally, if a is an expansive Zd-action by automorphisms of a compact, abelian group X, and if Y c X is a closed, a-invariant subgroup, then the action induced by a on the quotient group X/Y is again expansive (Corollary 6.15). In Section 7 we consider the dynamical systems arising from prime ideals I' C 91: d whose variety Vc(p) is finite, which provide the 'building blocks' for all Zd-actions by automorphisms of finite-dimensional tori and solenoids, and express the dynamics of a 9td / P in terms of Vc(p). This dynamical interpretation of V"x(x), where Xx = X for every x E X, and where the isomorphism of}C and XX d>"x(x) is given by f f-+ f(x) d>"x(x) for every f E }C. If the dimension of X is equal to k, then we can use Gram-Schmidt orthonormalization to find elements h, ... ,fk in }C such that h (x), ... , fk(X) is an orthonormal basis of X for every x E X, and we can use these vectors to calculate the trace T(x) (y) = tr( T/(x) (y)) for all x E X, Y E Y. Let [.,.] be the inner product on L2(y, >..y). Then [T(x)(.), T(x')(-)] is equal to the dimension of the space if linear operators A: X 1---+ X with AT/(x)(y) = T/(x')(y)A for all y E Y (cf. [1]). In particular, [T(x)(.), T(x')(.)] is a non-negative integer, and T/(x) and T/(x') are unitarily inequivalent if and only if 1 + [T(x) (-), T(x') (.)] ::::: [T(x) (.), T(x) (.)] = [T(x') (.), T(x') (. )]. We fix c > 0 and use Lusin's theorem (cf. [88]) to find a subset of positive measure B C X such that >..y( {y E Y : Ilh(xy) - h(x'y)11 < c}) > 1 - c for all i = 1, ... ,s and x, x' E B, where 11·11 denotes the norm on X. If c is sufficiently small we obtain that [T(x)(.),T(x')(.)] = [T(x)(-),T(x)(.)] for all x E B, i.e. that T/(x) and T/(x') are unitarily equivalent for all x, x' E B. Hence the set X'1) = {x EX: T/(x) is unitarily equivalent to T/} has positive measure, since it contains X , - 1 B for every x' E B. As X'1) is a closed, normal subgroup of X with positive Haar measure, X/ X'1) must be finite. We claim that X'1) is a-invariant. Indeed, if x E X'1)' then there exists a unitary operator V on X such that T/(x-1yx) = V-1T/(y)V for every y E Y. According to (1.1) we have, for every 'Y E rand y E Xc,

J:

J:

J:

J:

T/(a.:;l(x-lyX)) = U:;1T/(X-1yx)U-y = U:;lV-1T/(y)VU-y - -U-1V-1U U- ( )U U-1VU-y y -y-yT/Y-y-y 1

= u:;lv-1U-yT/(a.:;1(y))U:;lVU-y, so that T/(a:;l(x)) is unitarily equivalent to T/. We write 'ljJ: X 1---+ X/ X'1) for the quotient map and choose a map c': X/X'1) 1---+ X with cl(lx/x~) = Ix and 'ljJ. c'(v) = v for every v E X/X'1). Put i) = EBvEX/X~ T/(c'(v)), and observe that i) is a unitary representation of Y on the finite-dimensional Hilbert space £., = XX/x~, and that W = ker(i)) is a closed subgroup of Y which is normal not only in Y, but also in X. Furthermore, since i) is finite-dimensional, Y /W ~ i)(Y) is a Lie group. From (1.1) it is clear that there exists, for every 'Y E r, a unitary operator if,,! on £., such that

7

1. ERGODICITY AND MIXING

V')'-li](X)V')' = i](a,;-l(x)), and this implies (2) as in Lemma 1.2. The reverse implication follows from Lemma 1.2. 0 THEOREM 1.4. Let r be a countable group, and let a be an action of r by automorphisms of a compact group X. Then there exists a countable ordinal wand a collection {~ : ~ < w} of closed, normal, a-invariant subgroups of X, indexed by the set of ordinals ~ < w, with the following properties:

(1) Vo=X;

(2) If 0 ::; ~ < ~ + 1 < w then V~+1 yo, Z = Y n ker( fJ) is abelian, and Y I Z is zero-dimensional. The homomorphism 1f{ld: Y I--> C induces a surjective homomorphism from the zero-dimensional group YIZ to the Lie group CIA, and it follows that CIA is zero-dimensional and hence finite. This shows that Co C A, and the connectedness of A implies that Co = A, i.e. that Co is abelian. 0 We begin the proof of Theorem 2.4 with an approximation argument which we shall also require for the discussion of periodic points in Chapter 4. LEMMA 2.6. Let H be a compact group, K C H a normal subgroup, and let x: K I--> § be a continuous homomorphism which is central, i. e. which satisfies that X(hkh- 1) = x(k) for every k E K, h E H. Then there exists a homomorphism x: H I--> § and an integer 1 ::::: 1 such that X(k) = X(k)l for every k E K. PROOF. Let e: H I--> HI K be the quotient map, choose a Borel map c: HIK f---+ H with c(x) = x for every x E HIK, and put b(h,x) = C(hX)-lhc(x) for every h E H, x E HI K (cf. [78], Lemma 1.5.1). We denote by T the unitary representation

e.

(T(h)f)(x)

= X(b(h,h-1x»f(h-1x),

hE H, f E Je, x E HIK

of H on the Hilbert space Je = L2(HI K, )..H/K) induced by x. Since X is central, T(k) = X(k)I for every k E K, where I is the identity operator on Je. Choose an irreducible subrepresentation T' of T on a subspace X C Je and put X(h) = det(T'(h» for every h E H. Then x: H l---+ § is a continuous homomorphism, and X(k) = X(k)l for every k E K, where 1 is the dimension ofX. 0

12

1. GROUP ACTIONS BY AUTOMORPHISMS OF COMPACT GROUPS

LEMMA 2.7. Let A = ']['n, n ~ 2, F C Aut(A) a finite group, and let

{O A} =I- B

~ A be a closed, connected, F -invariant subgroup. Then there exists a closed, connected, F -invariant subgroup C C A such that B + C = A and B n C is finite.

PROOF. For every f3 E F we consider the dual automorphism /3 E Aut(A) = GL(n,Z) C GL(n,IR), and we set [u, v] = L,BEF(/3(U),/3(v)), u, v E IRn , where (.,.) denotes the Euclidean inner product on IRn. The annihilator B1. = {n E zn = 'fri : (b, n) = 1 for every b E B} c zn C IRn of B spans a subspace S C IRn of dimension m with 1 ::; m < n. The subspace S* = {v E IRn : [u, v] = 0 for all u E S} is obviously invariant under F = {/3 : f3 E F}, and we can find a finite, F-invariant subset T C S* nzn which spans S* (note that F C GL(n, Z) C GL(n, 1R)). If 3 c zn is the subgroup generated by T then B1. + 3 has finite index in zn, B1. n 3 = {O}, and the connected component C of 31. in A is an F-invariant, closed, connected subgroup of A such that B n C is finite and B + C = A. 0

LEMMA 2.8. Let H be a compact group, and let A C H be a closed, normal, abelian Lie group. Then there exists a closed, normal subgroup H' C H such that H' n A is finite, H' . A = H, and H / H' is abelian. PROOF. It obviously suffices to prove the lemma with AO replacing A, and we assume for simplicity that A itself is connected. Let G C H be the centralizer of A, i.e. G = {h E H : ha = ah for every a E A}. Since Aut(A) is discrete and X is compact, the quotient group K = H/G is finite, with cardinality n, say. As we are now assuming that A is connected, A ~ ']['r for some r ~ o. If r = 0 the lemma is trivial; if r ~ 1, we choose characters Xl, ... ,Xr in A such that the map a f-+ x(a) = (Xl (a), ... , Xr(a)) from A into §r is bijective. Lemma 2.6 implies the existence of an integer l ~ 1 and of continuous homomorphisms Xi: G ~ § such that Xi(a) = Xi(a)l for every a E A and i = 1, ... , r, and we set X = (Xl' ... , Xr): G ~ §r. Denote by ¢: H ~ K the quotient map, choose a map c: K ~ H with ¢. c(k) = k for every k E K, and set b(h, k) = c(hk)-lhc(k), h E H, k E K. Let 7 be the unitary representation of H induced by the representation X of G on ((7. Then 7 acts on the Hilbert space 9{ = £2 (Kt and can be written as 7 = 71 EEl··· EEl 7 r , where

(2.4) for every h E H, f E £2(K), k E K. For every k E K we write ek for the unit vector in £2(K) given by ek(k' ) = 1 if k = k', and ek(k' ) = 0 otherwise. Then {ek : k E K} is an orthonormal basis of £2(K), and we identify each 7i(h) with its representation as an n x nmatrix in this basis. This allows us to view 7 as a continuous homomorphism 7 = 71 EEl··· EEl 7 r : H ~ U(nt C U(nr), where U(m) is the group of unitary

2. EXPANSIVENESS AND LIE SUB SHIFTS

13

m X m-matrices. The restriction of T to A has finite kernel B = ker(T) n A, and T(a) E D(nr) for every a E A, where D(m) C U(m) denotes the subgroup of diagonal matrices. From (2.4) it is clear that there exist, for every h E H, unique matrices Ph and Dk) , i = 1, ... ,r, in U (n) such that Ph is a permutation matrix, D~i) is diagonal, and Ti(h) = D~i) . Ph for i = 1, ... , d. We set Dh = D~l) Ef) ... Ef)Dt) c D(nr), Qh = Ph Ef). "Ef)Ph , and observe that T(h) = Dh ·Qh, and that the map h f--> Qh is a continuous homomorphism from H into the group of permutation matrices in U(nr). For every h E H we denote by f3h the automorphism of D(nr) given by f3h(D) = T(h)DT(h)-l = QhDQ'f/. Since the group F = {f3h : h E H} c Aut(D(nr)) is finite and r ~ T(A) = {T(a) : a E A} c D(nr) ~ TnT is a closed, connected, F-invariant subgroup, Lemma 2.7 implies the existence of a closed, connected, F-invariant subgroup C C D(nr) such that C· T(A) = D(nr) and C n T(A) is finite. Put Q = {Qh : h E H} and A = T(H)· D(nr) = Q. D(nr) C U(nr). Then C and Q . C are normal subgroups of A, and we write ~: A f---t AI Q . C for the quotient map. Since Q . C . T(A) = A we can find, for every h E H, an a E A with ha E H' = ker(~ . T) C H, and we conclude that H' . A = H, and that HI H' is therefore abelian. Finally, since H' n A c C n A, H' n A is finite. 0 LEMMA 2.9. Let H be a compact Lie group. Then there exists an increasing sequence of closed, normal subgroups Hn C H such that Un>l Hn is dense in H, C(H~) is finite, HIHn is abelian, and Hn::) {a E C(H) :an = IH} for every n 2 1.

PROOF. Apply Lemma 2.8 with A = C(HO) to find a subgroup K such that K· C(HO) = Hand C(KO) = K n C(HO) is finite, and put Hn = {h E H : hn! E K} for every n 2 1. 0 LEMMA 2.10. Let A be a compact, abelian Lie group, X a closed, shiftinvariant subgroup of Ar , and let x(m) = {x EX: xm = Ix}, m 2 1. Then Um2':l x(m) is dense in X.

PROOF. Since A is isomorphic to a closed subgroup of Tn for some n 2 1, the dual group X is isomorphic to a quotient group of the direct sum of copies of indexed by f, and we can find a decreasing sequence (Vm, m 2 1) of subgroups of X such that X IVm is finite for every m 2 1 and nm>l Vm = {O}. Put Bm = V~ eX. Then Bm is finite, and Um>l Bm is dense in X. For every m 2 1 we can find an m ' 2 1 such that Bm x(m'), and this proves that Um2':l x(m) is dense in X. 0

zn,

C

LEMMA 2.11. Let H be a compact Lie group, and let X C H r be a full, shift-invariant subgroup. Then there exists an increasing sequence (Hn, n 2 1) of closed, normal subgroups of H such that C(H~) is finite for every n 2 1, and Un2':l Xn is dense in X, where Xn = X n H;.

14

I. GROUP ACTIONS BY AUTOMORPHISMS OF COMPACT GROUPS

PROOF. We choose (Hn' n ~ 1) as in Lemma 2.9, denote by 'l/Jn: H f----> H / H n the quotient map, and define a shift commuting homomorphism 'I/J n: H r f----> (H/Hn)r by ('l/Jn(x»-y = 'l/Jn(X-y) , for all x E X and, E r. Then Xn = ker('l/Jn), and 'l/JI embeds X/Xl injectively in (H/Hd r . An application of Lemma 2.10 shows that Un~l Xn is dense in X. D LEMMA 2.12. Let G be a compact, connected Lie group with finite centre,

f----> G' = G/G(G) the' quotient map, and define the shift commuting, surjective homomorphism 0: G r f----> G,r as in (2.3). If X c G r is a closed, shift-invariant, expansive subgroup, then O(X) C G,r is again expansive.

B: G

PROOF. Let 8 be a metric on G and denote by 8' the induced metric on G'. Since X is expansive there exists an e:l > 0 such that x = Ix = Ier whenever x E X and 8(x-y, Ie) < e:l for every, E r. Put e:2 = min{8(g, Ie) : Ie =Ig E G(G)}, let M denote the order of G(G), and choose e:3 > 0 such that {g E G : gM = Ie and 8(g, Ie) < e:3} = {Ie}. For every e: > 0 we put

X(e:)

=

{x EX: 8(x-y, G(G»)

< e: for every, E r}.

If e: < e:2/2 there exists, for every x E X(e:), a unique z(x) E G(G)r such that 8(x-y, z(x)-y) < e: for every, E r. If e: < e:2/4 then z(xx') = z(x)z(x') for all x,x' E X(e:), and if e: < e:2/2M then z(x M ) = z(x)M = Ier for all x E X(e:). We fix e: > 0 with e: < min{e:I/M,e:2/2M,e:3}' For every x E X(e:) we have that z(x M ) = z(x)M = Ix (since e: < e:2/2M), and 8(x!:(, Ie) < Me: < e:l for every , E r. Our choice of e:l implies that x M = lx, and since 8(x-y, G(G))

= 8(x-y,z(x)-y) = 8((xz(x)-I)-y, Ie) < e: < e:3

and (XZ(X)-I)!1

=

x!1

= Ie

for all , E r, we see that xz(x)-l = Ix and x E G(G)r. This proves that O(X(e:)) = {IB(x)}. We set B(e:) = {g E G' : 8'(g,I) < e:} and obtain that O(X) n B(e:)r = {Io(x)}, i.e. that O(X) is an expansive subgroup of G,r. D PROPOSITION 2.13. Let X be a compact, connected group, and let Y C X be a closed, normal subgroup with Y :J G(X). Then the following is true.

(1) G(Y) = G(X); (2) Y/G(X) is a connected subgroup of X/G(X); (3) IfY'={xEX:xy=yx for all yEY}, thenYnY'=G(X), and X/G(X)

= (Y/G(X») . (Y' /C(X))

~ (Y/G(X»

x (Y' /G(X);

(4) If Fe X is a finite, normal subgroup, then F C G(X); (5) X/G(X) has trivial centre.

2. EXPANSIVENESS AND LIE SUB SHIFTS

15

PROOF. We begin by proving (1). It is clear that C(Y) ::J C(X). In order to prove the reverse inclusion we define, for every x EX, an automorphism 'Y'; of Y by 'Y'; (y) = xyx- 1 for all y E Y, and we write 'Y;;(Y) for the restriction of 'Y'; to C(Y). The homomorphisms x I--> 'Y'; and x I--> 'Y;;(Y) from X into Aut(Y) and Aut( C(Y)) are both continuous. Since C(Y) is abelian, the group Aut(C(Y)) is zero-dimensional (cf. Section 1), and the connectedness of X implies that 'Y;;(Y) is trivial for every x E X. Hence xyx- 1 = y for all x E X and y E C(Y), so that C(Y) c C(X) and hence C(Y) = C(X). In order to prove (3) we recall that the group Out(Y) = Aut(Y)jInn(Y) is zero-dimensional (cf. Section 1). Since X is connected, 'Y'; E Inn(Y) for every x EX, and the map x I--> 'Y'; induces a continuous, surjective homomorphism 0: X 1---+ Y jC(X) ~ Inn(Y) with kernel Y' ::J C(X). If 0': XjC(X) 1---+ Y jC(X) is the homomorphism induced by 0, then every x' = xC(X) E XjC(X) can be written as x' = a' . b' with a' = x'O(X)-l E Y' jC(X) and b' = O(x) E YjC(X); in particular we can find an a E y' with aY = a'Y = xY. As x' E XjC(X) is arbitrary this shows that XjC(X) = (YjC(X)) . (Y' jC(X)), and since YnY' = C(X) we conclude that XjC(X) ~ (YjC(X)) X (Y' jC(X)), as claimed in (3). The assertion (2) follows from the fact that YjC(X) ~ (XjC(X))j(Y' jC(X)).

Next we turn to (4). If Fe X a finite, normal subgroup, then we define automorphisms 'Y; E Aut(F) as in the proof of (1) (with F replacing Y), and note that x I--> 'Y; is a continuous homomorphism from the connected group X into the finite group Aut(F). Hence 'Y; to be trivial for every x E X, so that Fe C(X). In order to prove (5) we claim that, if G is a compact, connected Lie group, then G jC( G) has trivial centre. According to Theorem XIII.1.3 in [30], G ~ (A x B) j D, where A is a finite-dimensional torus, B is a semi-simple group (which must have finite centre), and D is a finite central subgroup of A x B such that AnD and B n D are both trivial. If C' is the centre of the compact, semi-simple Lie group GjC(G) ~ BjC(B), then C' must be finite, so that it is the image of a finite, normal subgroup FeB under the quotient map B 1---+ BjC(B). As we have seen above, F C C(B) c C(G), and hence C' = FjC(G) = {I}. Note that the triviality of the centre of GjC(G) is equivalent to the statement that every a E G with a-1gag- 1 E C(G) for all g E G lies in C(G). Now assume that X is an arbitrary compact, connected group, and choose a decreasing sequence (Yn , n :::: 1) of closed, normal subgroups of X such that nn>l Yn = {I} and G n = XjYn is a Lie group for every n :::: 1. If XjC(X) has-non-trivial centre C(XjC(X)), choose an element a EX" C(X) with aC(X) E C(XjC(X)). Then a-1xax- 1 E C(X) for every x E X, and the result proved in the preceding paragraph shows that aYn E C(XjYn ) for every n :::: 1. Hence a-1xa E Yn for every x E X and n :::: 1, which implies that

16

I.

GROUP ACTIONS BY AUTOMORPHISMS OF COMPACT GROUPS

a E C(X). This contradiction to our assumption shows that C(XjC(X)) is trivial, as claimed in (4). D

COROLLARY 2.14. Let G be a compact, connected Lie gmup with trivial centre, and let X C G r be a full, connected, shift-invariant subgmup. Then there exists a continuous homomorphism ¢: G f-----+ X such that ¢(g hr = 9 for every 9 E G. PROOF. Since X c G r is full, C(X) is trivial, and we set V = ker( 1l'1 r) and observe that V is a closed, normal subgroup of X. Define V' as in Proposition 2.13, note that the restriction 1l" of 1l'{1r}: X f-----+ G to V' is an isomorphism of V' and G, and set ¢ = 1l',-1: G f-----+ V' eX. D PROOF OF THEOREM 2.4. Proposition 2.2 allows us to assume that X is a full, shift-invariant subgroup of H r , where H is a compact, connected Lie group. Lemma 2.11 yields an increasing sequence (Hn, n 2: 1) of closed, normal subgroups of H such that each Hn has finite centre and the union of the groups Xn = X n is dense in X. We fix n 2: 1 and put G = 1l'1 r (X~). Since X is full and X~ is normal in X, G is a connected, normal subgroup of H and hence of H n , the centre of G is finite, and Xn is a full, expansive subgroup of Gr. Lemma 2.12 shows that 8(Xn) C G,r is expansive, where 8 : G r f-----+ G,r is obtained from the quotient map e: G f-----+ G' = G jC( G) via (2.3). If G' -I- {I}, let ¢: G' f-----+ 8(Xn) be the homomorphism constructed in Corollary 2.14. For every, E r, 1/J"( = 1l'{lr } . a"( . ¢: G' f-----+ G' is a continuous homomorphism from G' into G'. Since G' is a compact, connected Lie group with trivial centre, the group Aut(G') is compact, and there exists a metric 6 on G' which is invariant under every (3 E Aut(G'). An elementary argument shows that 6(T/(g),T/(h)) ::; 6(g,h) for every continuous homomorphism T/: G' f-----+ G', and in particular 6(1/J"((g), 10') ::; 6(g, 10 1 ) for every, E r, 9 E G'. This shows that, for every c > 0, and for every 9 E G' with 6(g, 10') < c, we have that 6(¢(g)"(, 10') < c for all, E r. In view of the expansiveness of 8(Xn) this is impossible, and we obtain that G' = {l}, 8(Xn) = {I}, and that Xn is abelian. Since Un>l Xn is dense in X, X must be abelian, and the theorem is proved. D -

H;

Proposition 2.2 shows that every expansive action a of a countable group by automorphisms of a compact group is conjugate to a Lie subshift. The converse is obviously not true: if G is an infinite, compact Lie group, then the shift-action a of r on G r is not expansive. The next proposition shows that conjugacy to a Lie subshift amounts to a finiteness condition on the pair (X, a).

r

PROPOSITION 2.15. Let a be an action of a countable gmup r by automorphisms of a compact gmup X. Then (X, a) is conjugate to a Lie subshijt if and only if there exist finitely many continuous, irreducible, unitary representations Tl, ... ,Tn of X such that the family of representations {Ti . a"( : 'Y E r, 1 ::; i ::; n} separates the points of X.

2. EXPANSIVENESS AND LIE SUB SHIFTS

17

PROOF. If (X, a) is conjugate to a Lie subshift we assume for simplicity that X c C r , where C is a compact Lie group, and that a is equal to the shiftaction CT. Since C possesses finitely many irreducible, unitary representations PI, ... , Pn which together separate points (or, equivalently, since C has a finitedimensional, faithful, unitary representation), the representations Ti = Pi ·1T{lr} together separate the points of X. Conversely, if there exist irreducible representations T1, ... , Tn of X such that {Ti . a, : '"Y E r, 1 :::; i :::; n} separates the points of X, put T = T1 EB ... EB Tn and denote by J{ the Hilbert space on which T acts. Then C = T(X) is a closed subgroup of the group U(J{) of unitary operators on J{ and hence a compact Lie group. The homomorphism T: X f----+ C r , defined by (T(X)), = T· a,(x), '"Y E r, x EX, is injective, embeds X as a full, shift-invariant subgroup of C r , and satisfies that T . a, = CT, • T for every '"Y E r. 0 COROLLARY 2.16. Let a be an action of a countable group r by automorphisms of a compact, abelian group X. Then (X, a) is conjugate to a Lie subshift if and only if there exist characters Xl, . .. ,Xn in X such that X is generated by {Xj . a, : '"Y E r, 1:::; j :::; n}. PROOF. Proposition 2.15 and the Stone-Weierstrass theorem.

0

PROPOSITION 2.17. Let r be a countable group, and let a be a r-action by automorphisms of a compact group X. Then there exists a non-increasing sequence (Vn , n 2': 1) of closed, normal, a-invariant subgroups of X such that nn>l Vn = {Ix} and (X/Vn , a X / Vn ) is conjugate to a Lie subshift for every n 2':-1. PROOF. Choose a sequence of irreducible, unitary representations (Pn, n 2': 2': 1 we put Tn = PI EB ... EB Pn, denote by C n the compact Lie group Tn(X) (which is a subgroup of the unitary group of some finite-dimensional Hilbert space), and define a continuous group homomorphism Tn: X f----+ C~ by (Tn (X)), = Tn· a,(x), '"Y E r,X E X. Then Tn· a, = CT,· Tn for every '"Y E r, Vn = ker(Tn) is a closed, normal, a-invariant subgroup of X, VI ::) ... ::) Vn ::) ... , and nn;:l Vn = {Ix}. 0 1) of X which together separate the points of X. For every n

EXAMPLES 2.18. (1) Let a be an action of a countable group r by automorphisms of a compact Lie group C. Then a is obviously conjugate to the Lie sub shift Y c C r given by Y = {y = (y,) E C r : Y, = a,(Y1r) for all '"Y E r}. (2) Let a be the automorphism of ']['2 defined by the matrix (~ t), and let Y = {x = (x n ) E ']['Z : Xn + Xn+1 - Xn+2 = 0 (mod 1) for all n E Z}. Then Y c ']['Z is a closed, shift-invariant subgroup, and the coordinate map 1T{O,l}: Y f----+ ']['2 is bijective and satisfies that 1T{O,l} . CT = a· 1T{O,l}. Note that a is also conjugate to the shift on a closed, shift-invariant subgroup Y' c (']['2)Z, as described in Example (1).

18

I.

GROUP ACTIONS

BY

AUTOMORPHISMS OF COMPACT GROUPS

(3) Let p ?: 2 be a rational prime, and let Zp denote the compact ring of p-adic integers, i.e. the ring of all formal power series x = Ln>o xnpn with Xn E {O, 1, ... ,p - I} for all n ?: 0, furnished with the obvious operations of addition and multiplication, and with the compact topology which makes the bijection x = Ln~o xnpn I--t (xo, Xl, ... ) of Zp and Z /p a homeomorphism (we set Zjn = Z/nZ for every n ?: 2). The additive semi-group N is embedded in X as the set of all power series with only finitely many non-zero terms. Here we consider the case p = 2, regard X = Z2 as an additive group, and define an automorphism a of X as multiplication by 3. The dual group Y = X = Z; of X is (isomorphic to) the group {m2- n : n ?: 1,0 ::; m < 2n} under addition modulo 1, and 0: consists of multiplication by 3 (mod 1). For every n?: 1, the subgroup Y n = {m2- n : 0 ::; m < 2n} C Y (with addition modulo 1) is invariant under 0:, and this is easily seen to imply that Y = X is not finitely generated under 0: in the sense of Corollary 2.16. Hence a is not conjugate to a Lie subshift; since X is zero-dimensional, this also implies that a is not expansive (Corollary 2.3). For every n ?: 1 we set Vn = yn.l and note that Vn is a-invariant, X/Vn is finite, and Vn ~ Vn+ l . As in Example (1) we observe that (X/Vn , a XjVn ) is (trivially) conjugate to a Lie subshift, and it is clear that nn>l Vn = {Ix} (cf. Proposition 2.17). (4) Let r be the multiplicative group «:Y = Q" {O}, X = Q, and let a: r I--t a = be the r -action on X dual to the action f3 of r on X = Q defined by f3r(s) = rs for every r E r, SEQ. The group r = X consists of two f3-orbits, {O} and {f3r(l) : r E r} = r, so that (X, a) is conjugate to a Lie subshift by Corollary 2.16. In order to make this conjugacy explicit we set

r Sr

Y

= {Y = (Ys, s E r) E ']['[' : kys = lYrs (mod 1) for every r = ~,s E r}

(2.5)

and denote by (J the shift-action (2.1) of r on the shift-invariant subgroup Y C ']['['. If X: X ~ '][' ~ § is the surjective homomorphism corresponding to the character 1 E Q = X, and if X: X ~ ']['[' is the homomorphism defined by (X(x))r = X . ar(x), r E r, then X(X) = Y, and X· a r = (Jr . X for every r E r.

More generally, if lK is a countably infinite field, and if r = lK x = lK" {O}, then we obtain an ergodic r-action a on K which is dual to the action f3 on lK by multiplication, and which is conjugate to a Lie subshift. 0 CONCLUDING REMARK 2.19. The proof of Theorem 2.4 is a modification of the argument in [50], where Lam obtains the same result under the weaker assumption that r is a semi-group of continuous, surjective homomorphisms of X. For Z-actions the finiteness conditions in Proposition 2.15 and Corollary 2.16 were originally introduced by Rokhlin in [85] and have since reappeared in a number of papers (e.g. [103], [104], [55], and [45]).

3. THE DESCENDING CHAIN CONDITION

19

3. The descending chain condition Conjugacy to a Lie subshift is implied not only by expansiveness, as we have seen in Proposition 2.2, but also by another finiteness condition on (X, a), the descending chain condition. DEFINITION 3.1. Let r be a countable group, and let a be a r-action by automorphisms of a compact group X. The pair (X,a) (or the action a) satisfies the descending chain condition (d.c.c.) if there exists, for every nonincreasing sequence X :J Xl :J ... :J Xk :J ... of closed, a-invariant subgroups of X, an integer K ~ 1 that X k = XK for all k ~ K. If a satisfies the d.c.c., then a V and a X / w satisfy the d.c.c. for all closed, a-invariant subgroups V c X and all closed, normal, a-invariant subgroups WcX. In Section 4 we shall see that every action of a polycyclic-by-finite group (and, in particular, of Zd, d ~ 1) by automorphisms of a compact group X either satisfies the d.c.c., or is a projective limit of actions satisfying the d.c.c. For the moment we shall content ourselves with a much more basic example. EXAMPLE 3.2. Let a be an action of a countable group r by automorphisms of a compact Lie group X. Then a satisfies the d.c.c., since every decreasing sequence of closed subgroups of X must eventually become constant. In order to have something more specific one can set X = ,][,n for some n ~ 1 and take r to be a subgroup of GL(n,Z) = Aut(']['n). c:J PROPOSITION 3.3. Let r be a countable group, and let a be a r -action by automorphisms of a compact group X. If (X,a) satisfies the d.c.c. then it is conjugate to a Lie subshift. PROOF. Let (Vn' n ~ 1) be the sequence of closed, normal, a-invariant subgroups of X defined in Proposition 2.17. Since (X, a) satisfies the d.c.c., there has to exist an N ~ 1 with VN = {Ix}. D COROLLARY 3.4. Let r be a countable group, and let a be a r -action by automorphisms of a compact, zero-dimensional group X satisfying the d.c.c. Then a is expansive. PROOF. Proposition 2.2 implies the existence of a compact Lie group G such that (X, a) is conjugate to a full subshift of C r , and Corollary 2.3 does the rest. D Although the d.c.c. may appear unintuitive, it has a number of very useful dynamical consequences. The first of these concerns the structure of non-ergodic r -actions satisfying the d.c.c.

20

1.

GROUP ACTIONS BY AUTOMORPHISMS OF COMPACT GROUPS

PROPOSITION 3.5. Let r be a countable group, X a compact group, and let a be a r -action by automorphisms of X which satisfies the d. c. c. Then there exists a unique maximal, closed, normal, a-invariant subgroup X' c X such that X/X' is a Lie group and aX' is ergodic. In particular, if X is zerodimensional, then X' is open. PROOF. The d.c.c. implies that the ordinal w in Theorem 1.4 is finite, i.e. that there exist closed, normal, a-invariant subgroups X' = Vn C ... C Vo = X such that aX' is ergodic and, for every i = 0, ... ,n - 1, Vi/Vi+l is a Lie group with an a VdVi+l-invariant metric. Hence X/X' is a Lie group. If X is zerodimensional, then X/X' is a zero-dimensional Lie group and hence finite. 0 As a second application of the d.c.c. we consider the restriction of an ergodic action a of a countable group r by automorphisms of a compact group X to a closed, a-invariant subgroup Y C X. While it is clear that a XjY is ergodic, there is no guarantee that a Y will be ergodic. The next result describes a situation where the ergodicity of a does imply the ergodicity of a Y . THEOREM 3.6. Let a be an ergodic action of a countable group r by automorphisms of a compact group X which satisfies the d. c. c., and let XO be the connected component of the identity in X. Then axe is ergodic. PROOF. Suppose that axe is non-ergodic. According to Lemma 1.2 (2) there exists a non-trivial, continuous, irreducible, unitary representation T of XO such that r T has finite index in r. The closed, normal, a-invariant subgroup XrJ C X constructed in the proof of Lemma 1.3 (with Y replaced by XC) has finite index in X, and the ergodicity of a implies that XrJ = X, i.e. that 7](x) is unitarily equivalent to 7] for every x E X. Hence W = ker(7]) C XO c X is an a-invariant, normal subgroup of X, not just of XO, and XO /W is a compact, connected Lie group. We set X' = X/Wand Y = xo /W. If Y is non-abelian, let C(Y) be the centre of Y, and set X" = X' /C(Y) and Y' = Y/C(Y). Then Y' is a compact, connected Lie group with trivial centre which is normal in X", and a Y ' preserves a metric fj on Y', since a Y preserves the metric on Y ~ 7](XO) arising from the operator norm. For every x E X we consider the automorphism (3x of Y' defined by (3x (y) = xyx- 1 for every y E Y'. The map (3: x f---> (3x is a continuous group homomorphism from X into Aut(Y'), and {x EX: (3x E Inn(Y')} is a closed, normal, a-invariant subgroup of X" with finite index in X" (since Y' has trivial centre, the group of inner automorphisms of Y' has finite index in Aut(Y')). The ergodicity of aX" now implies that (3x E Inn(Y') for every x E X", and we denote by (3(x) E Y' the unique element with (3x(Y) = (3(x)y(3(x)-l for every y E Y'. It is clear that a~' ((3(x)) = (3(a?J" (x)) for all '"Y E r and x E X". In particular, (3-1(N) is an aX" -invariant subset of X" for every a Y ' -invariant subset N of Y'. Since Y' has a basis of a-invariant

3. THE DESCENDING CHAIN CONDITION

21

neighbourhoods of the identity we obtain a contradiction to the ergodicity of a X" and hence to the ergodicity of a. This shows that Y must be abelian, i.e. that Y = 1['8 for some s 2: 1. As before, we define a homomorphism (3: x f--t (3x from X' into the discrete group Aut(Y) by setting (3x(Y) = xyx- l for all x E X' and y E Y. The kernel of this homomorphism is an open, a-invariant subgroup of X' and hence-due to the ergodicity ofaX'~equal to X'. In other words, Y C C(X/). Proposition 3.3 allows us to assume that X' is a full, shift-invariant subgroup of H r for some compact Lie group H, and that a!f' = (J, for all '"Y E f. Then HO is a homomorphic image of the connected component Y of the identity in X', and Y c (Hol. We choose an increasing sequence (Fn' n 2: 1) of finite subsets of f with Un>l Fn = f, set Hn = 7rFn (X') C HFn, and define a shift commuting, injective -embedding 'l/Jn: X' f---t H~ by ('l/Jn(x)), = 7rFn ((J,(x)) for all x E X' and '"Y E f. For every n 2: 1 we have that 'l/Jn(Y) C (H~)r, and Yn = 'I/J;;l((H~)r) is a decreasing sequence of closed, normal, shift-invariant subgroups of X' with nn>l Y n = Y. Since (J = aX' satisfies the d.c.c., there exists an m 2: 1 with Ym ;, Y, and we assume without loss in generality that m = 1 and Y = X' n (HO)r. As X' C H r is full and Y is central we know that

7rlr(Y) = HO c C(H).

Lemma 2.8 implies the existence of a closed, normal subgroup H' C H such that A = H/ H' is abelian, H' n HO is finite, and H' . HO = H. We write 8: H f---t A for the quotient map and define a shift commuting homomorphism 0: X' f---t A r by (O(x)), = 8(x,) for all x E X','"Y E f. Since HI! = H' nHo is finite, the group Y' = ker(O)nY = ker(O)n(HO)r is a closed, zero-dimensional, and hence finite, subgroup of Y, and W = O(Y) ~ Y/Y ' is the connected component of the identity in the abelian group V = O(X/) cAr. Furthermore, the shift-action of f preserves a metric on Wand is therefore non-ergodic on W. The shift-action of f on V C Ar is algebraically conjugate to (Jx' /ker(9) and thus satisfies the d.c.c. Exactly as before, when we were dealing with X', H, and Y, we may change the group A, if necessary, and assume without loss in generality that W = Vn(AO)r. As A/Ao is finite, there exists an m 2: 1 with v m E V n (AO)r = W for every v E V, and the map (: v f--t v m from V to W is a surjective, shift commuting, group homomorphism. Since the shift-action of f on W is non-ergodic we see that facts non-ergodically on V. Hence the action aX' cannot be ergodic which, in turn, violates the ergodicity of a. This contradiction shows that axe must be ergodic. D The appearance of the d.c.c. in Theorem 3.6 is a little unexpected and raises the question whether axe is ergodic for every ergodic f -action a on X, i.e. independently of whether (X, a) satisfies the d.c.c. We postpone further discussion of this problem to Theorem 4.11 and Example 4.13 and turn instead to the third, and dynamically most interesting, aspect of the d.c.c.: its intrinsic connection with shifts of finite type.

22

1.

GROUP ACTIONS BY AUTOMORPHISMS OF COMPACT GROUPS

DEFINITION 3.7. Let r be a countable group, and let C be a compact group. A subgroup X C C r is of finite type if it is closed and shift-invariant, and if there exists a finite subset Fer such that

X

= {x

E

c r : 7rF(ai (x))

E 7rF(X) for every 'Y E r}.

(3.1)

In order to interpret condition (3.1) we can think of F as a window and of the subgroup H = 7rF(X) C C F as the set of allowed words; in order to decide whether a given point x E C r lies in X we look at x and all its translates through the window F, and x E X if and only if we always see an allowed word. This definition is consistent with the notion of a classical shift of finite type with finite alphabet, which is the set of all points in a shift space for which every string of coordinates of some fixed length n occurs in a previously specified list of allowed words (usually n = 2, but that is irrelevant). For example, the subshift of {O, l}z consisting of all sequences with no two adjacent '1'-s is of finite type, whereas the subshift of all sequences in which any two '1'-s have to be separated by an even number of 'O'-s is not of finite type; in the first case the window F could be chosen as {0,1} c Z and the set of allowed words as {(O, 0), (0, 1), (1, O)}, but in the second case no window F of finite size would work. THEOREM 3.8. Let r be a countable group, and let C be a compact Lie group. The shift-action a of r on C r satisfies the d.c.c. if and only if every closed, shift-invariant subgroup X C C r is of finite type. PROOF. Suppose that a satisfies the d.c.c., that X C C r is a closed, shiftinvariant subgroup, and that (Fn, n :::: 1) is an increasing sequence of finite subsets of r with Un>l Fn = r. For every n :::: 1 we set Xn = {x E C r : 7rFn (ai(x)) E 7rFn (X) -for every 'Y E r}, and we observe that Xn is a closed, shift-invariant subgroup of C r , Xn ::) X n+1 ::) X for every n :::: 1, and that nn>l Xn = X. The d.c.c. implies that there exists an integer N :::: 1 with X N- = X, i.e. that X is of finite type. Conversely, if every closed, shift-invariant subgroup of C r is of finite type, let (Xn' n :::: 1) be a non-increasing sequence of closed, shift-invariant subgroups of C r , and let X = nn>l X n . Then X is of finite type, and there exists a finite subset Fer satisfying (3.1). We set H = 7rF(X) and Hn = 7rF(Xn ), n :::: 1, and observe that

for every n :::: 1. Since (Hn, n :::: 1) is a non-increasing sequence of closed subgroups of the compact Lie group C F , there exists an N :::: 1 with HN = H and hence with X = Y N ::) XN ::) Xn ::) X for all n :::: N. This proves that a satisfies the d.c.c. 0

4.

GROUPS OF MARKOV TYPE

23

We have obtained the following implications for a r-action by automorphisms of a compact group X: expansiveness ===> conjugacy to a Lie subshift .;::= d.c.c.

(3.2)

As mentioned in Section 2, conjugacy to a Lie sub shift need not imply expansiveness. The next example shows that it need not imply the d.c.c., either, and that there is-in general-no connection between expansiveness and the d.c.c. EXAMPLES 3.9. (1) Consider the direct sum r = Ln>271.,jn of the cyclic groups 71.,/n = 71.,/n71., of order n, and let C = 71.,/2' We write a typical element I Eras I = (r2, 13, ... ) with In E Z/n for every n 2: 2 and 1m = 0 for all but finitely many m, and we set r n = {, E r : 1m = 0 for all m i= n}. Let X = C r , and let a be the shift-action of r on X. Then a is expansive on X. For every n 2: 2 we consider the full, shift-invariant subgroup Xn = {x E X : LI'Er", al'(x) = Ocr for every m :::; n}. Then X :;;2 X 2 :;;2 ... :;;2 Xn :;;2 ... :;;2 nn~2 Xn = Y, say, and Y c X = C r is obviously not of finite type. (2) Let r be a countable group with a strictly increasing sequence r 1 l Yk,m. The d.c.c. implies that there exists an M ~ 1 such that Yk,M = Yk for k = 1, ... ,n, which proves (1), and (2) is an immediate consequence of (1). 0 REMARKS 4.15. (1) Suppose that r = Zd, where d ~ 1. In the notation of Proposition 4.14 (1), consider the injective homomorphism morphism 8: Y 1---+ H Zd defined by 8(x)n = 7rp(an (x)) for every x E Y and n E Zd. Then W = Wo = 8(Y) is a full, shift-invariant subgroup of H Zd , and Wk

= 8(Yk) = {w

E H Zd : 7r{O,l}d(an

(w))

E 7r{O,l}d(Wk )

for every n E Zd}

(4.11)

for every k = 0, ... , n. In other words, by changing the alphabet C of the Lie subshift Y c C Zd , we may assume that F = {O,l}d. This argument is analogous to the re-coding of a classical subshift of finite type as a one-step Markov shift.

32

I. GROUP ACTIONS BY AUTOMORPHISMS OF COMPACT GROUPS

(2) If we apply Remark (1) to the groups Y and yo C Y in Corollary 2.10 (3), then (4.11) shows that 8(Y) = {w E Hz d : 7l"{O,l}d(O"n(w)) E K for every n E Zd} and8(YO) = 8(Y)n(HO)Zd = {w E H Zd : 7l"{O,l}d(O"n(w)) E KO for every n E Zd}, where K = 7l"{0,1}d(8(Y)) C H{O,l}d. We end this section with two examples. Since Zd is of Markov type for every d ~ 1 (Example 4.8 (1)), every closed, shift-invariant subgroup of G Zd , where G is a compact Lie group, is of finite type. In Remark 3.10 (2) we mentioned a peculiarity of subgroups of finite type with infinite alphabet. However, even if the alphabet G is finite, the subgroups of G Zd can have other, unexpected properties. EXAMPLES 4.16. (1) Let G = Zj2 F - {O -

,

1}2 -

X

Zj2,

{ (0,1) (1,1) } C Z2 (0,0) (1,0) ,

and let H

= {{ hk) (*,1)

(1,*) } . k (k,*) .,

l E Zj 2 } C G F ,

where * indicates that this location can contain an arbitrary element of Zj2' Then H is a full subgroup of G F , and

X

= X(F,H) = {x

E

GZd : 7l"F(O"n(X))

E

H for every n

E

Zd}

(4.12)

is a full, shift-invariant subgroup of G Zd . The shift-action 0" of Z2 on X has the following properties: (i) 0"(0,2) = idx; (ii) 0"(1,0) is ergodic and mixing, and is isomorphic to the full 4-shift. Hence 0" is ergodic and expansive, but not mixing. The subset

s = {{ i::Zj ~~:~j } : h, k E Zj2} (*,h) (k,l)

C

G{O,l} X{0,1,2},

satisfies that S n 7l"{0,1}X{0,1,2}(X) = 0, while 7l"D(S) = 7l"D(X) for every 2 x 2 square D C {O, I} x {O, 1, 2}. In other words, although the size of the defining window F is 2 x 2, there exist configurations of size 2 x 3 with the property that every subconfiguration of size 2 x 2 occurs in a point in X, whereas the configuration itself does not occur in any element of X. This is an indication that the extension problem for higher-dimensional subshifts of finite type, which is undecidable in general, is not completely trivial even in this very special setting, where we are dealing with closed, shift-invariant subgroups of G Zd for a finite group G. For a discussion of this and related problems involving decidability see [44]. (2) In Example (1) we replace Zj2 by a compact Lie group K with a distinguished closed, normal subgroup L C K and set G = K x K and

H = {{

(*,k) (k'I,*)} . k (*,k') (kl' ,*) .,

k'

E

K , l , l'

E

L}

C

GF .

4. GROUPS OF MARKOV TYPE

Then the group subgroup of G'Z?

X

=

X(F,H)

33

defined by (4.12) is again a full, shift-invariant

0

CONCLUDING REMARKS 4.17. (1) For finitely generated, abelian groups Theorem 4.2 was proved in [45]. Example 4.13 is due to Losert, and Example 4.16 (1) to Coppersmith. Losert has also shown me a proof that, if r is a countable, finitely generated group, and if 0: is an ergodic action of r by automorphisms of a compact group X, then o:xo is ergodic. (2) It is not known which countable groups r are of Markov type. Proposition 4.6 lends plausibility to the conjecture that r is of Markov type if and only if Z[r] is right Noetherian; however, the latter class of countable groups is just as mysterious as the groups of Markov type. As far as I am aware, all known examples of countable groups r for which Z[r] is right Noetherian are polycyclic-by-finite; furthermore, the ascending chain conditions on subgroups of r (in the sense of Example 3.9 (2)) does not imply that Z[r] is right Noetherian (or that r is of Markov type). (3) If G is a Lie group with finite centre, and if F = {a, 1}2 C Z2, the choice of full subgroups H c G F is rather limited, and in conjunction with Remark 4.15 this imposes severe restrictions on the possibilities for full, shiftinvariant subgroups of Gz 2 • In the Sections 5-6 we shall see that the situation is completely different if G is abelian.

CHAPTER II

zd-actions on compact abelian groups

5. The dual module According to Theorem 4.2, tl d is of Markov type for every d ;::: 1, and 7l d _ actions by automorphisms of compact groups enjoy the properties described in (4.10), Propositions 4.9-4.10, Remark 4.15, and Theorem 4.11. Just as compact, abelian groups like Tn = IRn /71 n have automorphisms with very intricate dynamical properties, there is an abundance of examples of interesting 7l d -actions by automorphisms of compact abelian groups. In this section we introduce a general formalism for the investigation of such actions which will also give us a systematic approach to constructing actions with specified properties. Let d ;::: 1, and let 0:: n ~ O:n be an action of tl d by automorphisms of X. For every n = (nl,"" nd) E tl d we denote by an the automorphism of X dual to O:n and write a: tl d f---> Aut(X) for the resulting 7l d -action dual to 0:. Under the action a the group X becomes a 7l d -module, and hence a module over the group ring 7l[71 d ]. In order to make this explicit we denote by (5.1)

the ring of Laurent polynomials in the (commuting) variables coefficients in 7l. A typical element f E 9'\d will be written as

f =

L cf(n)un ,

UI, ... ,Ud

with

(5.2)

nEZ d

where cf(n) E 7l and un = U~' . '" . U~d for all n = (nl,"" nd) E 7l d, and where cf(n) =1= 0 for only finitely many n E 7l d. Then 9'\d ~ 7l[71 d], 9'\d acts on X by

(f, a) ~ f . a =

L cf(n)an(a)

nEZ d

35

(5.3)

36

II. Zd-ACTIONS ON COMPACT ABELIAN GROUPS

for every

f

E 9td , a E

X,

and

X is an 9td-module.

Note that

an(a) = an(a) = un . a

(5.4)

for every n E Zd and a E X. Conversely, if9J1 is an 9td-module (always assumed to be countable), then Zd has an obvious action am: n f--+ a;;r' on 9J1 given by (5.5) for every n E Zd and a E 9J1. We write X obtain a dual action am:

=

Wt for the dual group of 9J1 and

n a;;t E Aut(X)

(5.6)

f--+

of Zd on X. For future reference we collect these observations in a lemma. LEMMA 5.1. Let a: n f--+ an be a Zd-action by automorphisms of a compact, abelian group X, and let a: n f--+ an be the dual action of Zd on the dual group X of x. If 9td is the ring defined in (5.1) then X is an 9t d-module under the 9t d -action (5.3). Conversely, if 9J1 is an 9td-module, then (5.5) and (5.6) define Zd-actions am = a and am = a by automorphisms of 9J1 and Xm = Wt, respectively. EXAMPLES

5.2. Let d

~

1.

(1) Let 9J1 = 9t d . Since 9td is isomorphic to the direct sum LZd Z of copies of Z indexed by Zd, the dual group X = 9td is isomorphic to the cartesian product ']['Zd of copies of '][' = ]R/Z. We write a typical element x E ']['Zd as x = (xn) = (xn, n E Zd) with Xn E '][' for every n E Zd and choose the following identification of X X is continuous, the Zd-action n f--7 On on V C X must also be non-expansive in the subspace topology, i.e. a is not expansive on V. We have proved that there always exists an infinite, a-invariant, but not necessarily closed, subgroup V C X on which a is non-expansive in the induced topology. This shows that a is not expansive and completes the proof that

W;W-o

(a)~(c).

The equivalence of (b) and (c) is seen by applying the implications (a) ~ (c) already proved to the Zd-actions a!Rd/pi, i = 1, ... , m. It is clear that (c)=}(d). Conversely, if Vc(Pi) n §d =f. 0 for some i E {l, ... , m}, choose iI,···,!k in ~d with Pi = iI~d + ... + fk~d, and define polynomials gj, hj, j = 1, ... , k, in

II. Zd-ACTIONS ON COMPACT ABELIAN GROUPS

54

by

and h j (a1, ... , ad, b1, ... , bd)

= Im(/j(a1 + b1R , ... , ad + bdR))

for all j = 1, ... ,k and (a1, ... ,ad,b 1, ... ,bd) E IR 2 d, where Re(z) and Im(z) denote the real and imaginary parts of z E Xa is an isomorphism of the discrete, additive group OC onto i(OC)J.. C OC1\. The resulting identification (7.2) depends, of course, on the chosen character X. In order to make the isomorphism (7.2) a little more canonical we consider, for every wE p"OC, the subgroup

O({w})' = {w = (w v ) E OC1\: Wv = 0 for every v

oc:

i= w}

~

OCw

of OC1\ and denote by XCw ) E the character induced by the restriction of X to O( {w})'. After replacing X by a suitable Xa, a E OC, if necessary, we may assume that the induced characters xC w) E w E Pf, satisfy that

OC:,

9(w

C

ker(x Cw »)

= {w

1T.:l9(w

E

Aw : XCw)(w) = 1},

(7.3)

ct. ker(x Cw »)

for every w E Pi'''. where 1Tw E 9(w is the prime element appearing in the preceding paragraph (cf. [109]). With this choice of X we have that

where

O(pf)' = {w = (w v ) E OC1\ : Wv = 0 for every v E p! = p"OC "pf"OC}. Now consider a finite subset Fe p"OC which contains P!" denote by (7.4) the diagonal embedding r

I-->

(r, ... , r), r E OC, put

RF = {a E OC: lalv :::; 1 for every v

~

F},

(7.5)

and observe that iF(RF) is a discrete, additive subgroup of ilvEF OC v . If

0= O(F) 0'

= {w = (w v ) E OC1\ : Iwvl v :::; 1

for every v E p"OC " F},

= O(p"OC " F)' = {w = (w v ) E OC1\ : Wv = 0 0" = 0 no',

then i(OC) + 0" = i(OC) (i(OC) + 0').1.. and

+ 0',

for every v E F},

and (7.3) implies that X E (i(OC)

+ 0").1..

Hence

(7.6)

64

II. Zd-ACTIONS ON COMPACT ABELIAN GROUPS

Let d 2': 1, C = (Cl, ... ,Cd) E (QX)d, and je = {J E rytd: f(c) = O}. We wish to investigate the dynamical system (X, a) = (XVld/i c , aVld/ic) determined by c. Denote by lK = Q(c) the algebraic number field generated by {Cl, ... , Cd} and put

F(c) = {v E

pl( : ICilv i- 1 for some i E {I, ... , d}},

(7.7)

which is finite by Theorem III.3 in [109], and

(7.8) where P(c) = p! U F(c). Then Re is an rytd-module under the action (f, a) f(c)a, and we define the Zd-action

f-+

(7.9) on the compact group

y(e) = Re = (

II

vEP(e)

lKv) /

iF(Re)

(7.10)

by (5.5)-(5.6), where we use (7.6) to identify Re and (IlvEP(e) lKv) / iF(Re). THEOREM 7.1. There exists a continuous, surjective, finite-to-one homomorphism ¢: y(e) I---> XVld/ic such that the diagram

y(e)

(c)

a", ----+

y(e) (7.11)

commutes for every m E Zd. PROOF. The evaluation map 'f]e: f f-+ f(c) induces an isomorphism 11 of the rytd-module rytdfje with the submodule 'f]e(rytd) C Re C lK; in particular (7.12)

for every a E rytdfje and m E Zd. We claim that Re/'f]e(rytd) is finite. Indeed, since lK = Q(c) is algebraic, every a E lK can be written as a = b/m with b E Z[c] = Z[Cl' ... ' Cd] and m 2': 1. In particular, since the ring of integers o(c) = ooc C lK is a finitely generated Z-module, there exist positive integers mo, Mo with moo(c) C Z[c] C 'f]e(rytd) and l:JcI'f]e(rytd)I :::; 10(c)/moo(c)1 = Mo < 00. According to the definition of F(c) there exists, for every v E F(c), an element av E 'f]e(rytd) such that lavl v > 1 and lavl w = 1 for all W E Pfoc "F(c). Then la~o(c)/1Je(rytd)1 :::; Mo and I(LvEF(e) a~o(c))/1Je(rytd)1 :::; M6 F(e) I for all

7. THE DYNAMICAL SYSTEM DEFINED BY A POINT

n >

o.

As n

-+ 00, EVEF(c) a~o(c) IF(c)1

IRc/1]c(9'ld) I :::; Mo


'lI'Zd by (K,(X))n = /'i,(Xn) for every x = (xm) E ijZd and n E Zd, and write (8.2) for the restriction of K, to Xrytd/p. The map K,rytd/p is surjective, and the diagram Xrytd/P

P -!Rd/ n

Q

~

~1 Xrytd/P

Xrytd/p

1~ ~

Q

!Rd/ P n

Xrytd/P

(8.3)

8. THE DYNAMICAL SYSTEM DEFINED BY A PRIME IDEAL

73

commutes for every n E 7l d . In order to explain this construction in terms of the dual modules we consider the ring 9t~Q) = lQ[ut 1, ... ,ujl] = IQ ®z 9td, regard 9td as the subring of 9t~Q) consisting of all polynomials with integral coefficients, and denote by p(Q) = IQ ®z P C 9t~Q) the prime ideal in 9t~Q) corresponding to p. Since p(p) = 0, every 9td -module 1)1 associated with p is embedded injectively in the 9t~Q) -module I)1(Q) = IQ ®z 1)1 by il)1:

a

f-+

1 ®z a, a E 1)1,

(8.4)

and I)1(Q) is associated with p(Q). Since 9td C 9t~Q), I)1(Q) is an 9td-module, and we can define the 7l d -action al)1«(!) on XI)1«(!) as in Lemma 5.1. Note that the set of prime ideals associated with the 9td-module I)1(Q) is the same as that of 1)1; in particular, al)1«(!) is ergodic if and only if (01)1 is ergodic and, for every n E 7l d , a~«(!) is ergodic if and only if a~ is ergodic. The homomorphism (8.5)

dual to (8.6)

is surjective, and the diagram

(8.7)

commutes for every n E 7l d . For

PROPOSITION 8.3. not of the form p = jc X = X 9t d/P is ergodic, primitive subg'rOup f = following p'rOperties. (l)f~71r;

(2) 0 E Q, and Q

1)1

= 9td/P

we obtain that

X( 9td/p)«(!)

= X 9td / p ,

a(9t d/p)«(!)

= a 9td / p ,

Z9td/P

= ",,9t d/P.

(8.8)

°

Let p C 9td be a prime ideal with p(p) = which is for any c E IQd. Then the 7l d -action a = a 9td / P on and there exists an integer r = r(p) E {I, ... , d}, a r(p) C 7l d , and a finite set Q = Q(p) C tl d with the

n (Q + m) =

0 whenever 0

i= m

E f,.

II. Zd-ACTIONS ON COMPACT ABELIAN GROUPS

74

(3) If f

= r+Q = {m+n : mEr, n E Q}, then the coordinate projection

7fr: X!l'td/P

t----t

coordinates in r -action n 1--+ on (ijQ)r.

ijr,

f,

which restricts any point x E

X!l'td/P C ijZd

to its

is a continuous group isomorphism; in particular, the /p, n E r, is (isomorphic to) the shift-action of r

a:':d

PROOF. The proof is completely analogous to that of Proposition 8.2. We find a matrix A E GL(d, Z) and an integer r E {I, ... , d} with the following I propert " les: f I Vi = UAe(j) an d viI = vi + p " lor.J = 1, ... " d then VI"'" VrI are algebraically independent elements of:R = !Rd/P, and there exists, for each j = r + 1, ... , d, an irreducible polynomial Ii (x) = L~=o g~) (xk) with coefficients in the ring Z[vtl, ... , vT-\J C !Rd such that Ii(VI, ... , Vi-I, Vj) E q and the supports of gaj) and gjj) are singletons. J We assume again that A is the d x d identity matrix, so that Vj = Uj for j = 1, ... ,d and r ~ zr is generated by e(l), ... ,e(r), set Q = {O} x ... x {O} x {o, ... ,lr+l -I} x ... x {O, ... ,ld -I} C Zd, and complete the proof in the same way as that of Proposition 3.4, using (8.1) instead of (6.19). The ergodicity of a:!l'td/P is obvious from the conditions (1)-(3), and from (8.3) we conclude the ergodicity of a!l'td/P. D REMARKS 8.4. (1) We can extend the definition of r(p) in Proposition 8.2 and 8.3 to ergodic prime ideals of the form p = je, C E (Q()d, by setting rOc) = O. Then the integer r(p) is a well-defined property of the prime ideal p, and is in particular independent of the choice of the primitive subgroup r c Zd in Proposition 8.2 or 8.3 (it is easy to see that there is considerable freedom in the choice of r): if r' , r', Q' are a positive integer, a primitive subgroup of Zd, and a finite subset of Zd, satisfying the conditions (1)-(3) in either of the Propositions 8.2 or 8.3, then r' = r(p). This follows from Noether's normalization theorem; a dynamical proof using entropy will be given in Section 24.

(2) If P c !Rd is an ergodic prime ideal with p(p) > 0, then the subgroup Zd in Proposition 8.2 is a maximal subgroup of Zd for which the restriction a r of a!l'td/p to r is expansive. In particular, r(p) is the smallest integer for which there exists a subgroup r ~ zr in Zd such that a r is expansive.

r c

(3) Even if the Zd-action a!l'td/P in Proposition 8.3 is expansive, the action a(!l'td/p)(Q) is non-expansive. By proving a more intricate version of Proposition 8.3 one can analyze the structure of the group X!l'td/P directly, without passing to X(!l'td/p)(Q): if X!l'td/p is written as a shift-invariant subgroup of ']['Zd (cf. (5.9)), and if r = r.:(p), r, Q are given as in Proposition 8.3, then the projection 7fr: X!l'td/P t----t ']['r is still surjective, but need no longer be injective; the kernel of 7fr is of the form yr for some compact, zero-dimensional group Y (cf.· Example 8.5 (2)).

8. THE DYNAMICAL SYSTEM DEFINED BY A PRIME IDEAL

75

EXAMPLES 8.5. (1) Let p = (2,1 + Ul + U2) C 9t2 (cf. Example 5.3 (5)). Then p(p) = 2, r(p) = 1, and we may set r = {(k, k) : k E Z} ~ 1£ and Q = {(O, 0), (1, On c 1£2 in Proposition 8.2. If X = X!1t2/P is written in the form (6.19) as X

= {x = (xm)

C

lF~d : x(ml,m2) + x(ml +1,m2) + X(ml ,m2+1) = OlF2 for all (ml' m2) E Z2},

then the projection 7rr: X

I---t

lF~ sends the shift a~~{f

= a(I,I) on X to the

shift on lF~ ~ (1£/2 X Z/2)'l'. Note that, although a(I,I) acts expansively on X, other elements of 1£2 may not be expansive; for example, a(1,O) is non-expansive. (2) Let p = (3 + Ul + 2U2) C 9t2. Then p(p) = 0, r(p) = 1, and rand Q may be chosen as in Example (1). Note that X!1t2/P = X = {x = (Xm) C 'JI'Zd : X(ml,m2) +X(ml +1,m2) +X(ml,m2+1) = O'Jl' for all (ml' m2) E Z2}; the coordinate projection 7rr: X I---t 'JI'r in Proposition 8.3 is not injective; for every x EX, the coordinates X(ml,m2) with ml ~ m2 are completely determined by 7rr(x), but each of the coordinates X(k,k+l), k E 1£, has two possible values. Similarly, if we know the coordinates X(ml,m2)' ml ~ m2 - r of a point x = (xrit) E X for any r ~ 0, then there are exactly two (independent) choices for each of the coordinates X(k,k+r+l), k E Z. This shows that the kernel of the surjective homomorphism 7rr: X I---t 'JI'r ~ ('JI'2)Z is isomorphic to Z~, where Y = 1£2 denotes the group of dyadic integers. If p is replaced by the prime ideal p' = (1 + 3Ul + 2U2) C 9t2, then rand Q remain unchanged, but the kernel of 7rr becomes isomorphic to (1£2 x Z3)r, where 1£3 is the group of tri-adic integers. Finally, if p" = (1 + Ul + U2) C 9t2, and if rand Q are as in Example (1), then 7rr: X!1t2/plf I---t ('JI'Q)Z is a group isomorphism. D CONCLUDING REMARK 8.6. The material in this section (with the exception of Proposition 8.1) is taken from [38].

CHAPTER III

Expansive automorphisms of compact groups

9. Expansive automorphisms of compact connected groups In the Sections 7-8 we investigated the structure of Zd-actions of the form a fRd / p , where p c 91d is a prime ideal. Although we can find, for every Zd-action a by automorphisms of a compact, abelian group X, a sequence of closed, a-invariant subgroups X = Yo J Y 1 J ... such that aYj/Yj+l is of the form a fRd / qj for every j ~ 0, where (qj) is a sequence of prime ideals in 91d (Corollary 6.2), the reconstruction of a from these quotient-actions is a problem of formidable difficulty. Only when d = 1 can one 'almost' re-build the action a from the quotient actions a fRd / q; (Corollary 9.4), due to the fact that Q ®z 91 1 = Q[ut1] is a principal ideal domain. The main tool in this reconstruction is the following Lemma 9.l. LEMMA 9.l. Let VJt be a Noetherian torsion 911 -module, which is torsionfree as an additive group. Then there exist primitive polynomials ft,· .. ,fr in 911 such that Ii divides Ii+1 for all j = 1, ... ,r-1, and an injective 911-module homomorphism 0: VJt ~ 91t!(ft) EB··· EB91t!Ur) = IJt such that IJtjO(VJt) is finite.

PROOF. Since VJt is torsion-free, none of the prime ideals associated with VJt contains a non-zero constant, and the embedding a ~ 1 ® a of VJt in VJt(Q) = Q ®z VJt is injective. This allows us to identify VJt with the subset {1 ® a : a E VJt} C VJt(Q) and to assume for simplicity that VJt c VJt(Q). We write 91iQ) = Q[u±1] for the ring of Laurent polynomials in the variable u with rational coefficients, regard 911 as a subring of 91iQ), and observe that VJt(Q) is a module over 91iQ) and hence over 91 1 . By Theorem XV.2.6 in [51] there exist polynomials ft, ... ,fr in 91iQ) such that fj divides fJ+1 for j = 1, ... ,r-1, and an 91iQ)-module isomorphism Of: VJt(Q) ~ 91iQ) j ft91iQ) EB· .. EB91iQ) j f r 91iQ ) = IJt(Q). We assume without loss in generality that each fi lies in 911, and that 77

78

III. EXPANSIVE AUTOMORPHISMS OF COMPACT GROUPS

Ii E 9\1 is primitive. Since 0'(9J1) C I)1(Q) is a Noetherian submodule over 9\1, there exists an integer k 2:: 1 with

O'(9J1)C~I)1,

(9.1)

where 1)1 = 9\1/ h9\1 EEl ... EEl 9\1/ fr9\1 C I)1(Q). Similarly we conclude the existence of an integer I 2:: 1 such that

1)1 C

~8'(9J1).

(9.2)

We claim that kl)1/11)1 is finite. Indeed, if f = h ... fn then every element a E kl)1/11)1 is annihilated by the ideal (kl, f) = kl9\1 + f9\1, so that every prime ideal associated with kl)1/11)1 must be non-principal. Since 9\1/p is finite for every non-principal prime p C 9\1 (Example 6.17 (3)), Proposition 6.1 implies that kl)1/11)1 is finite. Hence kl)1/8'(9J1) is finite by (9.2), and the proof is completed by setting O( a) = kO' (a) for every a E 9J1 and by noting that 8(9J1) C 1)1, by (9.1). D THEOREM 9.2. Let 0: be an expansive automorphism of a compact, connected group X. Then there exist primitive polynomials h, ... ,fr in 9\1 such that fj divides f)+1 for j = 1, ... , r - 1, and continuous, surjective, finiteto-one group homomorphisms "I: Y = X'."ftl/U,) x ... x X'."ftl/Ur) I----> X and "I': X I----> Y, such that 0: . "I = "I . 0:' and 0:' . "I' = "I' . 0:, where 0:' is the automorphism o:'."ftl/h'."ftl x ... x 0:'."ftl/fr'."ft 1 ofY. PROOF. The group X is abelian by Theorem 2.4, and we write 9J1 = X for the 9\1-module arising from Lemma 5.1. Then 9J1 is Noetherian by (4.10) and Proposition 5.4, and the connectedness of X implies that 9J1 is torsion-free as an additive group. We apply Lemma 9.1 to find the polynomials h, ... , fr and set Y = X'.l1, "I = iJ, and "I' = 8,-1.'IjJ, where 'IjJ: 1)11----> 8'(9J1) is the map 'IjJ(a) = la (cf. (9.2)). In the proof of Lemma 9.1 we have seen that II)1 C 8'(9J1) C kl)1, and that kl)1/11)1 is finite. Hence 1)1/8(9J1) ~ kl)1/8'(9J1) and 8'(9J1)/11)1 are finite, and duality shows that ker(TJ) and ker(TJ') are finite. D REMARK 9.3. Although the polynomials h, ... , fr in Lemma 9.1 and Theorem 9.2 are obviously not unique, the ideals (Ij) = iJ9\1' j = 1, ... ,r, are unique by Theorem XV.2.6 in [51]. Hence the automorphisms o:'."ftl/Ui), i = 1, ... ,r, are determined uniquely up to topological conjugacy (Theorem 5.9). COROLLARY 9.4. Let 0: be an automorphism of a compact, connected, abelian group X. The following conditions are equivalent. (1) 0: is expansive; (2) There exist primitive polynomials h, ... ,fr in 9\1 such that iJ divides iJ+1 for j = 1, ... ,r -1 and fr has no roots of modulus 1, and a finite, o:-invariant subgroup F eX, such that o:x/ F is algebraically conjugate to o:'."ftl/U') x ... x o:'."ftl/Ur);

9. EXPANSIVE AUTOMORPHISMS OF COMPACT CONNECTED GROUPS

79

(3) There exist primitive polynomials II, ... , fr in 9l l such that fj divides iJ+l for j = 1, ... ,r -1 and fr has no roots of modulus 1, and a finite, (a9'tt!(!l) x ... x a9'tt!(fr»)-invariant subgroup pI C X' = X9'tl/(f1l x ... x X9'tt!(fr), such that the automorphism induced by a ' = a9'tt!(!l) x ... x a9'tl/(fr) on X' / pI is algebraically conjugate to a.

PROOF. If a is expansive, a X / F is expansive for every closed, a-invariant subgroup P C X (by Corollary 6.15), and hence the automorphism a9'tl/(!l) x ... x a9't 1 /(fr) in (2) is expansive. According to Theorem 6.5 (4) this means that fr has no roots of modulus 1. If (2) is satisfied, then a ' = a9'tl/(!l) x· .. x a9'tt!(fr) is expansive by Theorem 6.5 (4), hence a X / F is expansive for some finite, ainvariant subgroup P eX, and this obviously implies that a is expansive. The equivalence (1){:=;>(3) is proved similarly. 0 Automorphisms of the form a9'tt!(f) with f C 9l l have already been discussed in Examples 5.2 and 6.17, and Corollary 9.4 shows how to obtain all expansive automorphisms of compact, connected, abelian groups from automorphisms of the form a9't t! (f) . In [52] there is a different realization of all expansive automorphisms of compact, connected, abelian groups in terms of rational matrices. In order to explain Lawton's description we consider the following generalization of Examples 5.6 (1) and 6.17 (2), where the underlying matrix was of the form (~). EXAMPLES 9.5. (1) Let AT E GL(n, Q) = Aut(Qn), and let X = Qri. Since Qn is torsion-free, the group X is connected. The automorphism a of X dual to f3 = AT on Qn is given by the transpose matrix A of AT. It is clear that a does not satisfy the d.c.c., since there does not exist a finite subset S = {Vl' ... , v m } C Qn such that Qn is generated by UkEZ (A k) T (S) (Corollary 2.16). In particular, a is non-expansive by (4.10). We write 9Jt* = Qn for the 9l l -module arising from a via Lemma 5.1. In order to determine the prime ideals associated with 9Jt* we consider, for every a E Qn, the annihilator ann(a) = {J E 9ll : f(A) . a = O}. Since XA has coefficients in Q, we multiply XA by the smallest integer k such that kXA E 9l l , and choose a prime decomposition of kXA = hl ... h m in 9l l . Then the set of prime ideals associated with 9Jt* is equal to {(hd, ... , (h m )). From Proposition 6.6 it is clear that a is non-ergodic if and only if one of the polynomials hi divides u l - 1 for some l 2: 1, i.e. if and only if A has an eigenvalue which is a root of unity. (2) In Example (1), let 1)1 C 9Jt* be an 9ll-submodule. Then the set of prime ideals associated with 1)1 is contained in {(ht), ... , (h m )). The group X'J1 = iJt is the quotient of X by a closed, a-invariant subgroup, and hence the automorphism a'J1 on X'J1 is ergodic whenever a is ergodic. We claim that a'J1 is expansive if and only if 1)1 is a Noetherian 9l l -module, and A has no eigenvalue of modulus 1. Indeed, if a'J1 is expansive, then (4.10) and Proposition

III. EXPANSIVE AUTOMORPHISMS OF COMPACT GROUPS

80

5.4 together imply that 1)1 is Noetherian, and Theorem 6.5 (4) shows that none of the polynomials hi can have a root of modulus 1. Conversely, if 1)1 is Noetherian, and if XA has no root of modulus 1, then am is expansive by Theorem 6.5 (4). The condition for ergodicity is unchanged from Example (1): am is non-ergodic if and only if A has an eigenvalue which is a root of unity. (3) In Example (2), let 9J7A = 1)1 = zn[A T , (A-1)T] be the subgroup of 9J7* = generated by

«r

Since 9J7A is invariant under left multiplication by AT, 9J7A is an ryh -submodule of 9J7*, and we write f3A for the automorphism of (the additive group) 9J7A defined by AT, and denote by a A the automorphism of x!Ut A = 9J7 A dual to f3A. In order to realize a A explicitly we consider the subgroup 3 c Z = LZ zn generated by all elements of the forme = (ek) E Lz zn such that ek = 0 for k tj. {l, l + 1} and BT el+1 = -mel for some integers l E Z and m ~ 1 for which B = mA has integer entries. The group X A = 3.L C (']['n)Z is closed, shift-invariant, and is given by

XA = {x = (Xk) E (']['n)Z : lXk+1 = BXk for all k E Z and all m E Z for which B = mA has integer entries}.

(9.3)

We claim that the shift a on X A is conjugate to a A . Indeed, the homomorphism 'if;: Z t----t 9J7A, given by 'if;(e) = LkEZ(Ak)Tek for every e E Z, is well-defined,

and ker( 'if;) = 3. Hence 'if; induces a dual isomorphism 'fl: X A = 3.L = ZfB t----t with 'fl. a = a A . 'fl. This also proves that a A satisfies the d.c.c., and that 9J7A C 9J7* is Noetherian (cf. Proposition 5.4). From Example (2) we see that a A is expansive if and only if A has no eigenvalue of modulus 1, and ergodic if and only if no eigenvalue of A is a root of unity. []

rotA

REMARKS 9.6. (1) If we apply Lemma 9.1 to the rytl-module 9J7A in Example 9.5 (3), we obtain the primitive polynomials ft, ... , fr in ryt1 described in the statement of Lemma 9.1. Choose mi E Z so that gi = mil fi E lR is monic for i = 1, ... , m. Then there exist integers l, l' such that ulgr(u) is the minimal polynomial of A, and u l' gl (u) ... gr (u) = XA (u) is the characteristic polynomial of A (cf. [51]). The presence of the monomials u l and u l ' is due to the fact that we are dealing with Laurent polynomials rather than polynomials. (2) The relation between 9J7 = 9J7A and rytd(ft) EB· .. EBrytd(Jr) in Lemma 9.1 and Remark (1) is completely analogous to that between the matrices A and B in Example 5.3 (2). THEOREM 9.7. An automorphism a of a compact, connected group X is expansive if and only if it is algebraically conjugate to an automorphism of the

9. EXPANSIVE AUTOMORPHISMS OF COMPACT CONNECTED GROUPS

81

form a A for some matrix A E GL(n, Q), n 2: 1, without eigenvalues of modulus 1 (cf. Example 9.5 (3)). Although Theorem 9.7 can be derived from Theorem 9.2, we give a different proof more closely related to the material in Section 4. The following definition and lemmas are more general than is necessary for the proof of Theorem 9.7, but we shall need them for the discussion of automorphisms of general compact groups in the Sections 10-12. DEFINITION 9.8. Let G be a compact group. A full, shift-invariant subgroup Y c G Z is called a Markov subgroup if there exists a (necessarily full) subgroup H c G x G such that Y

= YH = {y = (Yn)

E

GZ

:

(Yn, Yn+d E H for every n E Z}.

(9.4)

LEMMA 9.9. Let X be a compact group, and let a be a continuous automorphism of X which satisfies the d.c.c. Then we can find a compact Lie group G and a full subgroup H c G x G with the following properties. (1) If Y = Y H is the Markov subgroup defined in (9.4) and (Y is the shift on Y, then there exists a continuous isomorphism : X f----> Y with . a = (Y • ,(2) HO is a full subgroup of GO x GO, HO = H n (Go x GO), and yo = {y = (Yn) E Y : (Yn, Yn+1) E HO for every n E Z}. In particular, Y is connected if and only if H is connected.

PROOF. According to (4.10) we may assume that a is the shift on a full, shift-invariant subgroup X c G Z , where G is a compact Lie group. Proposition 4.14 and Remark 4.15 (1) allow us to assume furthermore that X = YH , where H c G x G is a full subgroup and YH is the Markov subgroup defined in (9.4), and that the connected components XO, GO, and HO, of the identity in X, G, and H, satisfy that XO = X n (GO)Z = {x E GZ : (xn,xn+d E HO for every n E Z}. Since y/y o is zero-dimensional and G and H are Lie groups, 7r{O}(YO) and 7r{O,I} (yO) are connected, open subgroups of G and H, respectively. Hence 7r{O} (yO) = GO, and 7r{O,I}(YO) = HO is a full subgroup of GO x GO. If HO i= H n (Go x GO) we choose a point (g,g') E H n (Go x GO) which does not lie in HO, and use the fullness of HO c GO x GO to find a point x = (xn) E GZ with Xo = g, Xl = g', and (xn,xn+d E HO c H n (Go x GO) whenever 0 i= n E Z. Then x EX n (GO)Z = XO, but 7r{O,l}(x) t/:. 7r{O,l}(XO), which is absurd. It follows that HO = H n (Go x GO). The last statement is obvious. 0 LEMMA 9.10. Let G be a compact Lie group, He G x G a full subgroup, and let

Fii

= {g

E

G: (10,g) E H}, Fii

= {g

E

G: (g, 10) E H}.

(9.5)

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III. EXPANSIVE AUTOMORPHISMS OF COMPACT GROUPS

Then Fil and Fii are closed, normal subgroups olG. lIthe group in (9.4) is expansive, then Fil and Fii are finite.

X = YH

C GZ

PROOF. It is clear that the groups Fjj are closed, and their normality is a consequence of the fullness of H. Now assume that YH is expansive. From (9.4) and the proof of Lemma 9.9 is is clear that GO = 7f{O} (YH), HO = 7f{O,l} (YH). In particular, HO is a full subgroup of GO x GO. Since YHis abelian by Theorem 2.4, HO and GO are abelian, and the finiteness of Fjj will follow from the finiteness of Fjjo. In order to establish this lemma it will thus suffice to assume from now on that G is connected and abelian, i.e. that G = 1['n for some n 2: 1, and that He G x G is a full, connected subgroup such that X = YH is expansive. The dual group of Z = (1['n)Z is (isomorphic to) the direct sum Z = Lz zn of copies of zn, indexed by Z, and the automorphism a of Z dual to the shift a on Z is again the shift. The annihilator X..L C Z is shift-invariant, X = Z/ X..L, and & is the automorphism of X induced by the shift a. As described in Lemma 5.1 and Proposition 5.4, X = ml is a Noetherian 9't 1-module under the action (/, a) 1--+ I * a, where I * a = I(&)(a) = LkEZ Cf(k)&k(a) for every I E 9't 1 and a E ml (cf. (5.2)). For every k E zn we write t 1--+ Xk(t) = (t, k) for the character of 1['n corresponding to k. If the group Fil is infinite, there exists a k E zn such that Xck does not annihilate Fil for any =I- C E Z. We define an element a E X = ml by x 1--+ (x,a) = Xk(7f{O} (x)), and claim that the annihilator ann(a) = {I E 9't 1 : I*a = o} is equal to zero. Indeed, ifO =I- I E 9't 1 and I * a = 0, then I = csu~ + Cs+l u~+1 + ... + ctui with -00 < s ::; t < 00, Ci E Z, CsCt =I- 0, and (x,1 * a) = (L~=sCjaj(x),a) = Xk(L~=sCjXj) = 1 for all x = (x m ) EX C (1['n)Z. According to the definition of X = YH there exists, for every t E Fil, an element x(t) = (x(t)m, m E Z) E X with x(t)j = O"][,n for j < t, and x(t)t = t. Then (x(t),J * a) = Xk(Ctt) = XCtk(t) = Uor all t E Fil, contrary to our choice of k. This shows that ann(a) = {O}, i.e. that {O} is one of the prime ideals associated with the Noetherian 9't 1 -module ml. By Theorem 6.5 (4) this is impossible in view of the expansiveness of a, so that Fil must be finite. Similarly one can show that Fii must be finite. D

°

PROOF OF THEOREM 9.7. Suppose that a is expansive. Then X is abelian by Theorem 2.4, and Lemma 9.9 and (4.10) allow us to assume without loss in generality that a is equal to the shift on a Markov subgroup X = YH C (1['n)Z, where H C 1['n X 1['n is a full, connected subgroup. The groups Fjj are finite by Lemma 9.10, so that H is an n-dimensional subtorus of 1['n x 1['n whose projection onto each copy of 1['n is surjective. An elementary argument shows that there exists a unique matrix A E GL(n, Q) such that H is the image of the subspace {(v, Av) : v E ]Rn} C ]Rn x ]Rn under the natural quotient map from ]Rn x ]Rn onto 1['n X 1['n. A brief glance at Example 9.5 (3) reveals that X = YH = X A, and that a is algebraically conjugate to a A . From Example 9.5 (3) we know that A cannot have any eigenvalues of modulus 1. This proves that (1)=}(2), and the reverse implication is contained in Example 9.5 (3). D

10. THE STRUCTURE OF EXPANSIVE AUTOMORPHISMS

83

CONCLUDING REMARK 9.11. The exposition in this section follows [45].

Theorem 9.7 is due to [52].

10. The structure of expansive automorphisms We fix a compact group 0 and a full subgroup H cOx 0, and define the Markov subgroup YH C OZ by (9.4). For every g E 0 and k E Z, put FH(g, k)

=

{Yk : Y

=

(Yn) E YH and Yo

= g}

C 0

(10.1)

and

(10.2) Then F H (±l) = FlI (cf. (9.5)), and FH(k) is a closed, normal subgroup of 0 with FH(k)

c

FH(k + 1) and F H ( -k)

c

F H ( -k - 1)

(10.3)

for every k :::: 0, since (Ie, Ie) E H. For k :::: 1, the sets FH(g, k) and FH(g, -k) are the usual k-th follower and predecessor sets of symbolic dynamics. From the definition of FH(g, k) it is clear that the map ()~: 0 f-----> OJ FH(k), obtained by setting (}~(g) = FH(g, k)j FH(k), is a well-defined, continuous group homomorphism with kernel F H ( -k), and that ()~ induces a continuous isomorphism

(10.4) LEMMA 10.1. Put A

= F H (-l)nFH (l),

0' = OjFH(l), H' = Hj(FH(l) 0' the quotient map. The shift O,Z given by 71(Y)n = ".,(Yn), nEZ, Y E

x FH(l)) c 0' x 0', and denote by".,: 0 commuting homomorphism 71: Y H Y H, has the following properties.

1---+

f----->

(1) 71(YH ) = YH" where YH' C O,Z is defined as in (9.4); (2) YH n FH(l)Z = ker(71) = AZ; (3) If FHI(k), k E Z, is defined as in (10.1)-(10.2) with YH' C O,Z replacing Y H , then FH'(k) = FH(k + l)jFH(l) for every k:::: 1; (4) There exists a H aar measure preserving Borel isomorphism '¢: Y H f-----> AZ X YH' which carries the shift on YH to the cartesian product of the shifts on AZ and Y H' ; (5) If OJ A is finite, the map '¢ in (4) can be chosen to be a homeomorphism; (6) If Y H is expansive, then A is finite and YH' is again expansive. PROOF. From (10.1)-(10.2) it is clear that hE F H (l) if and only if (Ie, h) E H, and h E F H ( -1) if and only if (h, Ie) E H. In particular, if".,': H f-----> H' is the quotient map, and if (g, h) E Hand ".,'(g, h) = (".,(g), ".,(h)) = (u, v) E H', then (g, hk) E Hand ".,'(g, hk) = (u, v) for every k E FH(l). It follows that there exists, for every (u, v) E H', and for every h E 0 with ".,( h) = v, an element g E 0 with ".,(g) = u and (g, h) E H. This implies (1).

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III. EXPANSIVE AUTOMORPHISMS OF COMPACT GROUPS

Now assume that (g, h) E ker(r7') c H. Then g, h E FH(l), (lG, h) E H (according to the definition of FH(l)), (g,lG) E H (since H is a group), and 9 E FH(-l) nFH(l) = A. Since AZ c YH this shows that Y = (Yn) E ker(1]) if and only if Y E A Z , as claimed in (2). In order to prove (3) we proceed by induction and assume that FH,(k) = FH(k + 1)/ FH(k) for k < K, where K :2: 1. If h E FH(K + 1), choose IG = ha, h l ,·.·, h k- l in G such that (hi, h i + l ) E H for i = 0, ... , K, where hK+1 = h. By setting h~ = TJ(hi ) we see that h~ = h~ = lG' and (h~, hi+1') E H' for all i = 0, ... , K, so that h' E FH,(K). Conversely, if h' E FH,(K), then we choose h~ E FH,(K - 1) with (h~, hi) E H', and the induction hypothesis allows us to find elements IG = ha, h l , ... , hK in G such that (hi, hi+l) E H and TJ(h i ) = h~ for i = 0, ... , K - 1, and TJ(h K ) = h~. There also exists an element (hK' h) E H with 1J(h K ) = h~ = 1J(h~) and 1J(h) = hi. Then kK differs from hK by an element of FH(1), so that (hK-l,k K ) E H, and by considering the sequence lG = h a, ... , hK-l, kK' h we see that h E FH(K + 1). This shows that FH,(K) = FH(K + 1)/ FH(l) and completes the induction step for (3). We denote by r: G ~ Gil = G / A the quotient map, choose a Borel map w: Gil ~ G with w(lG") = lG and r(w(u)) = u for every u E Gil, and define w: G" Z ~ GZ by w(w)n = w(wn ) for every W = (w n ) E G" Z. From (2) we see that the homomorphism r: Y H ~ G" Z, given by r(Y)n = r(Yn) for every Y = (Yn) C Y H, satisfies that ker(r) = ker(1]), and that yw(r(y))-l E AZ for every Y E YH. Since A Z C Y H we conclude that w(u) E YH for every u E r(YH), and that the map 9: YH ~ AZ X YH', given by 9(y) = (yw(r(y))-l, 1](y)), is a Haar measure preserving Borel isomorphism which sends the shift on YH to the cartesian product of the shifts on AZ and YH'. If Gil is finite, the maps w and ware both continuous, and we have proved (4) and (5). If YH is expansive, then A Z c Y H is again expansive, and A is finite. In this case the quotient map r: G ~ G / A is a homeomorphism of a neighbourhood N of the identity in G onto a neighbourhood N' of the identity in G / A, and by decreasing N, if necessary, we may assume that YH n N Z = {lGz} (the existence of such a neighbourhood NeG is equivalent to the expansiveness of Y H ). If Y' = r(YH ) c (G/A)Z is not expansive, there exists a point u = (un) E Y' n N'Z with u i- leG/A)Z. If u = r(y) for some y E Y H we can choose a point Z E AZ C YH with UnZ n E N for all n E Z. Since lGz i- UZ E YH n N Z we have arrived at a contradiction. Hence Y' is expansive. The continuous homomorphism 1]: YH ~ G'z induces a continuous, shift commuting isomorphism 1]': Y' = YH / AZ ~ YH', so that YH' is again expansive. This proves (6). D

°: :;

Let G be a compact group, H C G x G a full subgroup, G Z the Markov subgroup (9.4), and let (J be the shift (J(Y)n = Yn+l define FH(k), k E Z, by (10.1)-(10.2) and set, for every k :2: 0, FH(k)Z, Gk = G/FH(k), denote by 1J ek ): G ~ G k the quotient define a shift commuting map 1]e k ): Y ~ G~ with ker(1]e k ») =

PROPOSITION 10.2.

Y = YH C on Y. We Yk = Y n map, and

10. THE STRUCTURE OF EXPANSIVE AUTOMORPHISMS

85

Y k by 7J(k) (Y)n = 'T}(k) (Yn), n E Z. The maps 7J(k), k 2: 0, have the following properties. (1) 7J(k)(y) = YHk C G~, where Hk = H/(FH(k) x FH(k)), and where YHk is defined as in (9.4); (2) Yk+1/Yk ~ 7J(k) (Yk+l) = A~+1 for some closed, normal subgroup Ak+l C FH(k + 1)/ FH(k). (3) There exists a Haar measure preserving Borel isomorphism ¢: Y I - - t x··· x A~ X Y k +1 which carnes (1" to (1"(1) x··· x (1"(k) X (1"', where (1"(i) denotes the shift on A~ and (1"' is the automorphism of Yk+l induced by (1". The map ¢ can be chosen to be a homeomorphism if G is finite.

Af

If G is a compact Lie group such that C(GO) is finite, the following stronger assertions are true.

(4) There exists an integer K 2: 1 with FH(K) = FH(k) for all k 2: K, and an automorphism (3 E Aut(G K ) such that HK = {(u, (3(u)) : u E GK}, Y/YK ~ 7J(K)(y)

= YHK = {v E Gt : V n +l = (3(v n )

for all n E Z}

~GK'

and the isomorphism 7J(K) (Y) ~ G K sends the shift on 7J(K) (Y) to {3; (5) There exists a Haar measure preserving Borel isomorphism ¢: Y I - - t x··· x X GK which carnes (1" to (1"(1) x··· X (1"(K) X {3, where (1"(i) denotes the shift on A~, and ¢ can be chosen to be a homeomorphism if G is finite; (6) The restriction of (1" to Yk is ergodic for k = 1, ... ,K, and (1" is ergodic if and only ifYK = Y; (7) If YH is expansive, then G is finite.

Af

At

PROOF. In order to prove (1) we apply Lemma 10.1 to see that 7J(I)(y) = YH1 and note that the groups FHl (k), defined by (10.2) with YH1 and G1 replacing YH and G, satisfy that FHl (k) = FH(k + 1)/ F H (l) for all k 2: 1. Repeated application of Lemma 10.1 shows that 7J(k)(y) = Y Hk C G~ and FHk (m) = FH(m + k)/ FH(k) for k, m 2: 1. All the assertion in (1) are now obvious. For every k 2: 0, put Ak+1 = F Hk ( -l)nFHk(l) C FH(k+1)/FH(k) c G k , where Ho = H and Go = G. From Lemma 10.1 we know that 7J(k) (Yk+1) = 7J(k)(y) n F Hk (l)Z = A~+1' as claimed in (2), and the assertions in (3) follow from repeated applications of Lemma 10.1 (4)-(5). . If C(GO) is finite then G has only finitely many closed, normal subgroups, and the sequence FH(k), k 2: 1, must eventually become constant. Hence there exists a K 2: 1 with FH(k) = FH(K) for all k 2: K. Since the group G'K is a quotient of a compact, connected Lie group with finite centre, C(G'K) is again finite, and there exists an L 2: 1 with FHK (-l) = FHK ( - L) for all l 2: L. For every m 2: 1 we denote by ()m : GK/FHK(-m) I - - t GK/FHK(m) the isomorphism in (10.4) and put {3 = ()L+1()"i 1 : GK I - - t GK. From the definition

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of {3 it is clear that HK = {(u, {3(u)) : u E GK}, and the other statements in (4) are immediate consequences of this. The assertion (5) is obvious from (3), and (6) follows from (5) and from the fact that GK has no ergodic automorphisms (since Aut(GK) is compact, there exists a metric on GK which is invariant under every a E Aut(GK))' Finally, if Y is expansive, then Lemma 10.1 (6) implies that YHk is expansive and Ak is finite for every k = 1, ... , K, and that {3 is an expansive automorphism of G K. Since there exists a metric on G K which is invariant under every a E Aut(GK ) we conclude that G K must be finite. From (2) we conclude that YH is zero-dimensional, and Corollary 2.3 implies that G is finite. 0 COROLLARY 10.3. Let a be an automorphism of a compact group X such that C(XO) is zero-dimensional, and assume that a satisfies the d.c.c. Then there exist compact groups Al , ... , A K , G K, an automorphism {3 of G K, and a Haar measure preserving Borel isomorphism ¢: X t-----> Af x ... x A~ X GK which carries a to a(1) x ... X a(K) x {3, where a(i) denotes the shift on Af. The automorphism a is ergodic if and only if G K = {I}. If X itself is zero-dimensional, then a is expansive, the groups A l ,···, A K , G K, are finite, and the Borel isomorphism ¢ can be chosen to be a homeomorphism. PROOF. Lemma 9.9 allows us to assume that X = YH C G'L for some compact, connected Lie group G and some full subgroup H C G x G. Corollary 2.3 implies that aC(xO) is expansive, and that C(GO) = 7f{O} (C(XO)) is finite. Our assertion follows from Proposition 10.2. 0 COROLLARY 10.4. Let a be an automorphism of a compact group X which satisfies the d.c.c. If Z c X is a closed, normal, a-invariant subgroup such that C(ZO) is zero-dimensional and a Z is ergodic, then there exists a Haar measure preserving Borel isomorphism ¢: X t-----> Z x XjZ with ¢. a = (a Z x a X / Z ). ¢. If X is zero-dimensional, ¢ can be chosen to be a homeomorphism. PROOF. According to (4.10) and Remark 4.15 (1) we may assume that X = YH C G'L, where G is a compact Lie group and He G x G a full subgroup, and that Z = YL C K'L, where KeG is a closed, normal subgroup, L = H n K x K is a full subgroup, and YH and YL are given by (9.4). Since Z C K'L is full, the group C(KO) is finite by Corollary 2.3. The assertions follow exactly as in the proof of Proposition 10.2 by applying Lemma 10.1 repeatedly to YH , with FH(k) replaced by Fdk) c G for every k E Z. 0 EXAMPLES 10.5. (1) Let K be a compact group, G = K x K, and let H = {((k l ,k2), (k3,k4)) E G x G ~ K4 : k2 = k3}' Then H C G x G is a full subgroup. In the notation of Proposition 10.2 we have that F H (l) = {lK} x K, FH(2) = G, FH(-l) = K x {lK}, and FH(-2) = G. The group Al = FH(l) n FH(-l) is equal to {Ie}, G l = K X {IK} ~ K, and the

10.

THE STRUCTURE OF EXPANSIVE AUTOMORPHISMS

87

homomorphism 1J(1): YH 1---7 Gf has trivial kernel. Furthermore, HI = G l X G l , A2 = G l , and Y H ~ 1J(1) (YH ) = Y H1 = Gf ~ KZ. Note that YH is the two-block representation of the Bernoulli shift X = K Z , and that Proposition 10.2 has led us back from Y H to X.

(2) If G has infinite centre, the groups Ak in Proposition 10.2 may be trivial for every k ~ 1 even if Y H is ergodic. Consider the shift (J' = OI'JtdU) on X = X'JtdU) = {(xn) : 2x n = Xn+l (mod 1) for all n E Z}, where f = 2 - Ul E ~l' Then X = Y H with H = {(2t, t) : t E 'JI'} c 'JI' x 'JI' is full, PH(k) = {a}, and P H ( -k) = {l2- k : a :::::: l < 2k} for all k ~ 1. In particular, Gk = G, and 1J(k) is bijective for every k ~ 1. The ergodicity of YH was established in Theorem 6.5. 0 THEOREM 10.6. Let 01 be an ergodic automorphism of a compact group X satisfying the d. c. c. Then there exist an integer n ~ 1, a matrix A E GL(n, Q), none of whose eigenvalues is a root of unity, compact Lie groups AI,"" A K , K ~ 1, and a Haar measure preserving Borel isomorphism ¢: X 1---7 Af x ... x A~ X X A , such that ¢. 01 = (J'(l) X ... X (J'(K) X OI A , where (J'(i) and OI A are the shifts on Af and X A, respectively, and where X A is defined in (9.3). The automorphism 01 is expansive if and only if A has no eigenvalue of modulus 1 and every Ai is finite. PROOF. Lemma 9.9 allows us to assume that X is a Markov subgroup of G Z , where G is a compact Lie group, that 01 is equal to the shift on X, and that XO = X n (GO)z. We define PH(k), k E Z, as in (10.2), set P+ = Uk>l PH(k), and note that P+ is normal in G. From the definition of P+ it is clear that, for every x E X with Xn E P+ for some nEZ, x m+n E P+ for all m ~ a. If P+ is the closure of P+ and B -= {x EX: Xo E P+} then the continuity of 01 implies that OI-m(B) C B for all m ~ 1, i.e. that PH(9, k) c P+ for all 9 E P+ and k ~ 1 (cf. (10.1)). Let rJ: G 1---7 G' = GjP be the quotient map, and define 1J: G Z 1---7 G'z by (1J(x))n = rJ(xn) for all x = (xn) E Z and n E Z. The restriction to 1J(X) of the coordinate projection 7f{O} : G'z 1---7 G' induces a commutative diagram 1J(X) ~ 1J(X) 11"{O}

1

G'

l11"{O}

-----+

G'

T

where 7: G' 1---7 G' is a continuous, surjective group homomorphism, and the ergodicity of 01 forces 7 to be ergodic. If G' is not connected, then 7(G'O) = G'o and, since G' jG'o is finite, 7- 1 (G'O) = G'o. Hence G' must be connected or, equivalently, GO. P+ = G.

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If G' is non-abelian, then G' jC(G') is a compact, connected Lie group with trivial centre (cf. Proposition 2.13), r(C(G')) c C(G'), and r induces a homomorphism r': G'jC(G') f---> G'jC(G'). Since r' is surjective, ker(r') must be a finite, and therefore central, subgroup of G' jC(G'), and we conclude that r' C Aut(G' jC(G')). However, G' jC(G') has a metric which is invariant under Aut(G'jC(G')), and the ergodicity ofr' leads to a contradiction unless G' is abelian. We have proved that G' = G j P+ S::' Tm for some m ::::: 0, and by looking at dimensions of Lie algebras we see that there exist integers K ::::: 1 and n ::::: 0 such that G k = G j F H (k) S::' Tn for all k ::::: K. In the notation of Proposition 10.2 we obtain that '11(K)(X) = YHK C (Tn)'" and that F HK (±l) is finite, since G K = GjFH(K) S::' GjFH(K + 1) = G K +l, GjFH(-k) S::' GjFH(k) for all k ::::: 1, and FHK(l) S::' FH(K + l)jFH(K). In other words, HK C Tn X Tn must be isomorphic to Tn, which forces '11(K) to be of the form (9.3) for some matrix A E GL(n, Q). The remaining assertions all follow from Proposition 10.2. 0 If the automorphism a in Theorem 10.6 is non-ergodic, then Proposition 3.5 implies the existence of a maximal closed, normal, a-invariant subgroup X' C X such that aX' is ergodic and Xj X' is a compact Lie group. If a is

expansive, a stronger assertion can be made. THEOREM 10.7. Let a be an expansive automorphism of a compact group X. Then we can find an open, normal, a-invariant subgroup X' C X such that aX' is ergodic, and a Haar measure preserving homeomorphism 'ljJ: X f---> X' x Xj X' with 'ljJ. a = (aX' x aX/x') . 'ljJ. By combining the Theorems 10.6 and 10.7 we obtain the following corollary. COROLLARY 10.8. Let a be an expansive automorphism of a compact group X. Then there exist an integer n ::::: 1, a matrix A E GL(n,Q), none of whose eigenvalues has modulus 1, finite groups AD, AI"'" A K , K ::::: 1, and a Haar measure preserving Borel isomorphism ¢: X f---> AD xAfx ... xAt xXA such that ¢ . a = (J x a(1) x ... x a(K) x a A , where (J is an automorphism of AD, and where a(i) and (XA,a A ) are defined as in Theorem 10.6. For the proof of Theorem 10.7 we need a lemma. LEMMA 10.9. Let a be an ergodic automorphism of a compact group X. Then there exists, for every x E X and s ::::: 1, an element y E X with x = as(y)y-l. PROOF. We begin by proving the assertion under the assumption that X is abelian. Since a is ergodic, the automorphism 6: of X dual to a satisfies that 6: S (X) "I X whenever s::::: 1 and Ix "I X E X (Lemma 1.2 and Remark 1.7 (2)).

10. THE STRUCTURE OF EXPANSIVE AUTOMORPHISMS

89

Hence the homomorphism X f---> &S(X)X- 1 from X to X is injective, and the dual homomorphism x f---> o:S(x)x- 1 from X to X is surjective, as claimed. Next we assume that 0: is the shift on X = A'L, where A is a compact group. If x = (x n ) EX = AZ and s ~ 1 are fixed, there exists a unique point y = (Yn) E X with Xi = Yi for i = 0, ... ,s - 1 and Xn = yn+sy;;l for all nEZ, so that the lemma also holds in this case. Now assume that we have proved the lemma in general, but under the additional assumption that 0: satisfies the d.c.c. According to Proposition 4.5 there exists a non-increasing sequence (Vm' m ~ 1) of closed, normal, o:-invariant subgroups such that nm>l Vm = {Ix} and o:x/vm satisfies the d.c.c. for every m ~ 1. We fix x E X-and s ~ 1 and observe that the set Bm = {y EX: o:s(y)y-l E xVm } is closed and non-empty, and that Bm :J Bm+1 for all m ~ 1. The compactness of X implies that B = nm>l Bm -:f. 0, and x = o:s(y)y-l for every y E B. In order to prove our assertion for an automorphism 0: of a compact group X satisfying the d.c.c. we assume that X is a Markov subgroup of C Z , where C is a compact Lie group. We apply Proposition 10.2 and the proof of Theorem 10.6 and obtain an integer K ~ 1 such that the group C K (in the notation of Proposition 10.2) is isomorphic to ,][,n for some n ~ 0, and ry(K)(X) = YHK ~ XA for some A E GL(n,lQ) which has no root of unity as an eigenvalue. Furthermore, the closed, normal, o:-invariant subgroups Yk = X n FH(k)Z satisfy that {Ix} = Yo C Y1 C ... C Y K eX, X/YK ~ ry(K)(X), and Yk/Yk-l ~ A~ for k = 1, ... ,K. We fix x E X and s ~ 1, apply the first part of this proof, where the assertion was proved for abelian groups, to X/YK C (']['n)'L, and obtain a y(K+l) E X with o:s(y(K+l)(y(K+l)-l E xYK or, equivalently, with x(K) = o:s((y(K+l)-l)xy(K+l) E YK. Since YK/YK - 1 ~ A~, and since o:YK/YK-l corresponds to the shift on A~, the second part of this proof implies the existence of a point y(K) E YK such that o:s(y(K)(y(K)-l E x(K)yK _ 1. By repeating this argument we construct points y(i) E Yi, i = 1, ... , K, with o:S((y(1)-l) .... ·00s((y(K+1)-1)xy(K+1) ..... y(l) E Yo = {I}, which completes the proof of the lemma. D PROOF OF THEOREM 10.7. We choose a finite subset E E X such that {xX' : x E E} intersects each orbit of o:x/ x' in exactly one point. Fix x E E for the moment and denote by s(x) the smallest positive integer with o:s(x) (x) E xX'. Lemma 10.9 guarantees the existence of a y(x) E X' with o:S(X) (x)x- 1 = o:s(x)(y(x»y(x)-l, and we put x' = y(x)-lX and note that o:s(x)(x') = x'. The set E' = {o:k (x') : x E E, 0 ::::: k < s(x)} intersects each coset of X' in exactly one point, and we write z(v) for the unique element in E' n vX' for every vEX. Then o:(z(v» = z(o:(v» for every v E X, and the map v f---> 'l/J(v) = (vz(V)-l, z(v)X') from X to X' X X/X' is a homeomorphism with the required properties.

D

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III. EXPANSIVE AUTOMORPHISMS OF COMPACT GROUPS

REMARK 10.10. Corollary 10.3 gives a complete topological classification of ergodic and expansive automorphisms of compact, zero-dimensional groups (cf. [43]): two ergodic and expansive automorphisms a and a' of compact, zerodimensional groups X and X' are topologically conjugate if and only if they have the same entropy (cf. Chapter V). Indeed, if h(a) = h(a'), then Corollary 10.3 shows that h(a) = h(a') = logn for some n :::: 2, and that a and a' are topologically conjugate to the shift a on (71/n71..)'I·. Whereas the algebraic and topological classification of expansive automorphisms of compact, connected groups coincides by Theorem 2.4 and Theorem 5.9, the algebraic classification of expansive automorphisms of compact, zero-dimensional groups is much more complicated than the topological one, even if the groups are abelian. According to Corollary 5.10 and Example 6.17 (3), the latter problem is equivalent to the classification-up to module-isomorphism---of all Noetherian 9t 1-modules whose associated prime ideals are all of the form p = p9t1 for some rational prime p. The following examples may give some idea of the complexity of such a classification. EXAMPLES 10.11. Let p be a rational prime. (1) Let X be a compact group with px = 0 for every x EX, and let a be an ergodic automorphism of X. Then 9Jt = X is a module over the principal ideal domain 9t~p) (cf. Remark 6.19 (4)). By Example 6.17 (3) and the ergodicity of a, no non-zero ideal p C 9t~p) can be associated with 9Jt, so that 9Jt is a torsionfree 9t~p)-module. Theorem 2.2 in [51J implies that 9Jt is free, i.e. isomorphic to (9t~P))k for some k :::: 1, and duality shows that a is algebraically conjugate to the shift on (IF'~)'~. The integer k is obviously an algebraic conjugacy invariant; moreover, since a has entropy h(a) = k logp (cf. Chapter 5), k is also preserved by measurable conjugacies of such automorphisms. (2) If k :::: 2 then there exist infinitely many algebraically non-isomorphic 9t 1 -modules leading to algebraically non-conjugate ergodic group automorphisms with equal entropy k log p. In the following proof of this claim we denote by Mn(9t 1 ) the ring of n x n matrices with coefficients in 9t 1 . If 9Jt 1 = 9tt/pk 9tr, then a!m 1 is (algebraically conjugate to) the shiftaction of 7l on 7l'p k , and h(a!m 2 ) = klogp (cf. (5.5)-(5.6)). Note that a!m 1 is topologically, but not algebraically, conjugate to the 7l-action in Example (1). Now choose a non-zero element f = (11, h) E 9ti = 9t 1 x 9t 1 , consider the 9t 1 -module 9Jtf = 9ti/(pk 9ti + 9t 1 · f), and set a = (pk,h,h) = p k9t 1 + h9t 1 + h9t 1 c 9t 1 and b = (p, h, h) = p9t 1 + h9t 1 + h9t 1 c 9t 1 . We claim that the 9t 1-module 9Jtf is cyclic (and hence isomorphic to 9Jt1) if and only if a=b=9t 1 · Indeed, 9Jtf is cyclic if and only if there exists an element g = (gl, g2) E 9ti with 9t 1 . f + 9t 1 . g + p k 9ti = 9ti, i.e. if and only if the matrix A = (j~ ;~) E M 2 (9t 1 ) defines a surjective 9t 1 -linear map 4>A:

1)1 t--t 1)1,

where 1)1 = 9ti/pk 9ti.

10. THE STRUCTURE OF EXPANSIVE AUTOMORPHISMS

91

Suppose that ¢A is surjective. Since the kernel of ¢A is an 9t1-submodule of 1)1, an elementary rank- (or, by duality, entropy-) argument shows that ¢A is invertible. The adjoint matrix adj(A) =

(!f2 7/)

E

M2(9t1) of A satisfies

7/

that A . adj (A) == adj (A) . A = det A . (A ~ ), so that det A . ¢ is equal to the 9t1-linear map ¢adj(A): 1)11---+ 1)1 defined byadj A. Hence ¢A(I)1) C detA .1)1, which implies that multiplication by det A is invertible on 9tdpk9t1, det A . 9t1 + p k 9t 1 = 9t1, and a = b = 9t1. Conversely, if b = 9t1, then there exist elements 91, 92 in 9t1 such that 1192 - 9112 = 1 - ph for some h E 9t1. Hence there exist elements 9~, 9~, h' in 9t1 with 119~ - 9~h = 1 - pkh', which proves that a = 9t1 and that wtf has the cyclic element g = g + 9t1 . f + pk9t~ E wtf' Having verified that wtf is cyclic if and only if a = b = 9t1 we turn to the case where wtf is not cyclic. If f E p9t~, then wtf is obviously not cyclic, wtf jwt' is infinite for every cyclic submodule wt' C wtf, and h(o:!Dtf) > h(o:!Rl/pk!Rl) = k log p. Now suppose that wtf is not cyclic, but that f E 9t~ " p9t~. Then there exists an element g = (91, 92) E 9t~ such that det A E 9t1 "p9t 1 , where

A

= U~ ~~)

E

M 2 (9t 1 ). If wt'

c

wtf is the cyclic submodule of wtf generated

by g = g + 9t1 . f + pk9t~ E wtf, then the argument in the preceding paragraph shows that j{ = wtr/wt' ~ I)1/¢A(I)1), where ¢A: 1)11---+ 1)1 is the 9t1-linear map defined by A. By using the adjoint matrix adj(A) = (J~ ~~l) E M 2 (9td of A as above we see that every prime ideal p in 9t1 associated with the 9t1module j{ must contain the ideal q = p9t 1 + det A . 9t1, so that p induces a (principal) prime ideal jJ C 9t~p) generated by one of the irreducible factors of (det A) /p; moreover, every irreducible factor of (det A) /p occurs in this manner. In particular 9tl/p is finite for every prime ideal p C 9t1 associated with the Noetherian 9t1-module j{, so that j{ is finite and h(o:!Dtf) = h(o:!Dt') = klogp. Note, however, that o:!Dtf may not be ergodic: if f = (1 - U1, 1 - U1), then the polynomial 1 - U1 lies in a prime ideal associated with wtf, and o:!Dtf is non-ergodic by Theorem 6.5. It is now obvious how to construct infinitely many non-isomorphic modules wtf such that o:!Dtf is ergodic and h(o:!Dtf) = klogp: for any fixed, nonzero, irreducible element 1 =I=- h E 9t~p) we choose an I E 9t1 with I/ p = h and set f = (p, f) E 9t~. It is clear that p9t1 is the only prime ideal associated with wtf, so that o:!Dtf is ergodic by Theorem 6.5 and Proposition 6.6. Put g = (0,1) E 9t~ and consider the cyclic submodule wt' of wtf generated by g = g + 9t1 . f + pk9t~ E wtf. If A preceding paragraph shows that Iwtr/wt'l

= II)1jAI)1 I ~

= (~~ ~~),

Il)1jadj(A)AI)1I

then the discussion in the

= 19t1/(detA9t1 + p k 9t1W < 00,

and that the only prime ideal p C 9t1 associated with wtr/wt' satisfies that jJ = h9t~p). In particular, h(o:!Dtf) = h(o:!Rdpk!Rl) = klogp. Furthermore, ifwt"

92

III. EXPANSIVE AUTOMORPHISMS OF COMPACT GROUPS

is any other cyclic submodule of 9J1r such that 9J1r/9J1/1 is finite, then p is also associated with 9J1r /9J1/1. For different choices of h (and hence of j.l) we obviously obtain nonisomorphic modules 9J1r , which proves our assertion. [J CONCLUDING REMARK 10.12. Lemma 10.1 and Proposition 10.2 are taken from [111], [71] and [45], Corollary 10.3 for compact, zero-dimensional groups is due to [111] and [43], Example 10.5 (1) is taken from [43], and Theorem 10.6 can be found in [71] (cf. also [111]). Example 10.11 (2) is based on [22].

CHAPTER IV

Periodic points

11. Periodic points of Zd-actions

As we have seen in Theorem 5.7, every Zd-action by automorphisms of a compact, abelian group satisfying the d.c.c. has a dense set of periodic points (Definition 5.5), and Example 5.6 (1) shows that the d.c.c. cannot be dropped in general. In this section we investigate the density of the set of periodic points for a Zd-action a by automorphisms of compact group X satisfying the d.c.c. We begin with two examples which show that-if X is non-abelian-the d.c.c. does not necessarily imply that the set of a-periodic points is dense. EXAMPLES 11.1. (1) Let X = SU(2), and let hEX be an element of infinite order (i.e. h n =1= Ix for all n 2': 1). The inner automorphism a(x) = hxh -1, X EX, has no periodic points other than fixed points, and the set of fixed points of a is the closure of {h n : n E Z}, which is a maximal torus in X. It is clear that a satisfies the d.c.c., but that a is not ergodic. (2) Let X = SU(2)Z, and let h E SU(2) be an element of infinite order. We define (3 E Aut(X) by ((3(x))n = hxn h- 1 for every x = (xn) E X and nEZ, and consider the shift (J" on X. Since (3 and (J" commute, and since (J" is ergodic, (3 and (J" together define an ergodic Z2-action a with a(l,O) = (3 and a(O,l) = (J", which satisfies the d.c.c. by (4.10). Example (1) shows that the set of periodic points for this Z2-action cannot be dense. 0 We shall prove the following result. THEOREM 11.2. Let a be a Zd-action by automorphisms of a compact group X such that XO is abelian. If a satisfies the d. c. c., then the set of aperiodic points is dense in X. In conjunction with Theorem 2.4 and Corollary 4.7, Theorem 11.2 implies the following corollary. 93

94

IV. PERIODIC POINTS

COROLLARY 11.3. Let a be an expansive Zd-action by automorphisms of a compact group X. Then the set of a-periodic points is dense in X. For the proof of Theorem 11.2 we need several lemmas. If X is a compact group and a E Aut(X) we set

Fix(a)

= {x

Per(a) =

EX: a(x)

U Fix(a

= x}, n ).

(11.1)

n~l

LEMMA 11.4. Let X be a compact group, a an automorphism of X, and V a closed, normal, a-invariant subgroup of X. Suppose that the following conditions are satisfied.

(1) For every v E V and s 2: 1 there exists a point w E V with v = a 8 (w)w- 1 ; (2) Per(a V ) is dense in V; (3) Per(a x / V ) is dense in X/V. Then Per( a) is dense in X. PROOF. Let {j be a metric on X, and let {jf be the metric on X/V induced by {j. We fix x E X and c > 0, and choose a point U E Per(a x / V ) such that {jf(X, u) < c/2. If u = yV for some y E X then am(y)y-l = v E V for some m 2: 1, and assumption (1) implies the existence of awE V with am(w)w- 1 = v. Then am(w-1y) = w-1y, and u = w-1yV. By assumption (3), there exists an a-periodic point z E V with {j(w-1yz,x) < c, and w-1yz E Per(a). This shows that Per( a) is dense in X. 0 LEMMA 11.5. Let G be a compact group, H c G x G a full subgroup, Y = Y H the Markov subgroup defined by (9.4), and let a = a YH be the shift on Y H . Then Per(a Yk ) is dense in Yk for every k 2: 1, where Y k C Y is defined in Proposition 10.2. If G is a compact Lie group such that C( GO) is finite, and if a is ergodic, then Per( a) is dense in Y H . PROOF. We use the notation of Proposition 10.2. According to Proposition 10.2 (6), the sequence {ly} = Yo C Y1 C ... C YK = Y of closed, normal a-invariant subgroups of Y satisfies that Yk/Yk-l ~ A~ for every k = 1, ... ,K. Since aYk/Yk-l is the shift on Yk/Yk- 1, Per(aYk/Yk-l) is dense in Yk/Yk-l for every k, and Lemma 10.9 and the ergodicity of a Yk imply that there exists, for every k = 1, ... , K, every y E Yk , and every s 2: 1, an element z E Yk with a 8 (z)z-1 = y. Repeated application of Lemma 11.4 shows that Per(a Yk ) is dense in Yk for every k 2: 1. If G is a compact Lie group such that C(GO) is finite, and if a is ergodic, then YH = YK by Proposition 10.2 (6), so that Per(a) is dense in Y = Y K . 0

11. PERIODIC POINTS OF Zd-ACTIONS

95

LEMMA 11.6. Let G be a compact group such that C(GO) is finite, let He G x G be a full subgroup, and let Y = YH be the Markov subgroup defined by (9.4). The following conditions are equivalent.

(1) The set of shift-periodic points is dense in YH ; (2) The automorphism j3 of G K in Proposition 10.2 (4) has finite order.

PROOF. We use the notation of Proposition 10.2. The restriction of the shift u to YK is ergodic by Proposition 10.2 (6), and Lemma 10.9 implies that the shift u satisfies condition (1) in Lemma 11.4 (with u and YH replacing 0: and V). Lemma 11.5 shows that o:YK satisfies condition (2) of Lemma 11.4, and condition (3) is trivially satisfied if j3 has finite order. This proves that (2):::}(1). If j3 has infinite order, the finiteness ofInn(GK) in Aut(GK) implies that there exists an m ~ 1 such that j3m(g) = hgh- I for some element hE G K of infinite order. We denote by A the connected component of the identity in the closure of {h n : n E Z}. Every j3-periodic g E G K must commute with A, and hence the closure P of Per(,6) is not equal to G K , since An C(G K ) = {lG}. Since YYK E Per(,6) c P for every o:-periodic point y E Y, Per(o:) cannot be dense. This proves that (1):::}(2). 0 LEMMA 11.7. Let 0: be an expansive automorphism of a compact, zero dimensional group X, and let 8 be a metric on X. Then PerC 0:) is dense in X. Furthermore, if 0: is ergodic, there exists, for every e > 0, an integer N(e) ~ 1 such that Fix(o:n) is e-dense in X for every n ~ N(e) (a set Be X is e-dense if 8(x, B) < e for every x EX).

PROOF. By Lemma 9.9 and Corollary 2.3 we may assume that X = YH C G Z , where G is a finite group, He G x G a full subgroup, and YH the Markov subgroup defined in (9.4), and that 0: is the shift on X = YH. We use the notation of Proposition 10.2 and note that X/YK ~ G/ FH(K) = G K is finite, and that o:YK is ergodic. Since the automorphism j3 = o:X/YK is obviously of finite order, Lemma 11.6 shows that Per(o:) is dense in X. If 0: is ergodic, X = YK , and FH(K) = G. Hence we can find, for every s ~ 1 and x = (xn) E X, a point y E Fix(0:2s+2K) such that Xn = Yn for Inl :S s, and Ys+K = Y-s-K = IG. By choosing t sufficiently large we can ensure that Fix( o:m) is e-dense in X whenever m ~ 2t + 2K = N(e). 0 LEMMA 11.8. Let 0: be an automorphism of a compact, zero dimensional group, V C X be a closed, normal, a-invariant subgroup, and let "l: X ~ X/V be the quotient map. If o:V is ergodic, then "l(Fix(o:m» = Fix((o:X/V)m) for every m ~ 1.

PROOF. Let U E Fix(o:m), and choose x E X with u = "l(x). Then v = o:m(x)x- I E V, and Lemma 10.9 guarantees the existence of a point w E V with v = o:rn(w)w- I . Then w-Iv E Fix(o:m), and "l(w-Iv) = u. 0

IV. PERIODIC POINTS

96

LEMMA 11.9. Let a be an expansive Zd-action by automorphisms of a compact, zero dimensional group X, V c X a closed, normal, a-invariant subgroup, and let n E Zd be an element such that a~ is non-ergodic on V. Then there exists a closed, normal, a-invariant subgroup W 11m

-

m-+oo n-+oo

(V mEQA(m-2r)

Tm(U)

)

I detAI .,....".,-:-::--:---"'--,---,-':-:---....,....-,,...,. IG(QA(m - 2r), Q((m - 2r)n)1

.10gN(

V

mEC(QA (m-2r),Q((m-2r)n)

2: m-+oo lim lim IQ((ldetAI) )IIOgN( n-+oo m - 2r n

V

mEQ((m-2r)n)

T_m(U))

T-m(U))

= I detAI h(T,U).

By varying U we obtain that htop(T) = I det AI htop(TA).

0

PROPOSITION 13.2. Let T be a continuous Zd-action on a compact, metric space X, and let D.. C Zd be a subgroup of infinite index. Suppose that the restriction T(A): n 1-+ Tn, nED.., of T to D.. has finite topological entropy (cf. Remark 13.4). Then htop(T) = O. PROOF. We shall prove the proposition in the case where D.. = {m = (ml, ... ,md) E Zd : mk+1 = ... = md = O} for some k E {I, ... ,d - 1}the general case is only notationally more difficult. For every m, n 2: 0, put Q(m, n) = {-m, ... ,m}k x {-n, ... ,n}d-k. Let U be an open cover of X, and let Un = VmEQ(O,n) T_m(U). Then limm-+ oo ~ logN(VmEQ(m,o) T-m(Un )) =

V. ENTROPY

108

h(T(6.),Un ) :::; htop (T(6.») for every n 2: 0, and we choose, for every n 2: 0, an integer m(n) 2: n such that ~k logN(VmEQ(m,O) T-m(Un )) :::; 2htop (T(6.») for every m 2: m(n). Then (Q(m(n), n)) -+ 00 and IQ(mtn),n)l lOgN (

=

V

mEQ(M(n),n)

IQ(m(~),n)llOgN(mEQ(m(n),O) V T-m(Un)) 2

:::; n(d-k) htop(T as n

-+ 00.

T_m(U))

(6.)

)

-+

0,

Since this is true for every open cover U of X, htop(T)

= 0. 0

If T: m f--t T m is a measure preserving action of Zd on a probability space (Y, '1', /1) (i.e. a homomorphism from Zd into the group of measure preserving automorphisms of (Y, '1', /1)) we define the metric entropy of T by

h/-«T) = suph/-«T, P), P

(13.6)

where P ranges over the finite, measurable partitions of X,

h/-«T,P)

=

V

lim IQIIH/-« T-m(P)) , (Q)-->oo mEQ

(13.7)

and where H(Q) = H/-«Q) = - L.PEQ/1(P)log/1(P) is the entropy of a finite or countable measurable partition Q of X. The limit in (13.7) exists by subadditivity, and is equal to sUPQ h/-«T, Q) = sUPQ lim(Q)->oo Ibl H/-«V mEQ T_m(Q)), where Q ranges over the countable, measurable partitions of Y with finite entropy. We recall the notions of conditional information and conditional entropy. If P is a countable, measurable partition of Y and S c '1' a sigma-algebra, then the conditional information function I/-«PIS) is defined by

I/-«PIS)

=-

L

PEP

Ip log E/-«lpIS),

where Ip denotes the indicator function of P and E/-«·IS) is the conditional expectation with respect to S, and

H/-«PIS)

=

J

I/-< (PIS) d/1

is the conditional entropy of P with respect to S. If Q is a second countable, measurable partition of Y we write I/-«PIQ) and H/-«PIQ) instead of I/-«PIE(Q)) and H/-«PIE(Q)), where E(A) is the sigma-algebra generated by a family of sets A. In the special case where S is the trivial sigma-algebra {0, Y} we use the notation I/-«P) = I/-< (PIS) and note that H/-«PIS) = H/-«P).

13. ENTROPY OF Zd-ACTIONS

109

It is sometimes convenient to use another, equivalent definition of metric entropy. Let -< be the lexicographic order on 7J.,d. For any countable, measurable partition P of Y with finite entropy we set p:;. = Vm-(PIP:;.) =

i

I/1>(PIP:;.) dll,

(13.8)

and hence (13.9) If a countable, measurable partition P of Y is a generator for T, i.e. if the sigma-algebra

E( U T_m(P)) = V T_m(P) mEZ d

generated by

UmEZd

mEZ d

T_m(P) is equal to '1', then (13.10)

A sequence (Pn , n::::: 1) of countable, measurable partitions of X is increasing if the sigma-algebras E(Pn ) generated by P n form an increasing sequence. If (Pn , n ::::: 1) is such an increasing sequence, and if the sigma-algebra

is equal to '1', then (13.11) For background and details we refer to [84], [77], [105], [17] and [42]. Let a be a 7J.,d- action by automorphisms of a compact group X with normalized Haar measure .Ax, and let (j be a metric on X. We denote by Bo(C:) the c:-ball around Ix and set

hvo1(a) where Bo(Q,c:) THEOREM

group X. Then

=

1 lim limsuP--IQllog.Ax(Bo(Q,c:)),

0-->0 (Q)-->oo

(13.12)

= nmEQa-m(Bo(c:)). 13.3. Let a be a 7J.,d- action by automorphisms of a compact

and this common value is denoted by h(a).

V. ENTROPY

110

PROOF. Let 8 be a metric on X, and let E be a (Q, 8, c)-separated set of cardinality sQ(8,c). Then the sets XB6(Q,c), x E E, are disjoint, and hence sQ(8,c) ::; Ax(B6(Q,c))-1. By letting (Q) -> 00 and c -> 0 we see that (cf. Proposition 13.1)

htop(a)

= hsep(a) ::; e~O lim

1 liminf --IQllogAx(B6(Q,c)) ::; hvol(a). (Q)~oo

If P is a finite partition of X into Borel sets of diameter < c, then every set P in the partition Q = VmEQ a_m(P) is contained in a translate of B6(Q, c), and hence

H>,x (

V a-m (P)) = -

mEQ

L

Ax (A) log Ax (A)

AEQ

Ax(A)log Ax(B6(Q,c)) = -logAx(B6(Q,c)).

?:: - L AEQ

This proves that hvol (a) ::; h >'x (a). Finally we prove that h >'x (a) ::; h top (a). Let P = {P1 , ... , Pd be a finite, measurable partition of X, and choose c such that 0 < c < l/klogk. Since Ax is regular there exist compact sets PI C Pi, i = 1, ... , k, with Ax(Pi '-.. PI) < c. Let P' be the partition {P~, P{, ... , PG, where P~ = X'-.. U:=l PI- Then Ax (P~) < kc, and the conditional entropy H>,x (PIP') satisfies that . k

H>'x(PIP')

k

= - LLAx(PInPj)log(Ax(PfnPj)/Ax(Pf)) i=O j=l k

= -Ax(P~) L(Ax(P~ n Pj)/Ax(P~)) log(Ax(P~ n Pj)/Ax(P~)) j=l

::; Ax (P~) log k

< kdog k
X the product map 7f(x, y) = xy, write 7fi: X X X I-----> X, i = 1,2, for the coordinate projections, and consider the sigma-algebras 6 = 7f-1(!.B X ) c !.B xxx and 6 i = 7f;l(!.BX) c !.Bxxx, i = 1,2. Note that 6 and 6 i , i = 1,2, are invariant under the I£d-action a x a: n ~ an x an on X x X. Furthermore, 6 V 6 1 = 6 V 62

= !.Bxxx;

(13.14)

to prove this for i = 1, say, it suffices to note that the maps 7f1 and 7f2: (x, y) ~ y = 7fdx)-l7f(x, y) are both 6 V 61-measurable. Let MdX) and M1 (X)'" be the sets of probability measures and 0.invariant probability measures on !.B x, respectively. The convolution p, * v of

v. ENTROPY

112

two measures J.l,v E MdX) is defined by J.l*v(B) = J J1B(xy)dJ.l(x)dv(y) for every B E IB x. Since J.l * v is essentially the restriction of the product measure J.l x v on X x X to the a x a-invariant sigma-algebra 6 c IBxxx, it is clear that J.l * v E M1 (X), and that J.l * v E M1 (X)Q whenever J.l, v E M1 (X)"". We choose increasing sequences (Pn , n:::: 1) and (Qn, n :::: 1) of finite partitions in 6 c IBxxx and 6 1 C IBxxx, respectively, such that E(Pn ) / 6 and E(Qn) / 6 2 , where E(C) is the sigma-algebra generated by a family of sets C. From (13.8) it is obvious that h/1>xlI(a, P n V Qn) :s:; h/1>xlI(a, P n ) + h/1>xlI(a, Qn) for every n :::: 1, and (13.11) shows that h/1>xll(a, P n ) -. h/1>w(a) and h/1>xlI(a, Qn) -. hll(a) as n -. 00. According to (13.14) and (13.11), h/1>xlI(a, Pn V Qn) -. h/1>xlI(a) = h/1>(a) + hll(a) as n -. 00, so that

h/1>(a)

+ hll(a) = limh/1>xlI(a, P n V Qn) n :s:; limh/1>xll(a, Pn ) + limh/1>xlI(a, Qn) n n = h/1>w(a) + hll(a).

In particular, if hll(a) < 00, then h/1>(a) :s:; h/1>w(a), and by setting v noting that J.l * AX = Ax we have proved the proposition. 0

(13.15)

= Ax and

LEMMA 13.6. Let a be a Zd-action by automorphisms of a compact group X, and let (Vn' n:::: 1) be a decreasing sequence of closed, normal, a-invariant subgroups of X such that nn>1 Vn = {Ix}. Then h(a) = limn-->oo h(a x / Vn ) = sUPn21 h(a x / Vn ). -

PROOF. According to (13.6)-(13.7), h(a) = h)..x(a) = sUPp h)..x(T, P), where P ranges over the finite, measurable partitions of X. We write en: X f----+ XjVn for the quotient map. If P is a finite, measurable partition of XjVn then h(ax/Vn,P) = h(a,e;;1(p)), and we conclude that h(a) :::: h(a x / Vn ) for all n :::: 1. Conversely, if P is a finite, measurable partition of X, the increasing martingale theorem shows that limn-->oo H)..x (PllB n ) = 0, where IBn = e;;1(lB x / y J. Hence there exists, for every c: > 0, an n :::: 1 and a finite, measurable partition p' of XjVn such that H)..x (Ple;;1(p I)) < c:, and (13.7) shows that h(a, P) :s:; h(a, e- 1(PI)) + c:. Hence h(a) :s:; h(a x / Vn ) + c: for sufficiently large n, and the lemma is proved. 0 The final result shows that the topological entropy of the shift-action of c C Zd , where C is a compact group, can be calculated in terms of a maximum metric. Let C be a compact group, and let {) be a metric on C. If X C C Zd is a closed, shift-invariant subgroup, Q C Zd a rectangle, and c: > 0, we call a set E C X [Q, {), c:]-spanning for the shift-action (J" of Zd on X if there exist, for every x E X, ayE E with {)(Xlll' Ylll) < c: for all m E Q, and E is [Q, 0, c:]-separated if there exists, for every pair x =I y in E, an m E Q with {)(Xlll,Ylll) :::: c:. Let rQ({),c:) and sQ(o,c:) be the smallest cardinality of

Zd on a closed, shift-invariant subgroup X

13. ENTROPY OF Zd-ACTIONS

113

a [Q, '!9, E]-spanning set and the largest cardinality of a [Q, '!9, E]-separated set, respectively. PROPOSITION

13.7. The entropy of the shift-action fJ of 7!.,d on the closed,

shift-invariant subgroup X

c

G71 d satisfies that

1

1

1

1 lim liminf -IQllogsq('!9,E)

h(fJ) = lim lim sup -I-l logrq('!9, E) = lim liminf -IQllogrQ('!9, E) E--->O (Q)--->oo Q E--->O (Q)--->oo

= E--->O lim lim sup -IQllogsQ('!9,E) = (Q)--->oo =

E--->O (Q)--->oo

lim lim sup -IQ1110g >'x(B~(Q, E))

(13.16)

E--->O (Q)--->oo •

= E--->O lim liminf _11110g>'x(B~(Q'E))' _ (Q)->oo Q where B~(E) = {x EX: '!9(xo, Ie) < E} and B~(Q,E) = nmEQ fJ_m(B~(E)). Furthermore X is expansive if and only if there exists an E > 0 such that, for every x -j. Ix in X, '!9(x m , Ie) > E for some m E 7!.,d. PROOF.

Let 8 be the metric on G71d defined by

8(x, y) =

2:.::

mE71

r lml '!9(xm , Ym)

(13.17)

d

7l d

for all X,Y E G , where Iml = max{lmll, ... ,lmdl} for every m = (ml'"'' md) E 7!.,d. We claim that there exists, for every E > 0, an E' > 0 and an integer bE :::: 0 such that max '!9(x m , Ym) {mE71 d :Iml ~b,,}

> E whenever 8(x, y) > E'.

Indeed, let K = L:mE71 d 2- lm l < 00, choose bE such that L:{mE71 d :lml>b,,} 2- lml < E, and put E' = (K + l)E. If '!9(x m , Ym) ::; E for all m E 7!.,d with Iml ::; bE then

8(x, y) ::;

E

+

Erlml ::; (K

+ l)E = E'.

We fix E > 0 and note that there exist b = bE :::: 0 and E' > 0 such that, for every rectangle Q with sides of length l7, j = 1, ... , d, and for every (Q, 8, E)separated set E E X (cf. (13.3)), there exists a smaller rectangle Q' C Q with sides of length l7' = lj - 2b such that F is [Q', '!9, E']-separated. Conversely, if E is [Q, '!9, E]-separated, then E is obviously (Q, 8, E)-separated. This proves that

h(fJ) = lim lim sup IQ1110gSQ(8,E) = lim lim sup IQ1110gSQ('!9,E) E->O (Q)->oo

E->O (Q)--->oo

v.

114

ENTROPY

= 0lim liminf IQl1l0g SQ(8, E) = lim liminf IQl1l0g SQ('!9, E) ....... 0 (Q) ....... oo 0 ....... 0 (Q) ....... oo (cf. (13.4)). The other identities are proved similarly, and the final statement about expansiveness is an immediate consequence of the assertion at the beginning of this proof. 0 CONCLUDING REMARKS 13.8. (1) The exposition of entropy presented in this section follows [105] with some obvious changes necessitated by considering Zd-actions rather than Z-actions (cf. [42] or [63]). The proof of Proposition 13.5 is taken from [10].

(2) In the notation of the proof of Proposition 13.5, /1 * v = (/1 X v)1f- 1 , and 1f. (an X an) = an . 1f for every n E Zd. Hence h/J-*V(a) :::; h/J-(a) + hv(a) for all /1, v E Ml (X)

X

Y is the map T/(x)

=

X

(14.3)

ZxY

----->

Tm

ZxY

commutes for every m E Zd. This shows that the measure preserving Zd_ actions a and T on (X, Ax) and (Z x Y, Az x Ay) are measurably conjugate, and Theorem 13.3 implies that h(a) = h>,x(a) = h>,zx>,y(T). Theorem 14.1 thus reduces to proving that (14.4) The commutativity of the diagram (14.3) is preserved under quotients by closed, a-invariant subgroups of Y which are normal in X. Indeed, let V c Y be such a subgroup, and denote by T(V) the Zd-action TZ)(z, yV)

= (a!(z), b(m, z)a;;jV (yV)) = (a!(z), b(m, z)a;:'(y)V)

on (Z x Y/V, Az x Ay / v ). We define T/(V): X/V

I-->

(14.5)

Z x Y/V by

T/(V)(xV) = (O(x)V, c(O(X))-lXV), and note that the diagram

X/V ry(V)

1

Zx Y/V

X/V am

----->

X/V

1

ry(V)

-----> T(V) m

(14.6)

ZxY/V

again commutes for every m E Zd. Hence (14.7) for every closed, a-invariant subgroup V c Y which is normal in X. If P is a finite, measurable partition of Y and y E Y, let y . P be the partition {y . P : PEP}. We choose sequences of finite, measurable partitions (Pn , n :2': 1) and (Qn, n :2': 1) of Y and Z, respectively, which increase to the point partitions (i.e. which satisfy that E(Un>l P n ) = !By and E(Un>l Qn) = !B z ). For every rectangle Q C Zd, Z E Z, and- n :2': 1, we set -

Q~ =

V a~m(Qn), mEQ

P;:

=

V a~m(Pn)' mEQ

V. ENTROPY

116

V b( -m, z)· a~m(Pn)'

P~z =

mEQ

If Qm

X

Pn = {Q

X

P :Q

E

Qm, P E Pn }

then we obtain from (13.6), (13.7) and (13.11) that h(a z

X

a Y ) = sup sup h)..zx)..y(a z x a Y , Qm x P n ) n;::>:lm;::>:l = sup h)..z(a z , Qm) + sup h)..y (a Y , Pn) m;::>:l n;::>:l = h( a Z ) + sup

lim IQ11 n;::>:l (Q)-+oo

)

fZ H)..y (P;t)d>"z,

where a Z x a Y denotes the product action (m,(z,y)) Z x Y. For every rectangle Q C Zd we set

~ = {Q x Y : Q E Q~},

Pn

= {Z

X

(14.8)

P: P E Pn},

1---+

(a~(z),a;;'(y)) on

VT-m(Pn),

~=

mEQ

and note that I)..zx)..y (

V T-m(Qn x Pn) I~) = hzx)..Y(~I~)·

mEQ

Hence H)..zx)..y (

V T-m(Qm x Pn))

mEQ

V T-m(Qm x Pn) I Q~)

= H)..y(Q~) + H)..zx)..y ( Q + = H)..y(Qn)

and (13.11) implies that

1

ZxY

mEQ

="zxY,

h)..zx)..y(T) = sup sup h)..zx)..y(T, Qm x Pn ) n;::>:l m;::>:l = h)..z(a z ) = h(a z )

+ n;::>:l sup lim (Q)-+oo

+ n;::>:l sup lim (Q)-+oo

IQ11

IQ11

)

)

fZxY hzx)..Y(~I~~) d>"zxY

fZ }y f hy(P~z) d>"y d>..z(z)

(14.9)

lim IQ11 f H)..y(P~z) d>"z(z), n;::>:l (Q)-+oo )Z

= h(a z ) + sup

where ~~ = {B x Y : B E ~z}. The interchange of limits in (14.9) has to be justified, of course. The next lemma is motivated by a comparison of (14.8) and (14.9).

14.

YUZVINSKII'S ADDITION FORMULA

117

LEMMA 14.2. Let H be a compact Lie group. Then there exist a constant K > 0 and a sequence (Pn , n 2: 1) of finite, measurable partitions of H, which increases to the point partition, such that, for every n 2: 1, P E P n , and hE H, the number of non-empty sets in {P n pI : pI E h· P n } is less than or equal to K. If H is finite we may choose K = 1. PROOF. For a finite group H the assertion is trivial: put P n equal to the point partition of H for every n 2: 1. If H is infinite, let 5) be the Lie algebra of H. We choose and fix a positive definite, bilinear form B(·,·) on 5) x 5) which is invariant under the adjoint representation of H on 5), and denote by 8 the Riemannian metric on H defined by B(·, .). For notational convenience we identify (5), B) with (ll~m, (-,'J), where (-"J is the Euclidean inner product on .~m. For every E > 0 we set I(E) = [-E, E)m C ~m ~ 5), and we denote by P~(E) the partition of I(E) into disjoint translates of [-E2- n ,E2- n )m. The exponential map exp: 5) f---> H defines a homeomorphism of a neighbourhood N(O) of 0 E 5) onto a neighbourhood N(IH) of the identity in H; since the derivative of the exponential map at IH is the identity there exists a an EO with 0< EO < such that I(Eo) c N(O) and, for every n 2: 1 and P c P~, exp(P) contains a 8-ball of radius Eo2- n m- i and is contained in a 8-ball of radius Eo2- n m. Since the set NEO = exp(I(co)) contains an open neighbourhood of IH, there exist elements ho = IH, hi"'" hs in H such that He U;=i hjNEO '

4!n

Put Eo = NEo and E j = hjNEo " U{":~ hkNEo for every j = 1, ... ,s, denote by P4 the partition of E j induced by hj exp(P~), and set P n = U;=o P4· Then P n is a finite, measurable partition of H, and P n increases to the point partitions as n --+ 00. There exists a constant C > 0 such that C-iE m ::; AH(Bli(E)) ::; CEm for all E with 0 < E ::; 1, where Bli(E) denotes the open ball with 8-radius E and centre IH. Hence, if 0 < r ::; R < 1, at most C 22m(R/r)m disjoint open translates of Bli(r) can intersect Bli(R). We conclude that, for every h E Hand P E P~ = exp(P~), at most 22mm2m sets of the form pnQ, Q E h·exp(P~), are non-empty. From the definition of P n it is now clear that every PEPn can have non-empty intersection with at most 22mm2m(s + 1) sets in h . P n for every h E H. 0

c

c

PROOF OF THEOREM 14.1. We claim that it suffices to prove Theorem 14.1 under the additional hypotheses that h(aY ) < 00 and that a satisfies the d.c.c. If h(a Y ) = 00, then h(a) = hsep(a) 2: hsep(a Y ) = h(a Y ) = 00, and Theorem 14.1 holds trivially. Now suppose that Theorem 14.1 has been shown to hold for every Zd-action satisfying the d.c.c. If a is an arbitrary Zd-action by automorphisms of a compact group X with h(a) < 00 we apply Proposition 4.9 and choose a decreasing sequence (Vn, n 2: 1) of closed, normal, a-invariant subgroups of X such that nn>i Vn = {Ix} and a x / vn satisfies the d.c.c. for every n 2: 1. By assumption,- h(a x / Vn ) = h(a x / YVn ) + h(a Y / Vn ) for every

v. ENTROPY

118

n 2: 1. Lemma 13.6 shows that, as n ---+ 00, h(a X / Vn ) ---+ h(a), h(a X / YVn ) ---+ h(a x / y ), and h(aY/ Vn ) ---+ h(a Y ), and we obtain that h(a) = h(ax/Y)+h(aY ). Let therefore a be a Zd-action by automorphisms of X which satisfies the d.c.c., and let Y be a closed, normal, a-invariant subgroup of X with h(a Y ) < 00. According to (4.10) we may assume that X is a full, shift-invariant subgroup of G Zd , where G is a compact Lie group, that Y = {x E X C GZd : Xn E H for all n E Zd}, where H = 7r{OZd}(Y) is a closed, normal subgroup of G, and that a is equal to the shift-action of Zd on X. By Remark 4.15 we may take it that X = {x E G

Zd

d

: 7r{O,l}d(an (x)) E 7r{O,l}d(X) for every n E Z },

Y = {y E H ~ : 7r{O,l}d(an (y)) E 7r{O,l}d(Y) for every n E Z d }. (14.10)

Let (Qn, n 2: 1) be a sequence of finite, measurable partitions of Z = XjY which increases to the point partition. We apply Lemma 14.2 and choose a sequence (P~, n 2: 1) of finite, measurable partitions of H, which increases to the point partition, such that, for every n 2: 1, P E P~, and hE H, the number of non-empty sets in {P n pI : pI E h· P~} is ::::; K. Since the entropy of a partition of a probability space into K sets is less than or equal to log K we obtain that

(14.11) for every n 2: 1 and h E H. For every n 2: 1, let Pn = 7r{Olzd} (P~), and note that HAy (Pnly . P n ) ::::; logK and HAy(Y· PnlPn ) ::::; logK for every n 2: 1. From (14.8) and (14.9) we see that, for all n 2: 1,

h(a z x a Y ) = h(a z ) + sup hAY (a Y , P n ) n;:>:l

::::; h(a z ) + sup lim

n;:>:l (Q)->oo

IQ11 [ HAY (P;:IP;:,J d'\z(z)

JZ

+ sup

lim

n;:>:l (Q)->oo

::::; h AZ XAy (T)

+ log K,

hAZXAy(T)=h(az)+sup lim

n;:>:l (Q)->oo

Jz

'

IQ11 [HAy(P;:'zIP;:)d,\z(z)

+ sup

Jz

lim

n;:>:l (Q)->oo

::::; h(a z x a Y ) + log K, and hence

IQ11 [ HAY (P;: z) d,\z(z)

IQ11 [ HAy (P;:) d,\z

JZ

14. YUZVINSKII'S ADDITION FORMULA

119

If Y is zero dimensional, then H is finite (Corollary 2.3), and Lemma 14.2 allows us to choose K = 1. This proves equation (14.4)-and hence, by (14.3), Theorem 14.1-under the additional assumption that Y is zero dimensional. If Y is not zero dimensional then the constant log K in (14.12) will be non-zero, and we have to get rid of it. For every k 2: 1 we set t::.k

= {m = (ml, ... ,md)

Sk

= {m = (ml, ... ,md) E t::.k: 0:::; mj < k for every j =

mj

E 7l,d:

is divisible by k for some j E {1, ... , d}},

(14.13)

1, ... , d}.

It may be helpful to draw a picture for d = 2, where the sets Sk C t::.k and t::.k " Sk are marked by ® and x, respectively, and where (0,0) is located near the lower left hand corner of the picture: · x · · · x

x x x x x x x

. x . . . x

. x . . . x

. x . . . x

. x . . . x

x x x x x x x · E in at least one coordinate in Rk. Hence there exists a [Rk, {}, Ej-separated set in X of cardinality n(k-l)d, and by letting k --+ 00 we see from Proposition 13.7 that h( a) 2: log n. If Hm is infinite (i.e. uncountable), then we can find, for every M 2: 1, an E > 0 such that the group H' = 7fm(Hm) C C has an E-separated set of cardinality > M. By applying the argument in the preceding paragraph

v. ENTROPY

124

we obtain an [Rk,t9,Ej-separated set of cardinality> M(k-1)d, and by letting k ---+ 00 and 10 ---+ 0 we conclude that h(O") = 00. We return to the assumption that IHml = n < 00 and claim that h(O") ::; logn. If E eGis a finite, normal subgroup of G, then EnGo c C(GO) (Proposition 2.13). Hence there exists a unique, maximal, finite, normal subgroup E' c G, and E' n GO = C(GO). Since Hm is finite, the groups Yk = ker(71"6.k) and K k in the proof of Lemma 15.1 are zero dimensional and finite, respectively, and 71"m(Yk ) = 71"m(Kk ) C E' for every m E Rk. In particular, the coordinates in Bk of a point x E X determine completely the coset xmE' for every mE Rk. We write 1]: G f----t G' for the quotient map, define TJ: X f----t G''l/.,d by 'l/.,d (TJ(x))m = 1](x m ) for all m E 7l,d, and set X' = TJ(X) c G' . As we have just seen, the projection 71"Rk(Y) of a point Y E X' onto its coordinates in Rk is completely determined by 71" B k(Y). From (15.1) (applied to G') we show that there exists a constant c > 0 such that, for all x, y in X' and k ~ 1, c·maxmEBk t9'(xm, Ym) ~ maxmERk t9'(x m , Ym), where 19' is the invariant metric on G' induced by 19, and conclude as in the proof of Lemma 15.1 that the entropy of the shift-action 0"' of 7l,d on X' is zero. Put B~/(E) = {x E X' : t9'(xo,lol) < E}, and let, for every rectangle Q C 7l,d, B~/(Q,E) = n mEQ O"~m(B~/(E)). From Proposition 13.7 we know that

If 10 is small enough, then TJ- 1(B~, (Rk, E)) consists of at most IE'12dkd-l n(k-1)d disjoint translates of B~(Q, E) = nmERk O"_m(B~(E)), where B~(E) = {x EX:

t9(xo,lo) < c}. Hence Ax(B~(Rk,E)) ~ Axl(B~/(Rk,E))/IE'12dkd-ln(k-1)d, and

h( 0") = lim lim inf -(k - 1) -d log Ax (B~(Rk, E)) ::; log n. .::----+0

k-HXJ

0

THEOREM 15.6. Let 0: be a 7l,d-action by automorphisms of a compact group X whose connected component of the identity has zero dimensional centre. If h( 0:) < 00 then h( 0:) = log k for some integer k ~ 1. PROOF. Let (Vn, n ~ 1) be a decreasing sequence of closed, normal, 0:invariant subgroups of X such that nn>l Vn = {Ix} and o:x/vn satisfies the d.c.c. for every n ~ 1 (Proposition 4.9).-By Lemma 13.6, h(o:X/Vn) ---+ h(o:) as n ---+ 00, so that it suffices to prove the proposition under the assumption that 0: satisfies the d.c.c. The proof is completed by applying Lemma 15.5, together with the brief discussion preceding its statement. 0 COROLLARY 15.7. Let 0: be a 7l,d- action by automorphisms of a zero dimensional compact group X. If h(o:) < 00 then h(o:) = log k for some integer k~1.

16. MAHLER MEASURE

125

CONCLUDING REMARKS 15.8. (1) Lemma 15.5 implies that IHml = IHnl for all m, n E I. In order to prove this directly we assume for simplicity that d = 2, and consider the sets P = {(m, n) E 'Z} : n < O} and L = {(m,O) : m E Z} C Z2. Then 7rdker(7rp)) is a closed, shift-invariant subgroup of G L , and by identifying L = Z x {O} with Z we may assume that Y = 7r L (ker( 7r p )) c GZ . From the definition of X H in (15.2) we see that Y = YK , where YK is the Markov subgroup arising from K = {(k l , k 2 ) E G x G : U~ ~~) E H} as in (9.4). We set G' = 7r{O}(Y) c G and note that the groups FK(I) = {g E G' : n~ l~J E H} and FK ( -1) = {g E G' : (lc ~~) E H} are isomorphic to H(1,l) and Heo,l), respectively, and that G'/FK(I) ~ G'/FK (-I) by (10.1)(10.4). Since G' is normal in G, G' has finite centre, and the isomorphism of G'/FK(I) and G'/FK(-I) implies that IH(1,l) 1 = IFK(1) 1 = IFK(-I)1 = IHeo,l) I· Similarly one proves that IHeo,l) 1= 1Heo,D) 1= IHeo,l) I, and the proof in higher dimensions is completely analogous.

(2) Corollary 14.3 reduces the study of Zd-actions by automorphisms of compact groups with zero entropy to the analysis of zero entropy Zd-actions on zero dimensional groups, on connected groups with trivial centres, and on abelian groups, and Lemma 13.6 allows us to restrict our attention to actions satisfying the d.c.c. Zd-actions on compact, abelian groups will be discussed in Section 18, and the other two cases can be dealt with by investigating the closed, shift-invariant subgroups X c G Zd with zero entropy, where G is a compact Lie group such that GO has finite centre. By using both the notation and the arguments employed in the proofs of the Lemmas 15.1 and 15.5 one can easily show the following: the shift-action by Zd on a closed, shift-invariant subgroup X c G Zd satisfying (15.2) has zero entropy if and only if, for every k ~ 1, the projection of any point x E X onto its coordinates in Rk = {O, ... ,k}d is completely determined by its projection onto the boundary Bk = Rk n D..k of d Rk. The analogous statement for closed, shift-invariant subgroups Xc (']['n)Z is obviously incorrect (Remark 3.10 (2)). 16. Mahler measure In this section we discuss a quantity introduced by Mahler in [67], [68], which will yield the last remaining ingredient of equation (14.16) by providing an entropy formula for Zd-actions by automorphisms of compact, abelian groups. The Mahler measure of a polynomial 1 E 91d, d ~ 1, is defined as

M (f)

= {~XPU§d log 11(s)1 ds) if 1 i= 0, if 1 = 0,

(16.1)

where ds denotes integration with respect to the normalized Haar measure on the multiplicative subgroup §d C Cd, and where 1 is regarded as a function on Cd.

126

V.

ENTROPY

PROPOSITION 16.1. Let f = ao + ... + asu s be a polynomial with complex coefficients, with aoa s i- 0, and with roots 6, ... , ~s. Then

(16.2)

where log+ t

= log (max{l, t}) for every t

~ O.

PROOF. According to Jensen's formula ([2], p. 208),

Since ao/as

=

n;==1 ~j we have that s

= log lasl + L

log I~jl-

j==1

log I~j I {j:l::;j::;sand l~jl o.

PROOF. We assume without loss in generality that 9 has a non-zero constant term. Let A(a) = {z E § : Ig(z)1 :::::: a}. If k = 2 then A(A(a)) = A({Z E § : Icg(O) + zl :::::: a}) :::::: 2a;f, and we set C 2 = 211". Suppose that the lemma has been proved for k : : : K, where K 2': 2, and that 9 has K + 1 non-zero coefficients. Then g' = has K non-zero coefficients, and our induction hypothesis implies that

*

for every a > O. We set B(b)

= {z

A(A(a)) :::::: A({Z E b

E oo I§d X df.Lr = 0 for every non-trivial character X of §d, and Weierstrass' approximation theorem implies that lim(r)->oo I§d 9 df.Lr = I 9 dA§d for all continuous functions g: §d f----+ 0, a

y

-+

{

J§d

f dA§d

> 0 in IR with

{ log Ifrl dA§ = ( log If I dJ-tr > J Air(Y) J{ZE§d:lf(z)I-::;'Y}

-€

whenever (r) is sufficiently large. However, since fr has at most as many nonzero coefficients as f, this is precisely the statement of Lemma 16.5, so that the proposition is proved. 0 COROLLARY 00.

16.6. For every non-zero polynomial

f

E

!nd, 1 ::; M (f)


'(nl B;;jC(l)) =

: :; >.(nl

J=O

>.(nl

B;;jC(O))

J=O

A-jC(l))

= hm(A)

J=O

for all sufficiently large n. By letting 0 ~ 1 we see that hm is lower semicontinuous on MK(C). Upper semicontinuity is proved similarly by noting that, for 0 0, a neighbourhood N(A) c MK(C) and an mo ::::: 1 such that Ihm(B) - h(B)1 < c for all B E N(A) and m ::::: mo. We fix A, write the eigen~s with I~kl :::; 1 for k = -T, ... , 0 values of A without multiplicity as ~-T) and I~kl > 1 for k = 1, ... , s, and denote the multiplicity of ~k by Nk. Fix c > 0 and choose, > 0 such that the circles C k of radius, around ~k are disjoint, I~kl-, > 1 for all k = 1, ... , s, and 4Klog(1+,) < c. According to §I.5.3 in [36], ••• '

Pk

= Pk(A) = ~ 21fZ

1 Ck

(U - A)-l

d~,

where I is the K x K identity matrix, is the projection onto the generalized eigenspace Vk of A for ~k along the sum of the other generalized eigenspaces, and diml(Yk = N k . We fix k for the moment and set, for every y E Vk ,

The spectral radius formula shows that this series converges geometrically. If Ak denotes the restriction of A to Vk , then IIAkyilk < (I~kl + ,)llyllk for 0=1- y E Vk . We denote by Ek the unit ball in Vk in the norm II· Ilk and observe that (I~k I + ,) -1 Ek 1. As there are at most K other eigenvalues of B, each with modulus < 1 + ,,(, we obtain that s

L log Idet Bkl 2 2': h(B) - 2K log(l + "().

(17.3)

k=1

Since BE c E(B) we have that Bn7=~1 B-j E (17.3) imply that

c

n7=~1 B-j E(B), and (17.1)-

so that hm(B) 2': h(B) - c for all sufficiently large m and all B E N(A). For the opposite inequality we recall that (I~kl + ,,()-j Ek(B) 92 9.J1, and choose primitive elements h, ... , fr in 91 1 such that fJ divides fJ+l for all j = 1, ... , r -1, and 9.J1(IQI) is isomorphic (as an 91~1QI) -module) to the direct sum 91~1QI) / h 91~1QI) EB· .. EB 91~1QI) / fr91~1QI) (cf. Lemma 8.1). Then (17.17) Furthermore, if we regard 9.J1(IQI) as an 91 1 -module and denote bya9J!(Q) the automorphism of the group X9J!(Q) h(a9J!).

=~

defined by (5.5)~(5.6), then h(a9J!(Q»)

=

PROOF. The existence of the polynomials h, ... ,fr and the 91~1QI) -module isomorphism ()': 9.J1(IQI) f----+ 91~1QI) / h91~1QI) EB ... EB 91~1QI) / fr91~1QI) = l)1(IQI) was established in the proof of Lemma 9.1, and (14.1), Theorem 9.2, and Proposition 17.2, together imply (17.17). As in the proof of Lemma 9.1 we set l)1 = 91l/UI) EB··· EB91l/Ur) and consider the Noetherian 91 1-modules il)1 C l)1(IQI). Then h(a tfJ1 ) = h(a) for every k 2: 1, and the last assertion follows from Lemma 13.6 and duality by letting k -+ 00. 0

AT

THEOREM 17.7. Let A E GL(n, Q), and let X = Qn and aX = A = E Aut(X), where AT is the transpose matrix of A. If N E Qn is any subgroup which contains zn and is invariant under multiplication by AT, and if a v denotes the automorphism of V = IV = Qn / N.L induced by a, then n

h(a v ) = 2)og+ I(jl

11

+ log lal

j=1

=

(17.18) log IXA(e 27ris )1 ds

= log M (XA),

where XA (u) = uf + a n -1 u~-1 + ... + ao is the characteristic polynomial of A with roots (1, ... , (n, and where a is the lowest common multiple of the denominators of the coefficients ai, i = 0, ... , n - 1. Furthermore, h(a V h(a A ), where a A is the automorphism of XA defined in Example 9.5 (3).

)

=

PROOF. Consider the automorphism a A of XA and the dua191 1-module 9.J1A in Example 9.5 (3), and define 91~1QI), 9.J1* = Qn, and h, ... ) fr as in Lemma 17.6. Lemma 17.6 and Proposition 16.1 together imply that h(a A ) is given by (17.18). If N c Qn is any subgroup which is invariant under multiplication by A T and contains zn and hence 9.J1A, and if V = IV, then 9.J1A C N c 9.J1*, and

18. MAHLER MEASURE AND ENTROPY OF Zd-ACTIONS

139

by duality we obtain surjective homomorphisms X = 9Ji* f---7 V = IV f---7 X A. According to Lemma 17.6, h(o:X) = h(o:A), and (14.1) implies that h(o:V) = h(o:X) = h(o:A). 0 CONCLUDING REMARK 17.8. Lemmas 17.1 and 17.4 are taken from [63], as is the method used in the first proof of Proposition 17.2. If AT E GL(n, Z) = Aut(zn) and 0: = A E Aut(ll'n), then (17.18) is the well known entropy formula for toral automorphisms due to Sinai and Rokhlin [85], and for AT E GL(n, Q) and V = X = Qn, (17.18) is Yuzvinskii's formula in [111]. The reason for deriving (17.18) by the particular route used in this section is that the argument presented here anticipates the proof of the entropy formula for Zd-actions by automorphisms of compact, abelian groups in Section 18.

18. Mahler measure and entropy of Zd-actions We begin our discussion of Zd-actions by generalizing Proposition 17.2 to higher dimensions. THEOREM 18.1. For every If f

= 0,

f

E !J\d, h(o:'Ytd/(f»)

=

I log M (f)I.

then Theorem 18.1 is trivially true, since

If f = cum for some C E Z and m E Zd, then o:'Ytd/(f) is isomorphic to the shift-action of Zd on (Z/Icly£:d (cf. Example 5.2 (2)), and h(o:'Ytd/(f») = log Ici = log M (f) = Ilog M (f)I, as claimed in Theorem 18.1. Assume therefore from now on that f has at least two non-zero terms. We start by estimating h( o:'Ytd/(f») from below.

LEMMA 18.2. h(o:'Ytd/(f») 2: log M (f). PROOF. Let r = (rl,"" rd) E Zd be primitive, i.e. assume that the highest common factor of {rl,"" rd} is equal to 1. As in the proof of Proposition 16.2 we define a homomorphism 7]r: Zd f---7 Z by 7]r(m) = (r, m), and we note that 7]r is surjective since r is primitive. The map 7]r induces an injective homomorphism 'lj;r: ll'z f---7ll'Zd by 'lj;r(x)m = X'7r(m), and we claim that 'lj;r(X'Ytl/(fr») c X'Ytd/(f): indeed, if f = LnEZd cf(n)u n , then fr = LnEZd cf(n)u'7r (n), x E ll'z lies in X'Ytl/(fr) if and only if LnEZd cf(n)x m +'7r(n) = (mod 1) for all m E Z, and the last condition implies that 'lj;r(x) E X'Ytd/(f). For every n 2: 1 we set rn = (1, n, ... ,nd- l ). Then rn is primitive, (rn) = n, and 7]r n is a bijection of the rectangle Qn,m = {O, ... , n -1 }d-l x {O, ... , mI} C Zd onto Q~ m = {O, ... ,nd-lm - I}. In particular, if 'I9(s, t) = Is - tl is the metric on 11' defined in (17.6), E > 0, and if Fe X'Ytl/(frn) is [Q~m,'I9,E]­ separated (we are using the notation of Proposition 13.7), then 'lj;rJF) is a

°

v.

140 [Qn,m,

ENTROPY

19, c]-separated subset of X'Ytd/(f). From Proposition 13.7 it is now clear

that

(note that sQ;' ,,,(19, c) and sQn,'m (19, c) denote cardinalities of sets in different spaces!). According to the Propositions 16.2 and 17.2, h(a'Ytl/(frn») log M (frJ -+ log M (f), so that log M (f) ~ h(a'Ytd/(f»). D The proof of the opposite inequality h(a'Ytd/(f») ~ log M (f)

(18.1)

is much more involved, and it will be helpful to present it first for a specific polynomial of a particularly simple form. LEMMA

18.3. The inequality (18.1) holds for f

= 1 + Ul + U2

PROOF. According to Example 5.2 (2) we may assume that X = X'Yt2/(f) = {x E

1['Z2 : X(m,n)

+ X(m+l,n) + X(m,n+l) =

E 9't 2 .

°(mod 1)

for all (m, n) E 1:?}, and that a = a'Yt2/(f) is to the shift-action a of 'Z} on X. We define 19(8, t) = lis - til on 1[' by (17.7), fix c > and a rectangle Q = Q(M, N) = {O, ... , M I} x {O, ... ,N -I} c Z2, and construct explicitly an [Q,'t9,c]-spanning set Fe X of the desired cardinality (for notation we refer to Proposition 13.7). Let x E X. If y E X satisfies that 7l"Q(x) = 7l"q(y), then Ym = Xm for all mE {(ml,m2) E Z2 : ~ ml ~ M + N - 2, ~ m2 ~ M + N - 2 - md. Apart from this constraint we have considerable freedom in choosing y: for example, the coordinates Y(k,O) , k tJ. {a, ... ,M + N - 2}, are completely at our disposal, and we can choose them so that Y(k+L,O) = Y(k,O) for all k E Z, where L = M + N -1. It follows that there exists a point y E X with 7l"Q(Y) = 7l"Q(x) and a(L,O)(Y) = y. Denote by Z E £OO(Z) the point with -~ < Zm ~ ~ and Zm (mod 1) = Y(m,O) for all m E Z. There exist unique complex coefficients Ck, k = 0, ... ,L-1, such that Z = Lf~ci" CkV(k), where v(k) E £00 (Z) and v(k) (m) = e27rikm/ L for all k = 0, ... ,L - 1 and m E Z. It is clear that ick I ~ ~ for all k = 0, ... ,L - 1, and that Ck = CL-k for k = 1, ... ,L -1. For every k = 0, ... , L - 1, we set (k = _1_e27rik/L and define a map w(k): Z2 f----+ C by w(k)(m,n) = (k'v(k)(m) for all

°

°

°

°

(m, n) E Z2 if (k =1= 0, and w(k) == otherwise. Then L~~ci" CkW(k) (m) is real and L~~ci" CkW(k) (m) (mod 1) = Ym for all m E Z2. If we replace each coefficient Ck by a complex number of the form c~ = rk/ Ltk + iSk/ Ltk, where tk is a positive

18. MAHLER MEASURE AND ENTROPY OF Zd-ACTIONS

integer, rk and Sk are integers in the interval [- Ltk/2, Ltk/2], and c~

141

=

c~_k

for k = 1, ... , L - 1, then we obviously cannot expect y~ = 2:~~~ c~w(k) (m) (mod 1) to be equal to Yrn for every m E 7i}. Suppose, however, that we only wish to achieve that IY~-Yrnl < dor all mE Q' = Q(M, N)' = {O, ... , L-l}x {O, ... , N -I}. If we choose tk = Int((I+ l(kI N - 1 )/e) for every k = 0, ... , L-l, where lnt denotes the integral part, then we can indeed find such coefficients c~ = rk/ Ltk + iSk/ Ltk such that Y' = 2:~~~ c~w(k) (mod 1) satisfies that IY~ - Yrn I < e for all m E Q'. The total number of all Y' which can be written in this form is less than or equal to K(M, N, e) = TIk:i;1)/2[ L~1 . (1 + I(k IN -1)]2. Since Q = Q(M,N) c Q' = Q(M,N)' we have found an [Q(M,N),t9,e]spanning set of cardinality K(M, N, e), and Proposition 13.7 shows that h(o:)

=

h(a-)

1 :s: sup liminf MN log K(M, N, e). 0:>0 M,N->oo

Furthermore, liminf MIN logK(M, N,e)

M,N->oo

=

2 liminf -M M,N->oo N

(M+N)/2

"(log(M + N) -loge + log+(1 + l(kI N - 1 » ~

k=O

1

M+N-2

L

:s: M,N---+CX) liminf MN and by setting N

k=O

= lnt(log M)

(log(M +N) -loge+ (N -1)log+ I(kl), and letting M

L-1

2.. L L->oo L

h(o:):S: lim

log+ 11 +

e27rik/LI

k=O

where L

=M

---7

=

00

we obtain that

Jro log+ 11 + e 1

27riS

I ds,

+ N - 1. From Proposition 16.1 we know that log+ 11 + e 27ris I =

so that h(o:)

:s:

11 11

11

log 11 + e27ris + e 27rit I dt,

log 11 + e 27ris + e 27rit l ds dt = log M (f).

D

Although this is probably the most direct proof of Lemma 18.3, its extension to a general polynomial f E l)td is slightly awkward. For this reason we present a second proof of Lemma 18.3, based on hvol rather than h span . SECOND PROOF OF LEMMA 18.3. We use the same notation as in the first proof of this lemma. Let Q = Q(M, N) = {O, ... , M -I} x {O, ... , N -I}, L = M +N -1, Q' = Q'(M,N) = {O, ... ,L-l} x {O, ... ,N -I}, and let Y c X be the subgroup of points with a(L,O)(Y) = y. Then 7rQ(Y) = 7rQ(X), and we write 7rQ: Y f---t ']['Q for the restriction to Y of 7rQ: X f---t ']['Q. As in Proposition

142

V. ENTROPY

13.7 we set, for every E > 0, B(Q, E) = {x EX: IXml < E for all mE Q} and note that 7rQ -l(7rQ(B(Q, E))) ::::> {y E Y: IYml < E for all mE Q'} = G(Q',E), say. Since 7r /(7rQ(B(Q, E)) = B(Q,E) we have that

c

Ax(B(Q,E)) = A7rQ (x)(7rQ(B(Q,E));::: A7rQ(Y)(7r Q(G(Q',E)));::: Ay(G(Q', E)), and Proposition 13.7 shows that 1

h(o:) = h(CJ) = lim liminf - MN log Ax(B(Q(M, N), E)) E-+O

M,N-+OCl

1 ::; lim liminf ---log Ay(G(Q(M, N)', E)). £--+0 M,N--+oo MN In order to estimate Ay(G(Q(M, N)', E)) we consider the surjective homomorphism 1/

= 7r{0, ... ,L-1}X{0}: Y

f---+

']['L

and note that the diagram

']['L -----+ ']['L A

commutes, where

-1o -1 0 . . 0) ( A= ·· .. . . . . ... -1 -1 ...

·

.

o

0 -1 0

.

0

.

E

GL(L, Z) = Aut(,][,L).

... -1 -1 ... 0 -1

The matrix A induces a linear map on 0 M,N ->00

1

1

lim liminf -MN log )'(E(N,cL-'I))

0->0 M,N->oo

where N = Int(log M), L = M + N - 1, and M ~ this lemma we conclude that h(a) ~ log M (I). 0

00.

As in the first proof of

In order to prove the inequality (18.1) for an arbitrary polynomial J E 9td it helps to transform J into a more convenient form. If hE 9td and A E GL(d, Z) we put h A = LnEZ d ch(n)u An (cf. (5.2)). LEMMA

18.4. For every A E GL(d, Z) and m E Zd,

PROOF. Since (g) = (umg) and M (g) = M (umg) for every 9 E 9td we may assume that m = O. Example 5.2 (2) allows us to realize Xmd/(f) and Xmd/(fA) as closed, shift-invariant subgroups of ']['Zd, and to assume that amd/(f) and amd/(fA) are the restrictions of the shift-action a of Zd on ']['Zd to Xmd/(f) and Xmd/(fA), respectively. We define an isomorphism 'l/JA: ']['Zd f----+ ']['Zd by ('l/JA(X))n = XAn for every x = (xn) E ']['Zd and note that 'l/JA(Xmd/(f)) = Xmd/(fA), and that 'l/JA' an(x) = 'l/JA' a;;d/(f) (x) = a~~/(fA) . 'l/JA(X) = aAn' 'l/JA(X) for every x E Xmd/(f). In other words, 'l/JA is a conjugacy of the Zd_ actions amd/(f) and amd/(fA). A . n f-> amd/(fA) A n ' and Proposition 13 .1 shows that h(amd/(f)) = h(a~d/(fA)) = h(amd/(fA)). The second assertion follows from the fact that A E GL(d, Z) = Aut(,][,d) induces a (Haar measure preserving) automorphism A' of §d ~ ']['d with JA(z) = J(A'z) for every z E §d. 0

v.

144

ENTROPY

PROOF OF THEOREM 18.l. Let d > 1, and let I E 9ld be a polynomial with at least two non-zero terms (the other possibilities have already been taken care of). From Lemma 18.2 we know that h(a'.Rd/(f) ~ log M (f), but we still have to prove the opposite inequality (18.1). In view of Lemma 18.4 we can replace I by any polynomial un IA with n E 7l.,d and A E GL(d, 7l.,), without affecting the assertion of the theorem. In particular we may assume without loss in generality that I has the form

I = au:! + IK_lulj-l + ... + 10,

(18.2)

where a ~ 1, K ~ 1, and Ik E 7l.,[Ul, ... ,Ud-l] for every k = O, ... ,K - 1. According to Example 5.2 (2) we may take it that X = X'.Rd/(f) C 'll'Zd, and that a = a'.Rd/(f) is equal to the shift-action of 7l.,d on X. From (5.9) and (18.2) it is clear that the projection 7fz: X 1----+ 'll'z is surjective, where Z is the strip 7l.,d-l x {O, ... ,K -I} C 7l.,d. We proceed as in the first proof of Lemma 18.3, consider the rectangle Q = Q(M,N) = {O, ... ,M _l}d-l x {O, ... ,N -I} with N > K, and fix a point x EX. If y E X satisfies that 7fQ (y) = 7fQ (x) then y need no longer be completely determined on a set of coordinates which strictly contains Q, but there exists an integer q > 0 which is independent of M and N such that the coordinates of yare completely undetermined in Z . . . . Q', where Q' = Q(M,N)' = {O, ... ,M + qN _l)}d-l x {O, ... ,N -I} (this is equivalent to saying that 7fz-....Q' ·7fc;/({x}) = 'll'z-....Q'). The integer q is given by q = min{p : S(f) C ~(P)}, where S(f) = {m E 7l.,d : cf(m) #- O} is the support of I and ~(P) = {m = (ml, ... , md) E 7l.,d : 0 ~ md ~ K and 0 ~ mj ~ p(K-md) for all j=1, ... ,d-1}. Let Yn C X be the subgroup of points which have period n in the first d - 1 coordinates, i.e. which satisfy that aneW (x) = x for j = 1, ... , d - 1, where e(j) denotes the j-th unit vector in 7l.,d. Exactly as in the second proof of Lemma 18.3 we obtain that 7fq(Y) = 7fQ(X) with L = M + qN and Y = YL , and observe that h(a)

1

= 10--+0 lim liminf - Md-IN 10g>'x(B(Q(M,N),c:» M,N--+oo 1

(18.3)

~ lim liminf - Md-IN 10g>,y(C(Q(M,N)',c:», 10--+0

M,N--+oo

where

B(Q(M,N),c:) = {x EX: C(Q(M,N)',c:) = {y E Y:

IXml < c: IYml < c:

for all mE Q(M,N)}, for all mE Q(M,N)'},

II . II is given by (17.7). We put R = R(M, N) = {O, ... , L - 1}d-l, S = S(M, N) = R(M, N) x {O, ... , K - I}, and note that the projection 7fs: Y 1----+ 'll's is surjective. For every 0 < c: < ~ and every y E C (Q' , c:) = and where

18. MAHLER MEASURE AND ENTROPY OF Zd-ACTIONS

145

C(Q(M, N)" s) there exists a unique point ¢(y) = Z = (zn) E JR.Q' with IZnl < s and Zn (mod 1) = Yn for all n E Q', and we set D(Q',s) = ¢(C(Q',s)). Write a typical element v E JR.R as (v m) = (vm' mER), and define automorphisms Tj E GL(Ld-l,JR.) = Aut(JR.R) by (TjV)m = Vm+eU) (mod L) for every v E JR.R, where (mod L) indicates reduction modulo L in every coordinate. For every k = (k l , ... , kd - I ) E Zd-I and h E 9{d-1 we put Tk = T~l ... T;'!:.jl E GL(Ld-l, JR.) and h(T) = LkEZd- 1 ch(k)Tk E MLd-l (JR.), where Ms(JR.) is the set of s x s-matrices with real entries for every s ::::: 1, and where ch(k) is defined as in (5.2). The matrix 0

A = A(M,N) =

0

I

o

0

·

(

I

.. .

...

··

000

_a- 1 forT) _a- 1 ft(T) _a- 1 h(T)

induces linear maps both on JR.S = (JR.R)K and C S = (CR)K, which will again be denoted by A = A(M, N). Let w = e 27ri / L and define, for every n E R, a vector v(n) E C R by v(n)m = w(m,n) jVLd-1 for every mER (cf. (16.3)). The set {v(n) : n E R} is a basis of C R which is orthonormal in the Euclidean norm, and h(T)v(n) = h(w)v(n) for all n E Rand h E 9{d-I, so that the v(n) are simultaneous eigenvectors for all h( T), h E 9{d-l. Consider the bases {e( m) : m E 5} and {w( m) : m E 5} of C S defined by e(m)n = Dm,n and w(m)n = Dm,nV(m') for all m, n E 5, where m' = (ml,"" md-d E Zd-I for every m = (ml,"" md) E Zd, and Dm,n = 1 if m = n, and = 0 otherwise. For every k E Zd-I we consider the linear span Wk of {w(n): n E Zd and n' = k} in C S , write

for the maximum norm of an element t=

{nEZd:n'=k} in the basis {w(n) : n E Zd and n' = k}, set 5(I)k = {t E Wk : Iltllk < I}, and denote by Ak the Lebesgue measure on Wk, i.e. the Haar measure on Wk with Ak(5(I)k) = (27r)K. Then C S = EBkER W k , each Wk is invariant under A, and the restriction Ak = A(M, N)k of A to Wk has the form 0

o (

..

I 0

..

0 I

..

000

_a- 1 fo(w(k»

_a- 1 ft(w(k»

_a- 1 h(w(k»

v. ENTROPY

146

where w(k) (w k1 , .•. ,W kd - 1 ) for every k = (k1, ... , kd - 1 ) E generally we define, for every s = (81, ... , 8d-1) E §d-1 C Cd-I, 0

1

0

...

001 (

.. .. .. 006

_a- 1 fo(s) _a- 1h(s) -a-112(s)

o o. _a- 1

)

Zd-1.

More

,

f~-l(S)

and denote by (s,1, ... ,(s,K the eigenvalues of As, i.e. the roots of the polynomial

In particular we obtain that ~

= sE§ m~lEIax I(s,j I < 00, )-l, ... ,k

(18.4)

and since Ak = AW(k) for every k E R, every eigenvalue ( of A(M, N) satisfies that ( ~ ~, uniformly in M and N. Lemma 17.1, applied to the compact set {As: S E §d-l} c MK(C), implies that there exists, for every c' > 0, an N(c') :::: 1 with

1- ~

log Ak

(:0:

AkC(l)k) - 2

t,

log+ I(k,j II < c'

(18.5)

for all M :::: 1, N :::: N(c'), and k E R(M, N). We define a linear map W: C S f---+ C S by We(n) = w(n) for all n E S, and note that W is unitary with respect to the Euclidean inner product on C S , that AICS W = AICS and {z E C S : Ilzlloo < I} :) W{z E C S : Ilzlloo < ISI-!}, where Ilvll oo = maxnES IV n I is, as usual, the maximum norm both on ~s and on C S , and that {z E C S Since

AICS

:

n

j=o~~_lIIAj zlloo < c} :) cISI-! EB [N

1 AiC (l)k].

kER )=0

is the product of the measures Ak, k E R, we obtain that

1

s·

- N logAlCs({z E C : IIA)zlloo

1 1 1 ~ - N log(cISI-2) - N and (18.5) shows that

1- ~

L

kER(M,N)

log Ak

CrY j=O

-2

< c for j = 0 ... ,N -I})

L logAk (N-1.) n Aj.C(l)k ,

kER

(18.6)

j=O

Ai C (l)k)

(18.7)

K

L L log+ IC(k),j II < L

kER(M,N) j=l

d - 1 c'

18. MAHLER MEASURE AND ENTROPY OF Zd-ACTIONS

147

for all M :::: 1 and N :::: N(e'). We can now determine Ay (C (Q' , e)) exactly as in the first proof of Proposition 17.2. Let e < (4Ial(~+1))-1, where ~ is given by (18.4), and let v = (vn ) E ]RQ'. For every k = 0, ... , N - K we define a point 1l"k(V) = v(k) E ]Rs by setting v(k)n = Vn+ke(d) for all n E S. The maps 1l"k: ]RQ' I--+]Rs are linear, and 1l"k+1(Z) = z(k + 1) = Az(k) = A1l"k(Z) for every z E D(Q',e). Hence

1l"o(D(Q',e))

= {v =

Ay(C(Q',e))

max IIAkvlloo < e}, k=O, ... ,N-K lal-(N-K)ISI AIRS (1l"o(D(Q', e))), E]Rs :

(18.8)

and

It follows that

2h(o:) = lim liminf d:1 logAx(B(Q(M,N),e)) 0: ...... 0 M,N ...... oo M N :::; lim liminf - Md:1 0: ...... 0

M, N ...... 00

N

(by (18.3))

logAYL(C(Q(M,N)',e)) (by (18.8))

2

= lim0 M,N liminf - Md-1N log[lal-(N-K)ISIAIRS(M,N) (1l"o(D(Q(M,N)',e)))) ...... oo 0: ......

(by (18.9))

1 :::; 2 log lal + 0:lim lim inf Md - 1N ...... 0 M,N ...... oo

. log ACS(M,N) ({ z E rcS(M,N) :. max IIAj zlloo < e}) J=0, ... ,N-1 K (by (18.6)-(18.7))

L

= 2 log lal + M,N liminf :-1 ...... ooM (by letting M :::; 2 log lal + i~moo

= 2 log lal +

=2

2

2 Ld-1

}Sd-l

r }srlog

}Sd-l

r

--+ 00

Llog+ I(w(k),jl kER(M,N) j=l and setting N = Int(log M), L

= M + qN)

K

L L log+ I(w(k),j I

kER(M,N)j=l

(by Lemma 17.1)

log+ I(s,j I dASd-l (8)

Ifsl dAsdASd-l

(by Proposition 16.1)

= 2 log M(f).

0

We state some consequences of Theorem 18.1. For notation we refer to Lemma 5.1.

v. ENTROPY

148

COROLLARY 18.5. Let d 2:: 1, and let p C

~d

be a prime ideal. Then

if P = (f) is principal, if p is not principal. PROOF. Use Lemma 17.4.

D

PROPOSITION 18.6. Let d 2:: 1, and let 911 be a Noetherian ~d-module. If 911 = 'Jl:s ::J ... ::J 'Jl:o = {O} is a prime filtration of 911, and if qi C ~d is the prime ideal satisfying that 'Jl:d'Jl:i - l ~ ~d/qi' i = 1, ... , s, then s

h(Q'JJt)

= Lh(Q!ltd/qi),

(18.10)

i=l

where h(Q!ltd/qi) is given by Corollary 18.5 for every i = 1, . .. , s. If ,c is an arbitrary countable ~d-module, there exists a increasing sequence ('cn, n 2:: 1) of Noetherian submodules of ,C such that Un>l'cn = 'c. Then h(Q'c) = lim n -+ oo h(Q'cn), and h(Q'cn) is determined by (18.H)) for every n2::1.

PROOF. The first assertion is clear from the addition formula (14.1), and the second follows from Lemma 13.6 by setting Vn = ,C* C = for all n2::1.

x,c

D

:c

EXAMPLES 18.7. (1) Let d 2:: 1, denote by Mn(~d) the ring of n x n matrices with entries in ~d, and let A E Mn(~d). We claim that (18.11)

In order to prove (18.11) we set X = X!lt:i/A!lt:i = ~~~ and Q = Q!lt:i/A!lt:i, and assume first that det A = o. Put 'Jl: = {w E ~~ : Akw = 0 for some k > O}. Since ~~ is a Noetherian ~d-module, there exists a K 2:: 1 with 'Jl: 2 A'Jl: 2 ... 2 AK'Jl: = {O}, and we choose an element w E 'Jl: with'AK-lw =f.: O. Then fAK-lw = Ak-l fw =f.: 0 for every non-zero f E ~d. It follows that fw E 'Jl:"A~~ for every non-zero f E ~d, and that ~d ~,C = ~dw/A~n C ~~/A~~. We set Y =,C.1. C ~~~ = X and obtain that h(Q) 2:: h(Qxty ) = h(Q'c) = h(Q!ltd/(O») = 1log M (0)1 = 00 by Theorem 18.1. Now assume that B, C E Mn(~d) have non-zero determinant, and that (18.11) holds for Band C. Consider the exact sequence

o -+ B~~/BC~~

-+ ~~/BC~~ -+ ~~/B~~ -+

o.

18. MAHLER MEASURE AND ENTROPY OF Zd-ACTIONS

149

Since det B =I- 0, the map B is injective on 9td, so that B9td/ BC9td 9td/C9t d, and the addition formula (14.1) shows that

h( a'.R;t / BC'.R;t)

= h( a'.R;t / B'.R;t) + h( a'.R;t /C'.R;t) = log M (det B) + log M (det C)

~

(18.12)

= log M (det BC), so that (18.11) holds for BC. We regard A as a matrix over the field of fractions 9td and note that there exists a non-zero polynomial f E 9td such that fA can be written as a product of elementary matrices of the form

o

11

o

, and

(

:". J

,(18.13)

°

where =I- 9 E 9t d . It is clear that (18.11) holds for any of the elementary matrices in (18.13), and hence for their product fA. We conclude that h(a'.R;t/fA'.R;t) = log M(detfA) = nlog M(f) + log M(detA), and (18.12) (with B = fI and C = A) shows that

h( a'.R;t / f A'.R;t) Since M (I)
1, C E (QlX)d, and consider the 7!.,d- actions a'.Rdhc and a(c) in Theorem 5.1. Since the homomorphism 1> in (5.8) is finite-to-one, we know that h( a(c)) = h( a'.Rdh c ), and Corollary 18.5 implies that h( a'.Rdh c ) = 0, since the ideal jc is prime and non-principal. An alternative proof of this fact is based on Example 17.3 (2), where we saw that h(a~)) = h(a;;dhc) < 00 for all n E 7!.,d; a glance at the definition of hcover(·) = h(·) now shows that h( a(c)) = h( a'.R dh c ) = 0. [] CONCLUDING REMARK

[63].

18.8. All the material in this section is based on

CHAPTER VI

Positive entropy

19. Positive entropy We begin this section with a brief discussion of entropy for a single automorphism Q of a compact group X, in which we prove that every ergodic automorphism of an infinite, compact group has positive entropy, and that automorphisms with zero entropy have a very degenerate structure. LEMMA 19.1. Let f E!Jt 1 be a Laurent polynomial with M (f) = 1. Then there. exist integers r 2: 1, t E Z, and cyclotomic polynomials C1, . .. , cr , such that feu) = ±ut I1~=1 er(u). In other words, ±f is a finite product of generalized cyclotomic polynomials (cf. Corollary 6.11). PROOF. It is obviously enough to prove the lemma under the assumption that f is irreducible. We choose an integer t such that utf(u) = ao + a1U + ... + asu s with aoa s =I- o. By Proposition 16.1, lasl = 1, and the roots 6, ... ,~s (counted with multiplicity) must all have modulus:::::; 1. Hence ao = 6 ... ~s = 1, and I~jl = 1 for j = 1, ... , s, so that ~j1 = ~j E {6, ... ,~s} for every j = 1, ... ,8. Let IK = Q(6), and let pft{ be the set of places of IK (we are using the same notation as in Section 7). Since both 6 and (["1 are algebraic integers, we have that I~j Iv = 1 for every v E pft{. As in Section 7 we write IKA for the adele ring of IK and consider the diagonal embedding i: IK t---t Aft{. Then the elements i(~j), k 2: l,j = 1, ... ,8, lie in the compact subset I1vEPdw E IKv : Iwlv : : :; I} of IKA. In particular the set {i(~j) : k 2: l,j = 1, ... ,8} is finite, and there exists an integer k 2: 1 such that i(~t) = i(I). It follows that ~t = 1 and that either f or - f is a generalized cyclotomic polynomial. 0 THEOREM 19.2. Let Q be an automorphism of a compact group X with zero entropy. Then there exists a decreasing sequence of closed, normal, Qinvariant subgroups X = Yo :) Y 1 :) ... :) Y n :) ... such that i >1 Yi = {Ix}, and Yi-dYi is a compact Lie group with an QYi-dYi -invariant m~tric for every

n

151

152

VI. POSITIVE ENTROPY

i 2: 1. In particular, there exists no infinite, closed, a-invariant subgroup Y c X such that a Y is ergodic. Finally, X is zero-dimensional and a satisfies the d.c.c., then X is finite.

PROOF. First assume that a satisfies the d.c.c. If a is ergodic, we apply Theorem 10.6 and conclude that, in the notation of that theorem, Ai = {I} for i = 1, ... , K, and that log M (XA) = 0 (Theorem 17.7), where A is the n x nmatrix appearing in Theorem 10.6 and XA is the characteristic polynomial of A. From (17.18) we see in particular that XA E 9l 1 , and Lemma 19.1 implies that every eigenvalue of A is a root of unity, contrary to Example 9.5 (3). This shows that a must be non-ergodic. Since 0 = h(a) 2: h(aV ) = 0 for every closed, a-invariant subgroup V eX, a v will again be non-ergodic by the first part of this proof, and Theorem 1.4, combined with the d.c.c., yields that there exist closed, normal, a-invariant subgroups X = Va :J ... :J Vr = {I} such that Vi-dVi is a compact Lie group with an a-invariant metric for every i = 1, ... ,r. In particular, if X is zero-dimensional, then Vi-dVi must be finite for i = 1, ... ,r, so that X is finite. For an arbitrary automorphism a of a compact group X with h(a) = 0 we choose a decreasing sequence (Vn, n 2: 1) of closed, normal, a-invariant subgroups such that nn>l Vn = {Ix} and a X / Vn satisfies the d.c.c. for every n 2: 1 (Example 4.8 (1) ~nd Proposition 4.9). Then h(aVn-l/ Vn ) = 0 for every n 2: 1, where Va = X, and by applying the preceding paragraph to each aVn-l/ Vn we obtain the promised sequence (Yn , n 2: 1). 0 If d > 1, and if a is a Zd-action by automorphisms of a compact, abelian group X, we assume without loss in generality that a = a!m and X = X!m = 9R for some countable 9ld-module 9J1 (Lemma 5.1). We already know how to characterize ergodicity of a!m in terms of the associated primes of 9J1 (Proposition 6.6), and we now prove that positive entropy is again determined by the prime ideals associated with 9J1.

DEFINITION 19.3. A prime ideal q C 9ld, d 2: 1, is positive if h(arytd/q) > 0, and null otherwise. PROPOSITION 19.4. Let 9J1 be a countable 9ld -module, and let a and X = X!m = 9R.

=

a!m

(1) h(a!m) = 0 if and only if every prime ideal p associated with 9J1 is null; (2) If 9J1 is Noetherian, then h(a!m) < 00 if and only if every prime ideal associated with 9J1 is non-zero.

PROOF. For every prime ideal p associated with 9J1 there exists an element 9J1 with ann(a) = p. Put 1)1 = 9ld . a and Y = 1)1..L C X, and note that !)1 = X/Y and a'Jl = aX/Yo In particular, h(a) 2: h(a x / Y ) = h(arytd/P), so a E

19. POSITIVE ENTROPY

153

that p must be null if h(a) = 0, and p is non-zero if h(a) < 00 (otherwise 00 = h(a!Jt d ) = h(a!Jt d / P ) = h(a x / Y ) ::::; h(a) < 00). Conversely, if every prime ideal p C SJtd associated with 9J1 is null, we choose an increasing sequence (9J1 n , n ~ 1) of Noetherian submodules of 9J1 with 9J1 = Un>l 9J1n . Fix n ~ 1 for the moment and choose a prime filtration 9J1n = 918 :J .~. :J 910 = {O} of 9J1. Then 91j /91j _l ~ SJtd/qj for some prime ideal qj C SJtd containing one of the associated primes of 9J1n , and hence one of the prime ideals associated with 9J1, and Lemma 17.4 shows that qj is null for j = 1, ... , s. According to Proposition 18.6, h(a 9Jtn ) = 0, and Lemma 13.6 implies that h(a) = 0, as claimed in (1). In order to complete the proof of (2) we observe that, if every prime ideal p C SJtd associated with 9J1 is non-zero, then h(a!Jt d / P) < 00 (Corollary 18.5), and Proposition 18.6, Lemma 17.4, and Corollary 6.2 together imply that h(a) < 00. 0 By Corollary 18.5, every non-principal prime ideal p C SJtd is null, and the characterization of all null prime ideals p C SJtd which are principal is equivalent to the problem of determining the polynomials f E SJtd with M (f) = 1. THEOREM 19.5. Let f E SJt d , d ~ 1. Then M (f) = 1 if and only if ±f is a product of generalized cyclotomic polynomials (cf. Corollary 6.12).

For d = 1, Theorem 19.5 is Lemma 19.1. For d > 1, the proof of Theorem 19.5 depends on an inequality in [100], which compares the Mahler measure of a non-zero polynomial f E SJtd with that of certain polynomials associated with the faces of the convex hull of the support of f. In order to formulate Smyth's inequality we have to introduce a little bit of terminology. Let C C ]Rd be a closed, convex set. A hyperplane H C ]Rd supports C if C n H =I=- 0, and C is contained in one of the two closed half-spaces defined by H. A face of C is an intersection of the form = en H, where H is a supporting hyperplane of C (in particular, a face may consist of a single extremal point). Now assume that 0 =I=- f = L:mEZd cf(m)u m E SJtd , denote by S(f)

= {m E Zd : cf(m)

=I=-

(19.1)

O},

the support of f, and write

0, and choose a maximal [Q(m),{},c:l-separated set FeY (for notation see Proposition 13.7). The discussion in the preceding paragraph shows that there exists a [Q(m,n),{},c:l-separated set in X whose cardinality is ~ IFln, where IFI is the cardinality of F. By letting first n ---+ 00 and then m ---+ 00 we see from Proposition 13.7 and Theorem 18.1 that log M (I) = h(a) ~ h(a'Y.d-!/(fif») = log M (leI», as claimed. 0

°

PROOF OF THEOREM 19.5. The case where d = 1 is dealt with by Lemma 19.1. Assume therefore that d > 1. If f E !)td is a product of generalized cyclotomic polynomials, then the same is true for every polynomial of the form urnfA with mE Zd and A E GL(d, Z) (the notation is taken from Lemma 18.4). Hence the assertion of the theorem is unaffected if we replace f by urn fA. If f is a generalized cyclotomic polynomial, then f = urnc(u n ), where m, n E Zd, n -=f. 0, and where c is a cyclotomic polynomial in a single variable. From (16.1) and a change of variable it is clear that M (I) = M (c), and Proposition 16.1 shows that M (I) = M (c) = 1. If f is a product of generalized cyclotomic polynomials f = hI'" h s , then M (I) = n;=1 M (h j ) = 1, which completes the proof of one of the implications in Theorem 19.5. In order to prove the reverse implication we have to show that every polynomial f E !)td with M (I) = 1 is a product of generalized cyclotomic polynomials. We use double induction on the number of variables d and the number of irreducible factors r of f in qui=1 ... uyll (excluding monomials). The result is clearly true if d = 1, or if r = 0, in which case f = au rn for some a E Z and m E Zd. Now suppose that the assertion is true if either d < D or r < R, where D ~ 2 and R ~ 1. Let f E !)tD, and assume that M (I) = 1 and that f has R irreducible factors in qui=1 ... uYll. If S(I) lies on a straight

19. POSITIVE ENTROPY

155

line we can find elements m E ZD and A E GL(D, Z) such that 9 = u m fA is a polynomial of the single variable Ul' Since M (f) = M (g), Lemma 19.1 implies that ±g is a product of cyclotomic polynomials, and hence that ±f is a product of generalized cyclotomic polynomials. Assume therefore that S(f) does not lie on a straight line or, equivalently, that e(f) has a face s;: e(f) which consists of more than one point. As in the proof of Lemma 19.6 we may replace f by u m fA for some m E ZD and A E GL(D,Z) and assume that S(f) C {m = (ml"'" mD) E ZD : mD ~ O}, and that c {m = (ml, ... ,mD) E ZD: mD = O}. After multiplying f by a monomial we may thus take it that f = L:f=o ubfJ, where fj E 9lD-l for j = 0, ... , L, foh =I- 0, and S(fd consists of more than one point. Lemma 19.6 implies that M (h) = 1 and, since h E 9l D- l , our induction hypothesis implies that h is a (necessarily non-trivial) product of generalized cyclotomic polynomials in the variables Ul,"" UD-l. In particular, fL can be written-in qur\ ... ,ui)~ll-as a product of the form h = cum Il~=l(uaj - aj), where =I- C E C, al, ... , at E § c C, n E ZD-l, and where al, ... , at are primitive · ~ '77 D-l ( 1 u a = u al 1 eement s In as usua, l ",u aD-l" D - l lor a 11 a = (al, ... ,aD-l ) E

°

zD-l ).

We claim that h divides fJ for j = 0, ... , L - 1. Indeed, if h does not divide fk for some k E {O, ... , L - I}, then fk is not divisible by some (u aj - aj). Since aj is primitive, we can apply Lemma 11.3 to find a matrix B E GL(D, Z) such that Be{D) = e{D) and Baj = e{D-l): where e{t) is the i-th unit vector in ZD. After replacing f with fB, if necessary, we can write !k as fk = (uD-l-aj)g+h, where h is a non-zero complex polynomial in the variables urI, ... , ui)~2' There exists a point c = (Cl"'" CD-2) E §D-2 with h(c) =I- 0, and a neighbourhood N(Cl,,,.,cD-2,aj) C §D-l of (Cl, ... ,cD-2,aj) such that Ifk(t)/ h(t)1 > 1 +

(L-2) k _ 2

(19.4)

for all t E N(Cl, ... , CD-2, aj). We fix t = (tl, ... , tD-l) E §D-l and regard

as a function of the single variable UD. If every root of F t has modulus ~ 1, then the coefficient of ut in F t must have modulus ~ (~=~), and by comparing this with (19.4) and (16.2) we see that J§log 1Ft I dA§ > Ih(t)1 for every t E N(Cl"'" CD-I, aj). According to (16.2), Js log 1Ft I dAS ~ Ih(t)1 for every t E §D-l, and we obtain that

°

r r 1ft IdAS dASD-l > r IhldASD-l=logM(h)=O, iSD-l

= log M (f) =

i§D-l is

VI. POSITIVE ENTROPY

156

which is absurd. This contradiction implies that h divides each Ii, j = 0, ... , L - 1, and hence f. We can write f as a product f = gh, where h is a non-trivial product of generalized cyclotomic polynomials. Hence 9 has fewer than R irreducible factors, and 1 ::; M (g) ::; M (f) = 1. Our induction hypothesis (on polynomials with fewer than R irreducible factors) allows us to write 9 as a product of generalized cyclotomic polynomials, so that f is a product of generalized cyclotomic polynomials. 0 Having determined the principal prime ideals p C rytd which are null, we turn to the problem of calculating the Mahler measure M (f) of a polynomial f E rytd' An explicit calculation is, of course, possible only in some special cases, such as when d = 1 (Proposition 16.1), or for certain polynomials discussed in [13] and [101]. We present some of Smyth's results, for which we shall use the following notation: a character (mod q) is a homomorphism X: Z f-----+ h(a) ;::: h(a SJI ) ;::: h(aE9;~l .It;) =

aE Jt;

00, which is absurd.

We have proved that the existence of an ergodic automorphism a of 11'00 with finite entropy implies that 1 cannot be isolated in {M (f) : f E !n l }. Conversely, if 1 is not an isolated point in {M (f) : f E !nl}, then we can choose a sequence (fn, n ;::: 1) of irreducible polynomials in !nl with 0 < log M (fn) < 2- n log 2 for every n ;::: 1. According to Lemma 19.1, fn is not a generalized cyclotomic polynomial, and Example 6.17 (3) shows that an = a9'tl!(fn) is ergodic-and hence mixing-for every n ;::: 1 (Theorem 1.6). Furthermore, since log M (fn) < log 2, fn is monic by Proposition 16.1, and the product ofthe roots of fn must have modulus 1 (since it is an integer). After multiplying fn by a

VI. POSITIVE ENTROPY

162

power of the variable u we may assume that ±f(u) = CO+ClU+·· ·+us for some s ~ 1, where Icol = 1, and Example 6.17 (3) shows that Xn = X Trn be a measure preserving Zd-action on a probability space (Y, 'I, /1). The Pinsker algebra !,p(T) is the smallest sigma-algebra containing all finite partitions P c 'I with hJ1.(T, P) = o. It is clear that !,p(T) is T-invariant. The action T has completely positive entropy if !,p(T) is the trivial sigmaalgebra {0,X} or, equivalently, if hJ1.(T,P) > 0 for every non-trivial, finite, measurable partition P c '1'. If oo J.L(A6An) = O. We set P = {A, Y " A}, Pn = {An, Y " An}, and note that HI"(PIQ) = limn-->oo HI"(PnIQ) = limn-->oo H/.L(Pn ) = H/.L(P) for every finite partition Q C ~(T). If ~(T) i= {0, Y} we obtain a contradiction by choosing A E ~(T). Hence ~(T) must be trivial, i.e. T has completely positive entropy. 0

r

r

20. COMPLETELY POSITIVE ENTROPY

165

LEMMA 20.4. Let X be a compact group, and let 6 c ~ x be a left and right translation invariant sigma-algebra (i. e. xB = {xy : y E B} E 6 and Bx = {yx : y E B} E 6 for every B E 6 and x E X). Then there exists a closed, normal subgroup Y C X such that 6 = ~x/Y (mod Ax), where ~x/Y = {BY: B E ~x} is the sigma-algebra of Borel subsets of X/Yo If 6 is invariant under an action a of a countable group f by automorphisms of X, then Y is a-invariant.

For every f E L 2(X,6,Ax) and every continuous function h: X f-----+ C, the convolution x 1-+ f * h(x) = Ix f(y-Ix)h(y) dAx(Y) is continuous and lies in L2(X, 6, Ax). By varying h we see that L2(X, 6, Ax) is spanned by its continuous elements. We write A for the set of continuous, 6-measurable, complex valued functions on X, and note that A is invariant under left translation: for every f E A and x' EX, the map x 1-+ f (x' x) is again an element of A. We define an equivalence relation rv on X by setting x rv x' if and only if f(x) = f(x') for all f E A, denote by [x] the equivalence class of a point x E X, and consider the space Z = {[x] : x E X} of equivalence classes. Then Z is compact and metrizable in the smallest topology in which every element of A, regarded as a function on Z, is continuous. As A is invariant under left translation we obtain a continuous action x 1-+ ¢x of X on Z by ¢x([x']) = [xx'] for every x, x' EX. This action is obviously transitive, i.e. Z consist of a single ¢-orbit. We set Y = {x EX: ¢x([lx]) = [x] = [Ix]} and note that Y is a closed subgroup of X, and that the continuous, surjective map x 1-+ [x] induces a homeomorphism X/Y f-----+ Z which allows us to set Z = X/Yo Since A is dense in L2(X, 6, Ax), the collection of open subsets of Z = X/Y generates 6. The normality of Y follows from the fact that, for every f E A and x E X, the map x' 1-+ f(x-Ix'x), x' E X is also an element of A. Hence the map [x'] 1-+ 1/'x([x']) = [xx'x- I ] is a homeomorphism of Z for every x E X, and xYx- 1 = 1/'x([lx]) = [Ix] = Y for every x E X. This proves the first assertion of the lemma. The second assertion follows from the observation that f . a, E A for every f E A and '"Y E f, so that a induces a continuous f-action Ci on Z. Since Ci,([lx]) = [Ix] = Y for every '"Y E f, the group Y is a-invariant. D PROOF.

PROPOSITION 20.5. Let a be a Zd-action by automorphisms of a compact group X such that the set of a-periodic points is dense in X (Definition 5.5). Then there exists a closed, normal, a-invariant subgroup Y C X such that the Pinsker algebra \l}(a) is equal to ~x/y (mod Ax). In particular, Y is the unique minimal, closed, normal, a-invariant subgroup of X such that h(a x / Y ) = 0, and a has completely positive entropy if and only ifY = {Ix}. PROOF. If A C Zd is a subgroup of finite index we denote by a(A) the restriction of a to A. A glance at the definition of h>.x (a, P) for any finite, measurable partition P of X in (13.7) shows that \l}(a(A)) = \l}(a) for every subgroup A C Zd of finite index. Furthermore, if A is such a subgroup, and if

166

VI. POSITIVE ENTROPY

= {X EX: an(x) = X for every n E A}, then an commutes both with left and right translation by X for every n E A and x E FixA(a), and l,JJ(a) = l,JJ(a(Al) is invariant under left and right translation by every x E FixA(a). By varying A we see that I,JJ( a) is invariant under left and right translation by every a-periodic point in X, hence under left and right translation by every element in a dense subset of X, and therefore under all translations. An application of Lemma 20.4 yields that l,JJ(a) = IB x / y for some closed, normal, a-invariant subgroup Y c X. In particular, h(a x / Y ) = O. If Y' is a second closed, normal, a-invariant subgroup with h(a x / y ') = 0 and Y" = Y n Y", then h(a x / y ") = o by (14.1), which proves that Y is the unique minimal, closed, normal, ainvariant subgroup of X with h(a x / Y ) = o. The last assertion is obvious. D FixA(a)

COROLLARY 20.6. Let a be an ergodic automorphism of a compact group X. Then a has completely positive entropy. PROOF. Corollary 20.3 and Proposition 4.9 allow us to assume without loss in generality that a satisfies the d.c.c. In this case Theorem 11.1 implies that a has a dense set of periodic points, and Proposition 20.5 and Theorem 19.2 complete the proof. D COROLLARY 20.7. Let a be a Zd-action by automorphisms of a compact group X, and assume that XO is abelian. Then there exists a closed, ainvariant subgroup Y C X such that the Pinsker algebra l,JJ(a) is equal to IB x / y (mod Ax), and a Y has completely positive entropy whenever Y = {Ix}. PROOF. Choose a decreasing sequence (Vn' n ;::: 1) of closed, normal, ainvariant subgroups of X such that nn>l Vn = {Ix} and a X / Vn satisfies the d.c.c. for every n ;::: 1 (Proposition 4.9)~ By Theorem 10.2, the set ofa x / vn _ periodic points is dense in XjVn for every n ;::: 1, and Proposition 20.5 implies the existence of closed, normal, a-invariant subgroups Yn :J Vn such that l,JJ(a x / Vn ) = IB x / Yn for every n ;::: 1. The sequence (Yn , n ;::: 1) is obviously non-increasing, and we set Y = nn>l Yn and note that YnjVn = Y jVn for every n;::: 1, and that l,JJ(a) :J IB x / y . Proposition 20.5 implies that Y is the unique minimal closed, normal, a-invariant subgroup of X such that h(a x / Y ) = O. Now assume that a Y does not have completely positive entropy. Since yo C XO is abelian, the set of a Y -periodic points is dense in Y by Theorem 10.2, and Proposition 20.5 implies the existence of a unique minimal, closed, normal (in Y), aY-invariant subgroup Z s;;: Y with h(a Y/ Z ) = o. As h(aY/xZX~l) = 0 for every x E X, the uniqueness of Z implies that Z is normal in X, and (14.1) yields that h(a x / Z ) = 0, in violation of the minimality of Y. This shows that a Y has completely positive entropy. D If the group X is abelian, our ability to translate the dynamical properties of a Zd-action a by automorphisms of X into algebraic properties of the 9tdmodule m = X gives us a characterization of the Pinsker algebra l,JJ(a) in terms of the prime ideals associated with m.

167

20. COMPLETELY POSITIVE ENTROPY

THEOREM 20.8. Let a be a Zd-action by automorphisms of a compact, abelian group X, and let VJt = X be the rytd-module defined in Lemma 5.l. Then the Pinsker algebra l.lJ(a) is given by l.lJ(a) = ~x/'.rt.L, where 1)1 c VJt is the unique maximal submodule such that every prime ideal p C rytd associated with 1)1 is null. In particular, a Y has completely positive entropy, where Y = 1)1.1 C X, h(a x / Y ) = 0, and a has completely positive entropy if and only if every prime ideal p C rytd associated with VJt is positive.

PROOF. Consider the collection 'N of submodules VJt' c VJt with h(a!m') = h(ax/!m'.L) :::::: 0, partially ordered by inclusion. Then {O} E 'N =I- 0 and, for every totally ordered subset 'N' c 'N, VJt" = U!m'E:N' VJt' E 'N, since h(a!m = SUP!m'EN' h(a!m') = 0 by Lemma 13.6 (as VJt is countable, 'N' is a countable set). According to Zorn's lemma, 'N has a maximal element. For every pair 1)1, 1)1' E 'N, the injective homomorphism X'.rt+'.rt' = ~, 1---+ X'.rt X X'.rt' dual to the addition map 1)1 x 1)1' 1---+ 1)1 + 1)1' carries a'.rt+'.rt' to the restriction of the zero entropy Zd-action a'.rt x a'.rt' to a closed, invariant subgroup, so that h(a'.rt+'.rt') = 0 and 1)1+1)1' E 'N. If 1)1 and 1)1' are both maximal elements in 'N, then 1)1+1)1' E 'N, which implies that 1)1 = 1)1', i.e. that 'N contains a unique maximal element 1)1. Proposition 19.4 shows that every prime ideal p C rytd associated with 1)1 is null, and that 1)1 is the maximal submodule of VJt with this property. By Corollary 20.7 there exists a unique minimal, closed, a-invariant subgroup Y c X such that h( a X / Y ) = 0 and a Y has completely positive entropy, and it is clear that 1)1 = y.i. The last assertion follows from the trivial observation that 'N =I- {{O}} if some prime idealp C rytd associated with VJt is null, since there exists an a E VJt with rytd/P ~ rytd . a = 1)1' E 'N. 0 ll

)

COROLLARY 20.9. Let a be a Zd-action by automorphisms of a compact, abelian group, and let Y C X be a closed, a-invariant subgroup. If both a Y and a X / Y have completely positive entropy then a has completely positive entropy. PROOF. Let VJt = X be the rytd-module defined in Lemma 5.1, and consider the submodule y.i = 1)1 C VJt. If P C rytd is a prime ideal associated with VJt there exists an element a E VJt such that p = {f E rytd : f· a = O}. For a E 1)1, h(a fRd / P) > 0 by Theorem 20.8 (applied to a'.rt)j for a¢. 1)1 we set a = a + 1)1 E VJt/1)1 and consider the submodule ,c = rytd . a E VJt/I)1. Every prime ideal q associated with ,c is also associated with VJt/I)1, and must contain p. According to Lemma 17.4 and Theorem 20.8 (applied to a!m/'.rt) , 0< h(a fRd / q ) ~ h(a fRd / P ). 0

In order to derive a further corollary of Theorem 20.8 we need a lemma. A much more general version of this result, due to Kaminski, will be proved below (Theorem 20.14).

VI. POSITIVE ENTROPY

168

LEMMA 20.10. Let 0: be a 7/.,d- action by automorphisms of a compact group X. If 0: is not mixing, then there exists a closed, normal, o:-invariant subgroup y S;; X such that h(o:x/Y) = O. In particular, if 0: has completely positive entropy, then it is mixing. PROOF. Suppose that 0: is not mixing. Then O:n is non-ergodic for some 0 "ln E 7/.,d (Theorem 1.6 (2)), and we may assume that n is primitive. Lemma 11.3 allows us to find a matrix A E GL(d,7/.,) such that An = e(d) = (0, ... ,0,1) E 7/.,d, and by replacing the 7/.,d- action 0: by the action m f-+ O:Am, if necessary, we can ensure that n = e(d). Lemma 1.2 implies that there exists a closed, normal, O:e(d)-invariant subgroup V S;; X such that C = X/V is a compact Lie group and O:~d) preserves a metric {j on C. We write 17: X f---+ C for the quotient map and define a homomorphism "1: X f---+ C Zd - 1 by setting ("1(x))m = 17' O:(ml ,... ,md-l,O) (x) for every m = (m1' ... ,md-d E 7/.,d-1. Let ( E Aut("1(X)) be given by (((Y))m = O:e(d) (y(m)) for every m E 7/.,d-l, and define a 7/.,d- action (3 by automorphisms of "1(X) by setting ((3m(Y))n = (md(Y(m,+n" ... ,md-,+nd-,)) for every y E "1(X), m = (m1,"" md) E 7/.,d, and n = (n1,"" nd-d E 7/.,d-1. Then "1 . O:m = (3rn . "1 for every m E 7/.,d, and (3e(d) preserves a metric on "1 (X) . Therefore h)..'I1(x) ((3e(d)) = 0 by (13.12) and Theorem 13.3, and h((3) = 0 by Proposition 13.2. If Y = ker("1) then {0,X} "I- I.B x / y C s.p(o:) , so that 0: cannot have completely positive entropy. D COROLLARY 20.11. For f E 91 d , o:'Jtd/U) is mixing if and only if it has completely positive entropy. PROOF. The prime ideals associated with 9)1 = 91 d/ (f) are the principal ideals arising from the irreducible factors of f. If o:'Jtd/(f) is mixing, Theorem 6.5 (2) implies that f cannot be divisible by any generalized cyclotomic polynomial, so that every prime ideal associated with 9)1 is positive by Theorem 19.5. An application of Theorem 20.8 shows that o:'Jtd/U) has completely positive entropy. The reverse implication follows from Lemma 20.10. D To illustrate the difference between the prime ideals associated with a Noetherian 91d -module 9)1 and the prime ideals occurring in a given prime filtration of 9)1, we present an example of two Noetherian 91d -modules 9)1 and 9)1' which have prime filtrations with term-wise isomorphic quotients, but which satisfy that 0:9)1 has completely positive entropy, but not 0:9)1'. EXAMPLE 20.12. Let d = 2, and let f(ud E 7/.,[U1J C 912 be irreducible and non-cyclotomic. We set g = U2 - 1 and define ideals in 912 by

P = (f), a1 = (f,g2), Let

9)1

b1

q = (f2, fg),

= (f2,g),

a2 = (f,g).

= 912 /p and 9)1' = 912 /q, and consider the filtrations

o C a1/p C a2/p C 912 /p = 9)1,

20. COMPLETELY POSITIVE ENTROPY

oc

bdq

c

169

a2/q C 9't2/q = 9)1'.

The first quotients adp and bdq are both isomorphic to 9't2/p, and all the other quotients are isomorphic to 9't/a2' Since p and a2 are both prime, these are prime filtrations with term-wise isomorphic quotients. However, p is the only prime ideal associated with 9)1, so that a 9Jt has completely positive entropy by Theorem 20.8, but the set of associated primes of 9)1' is equal to {p, a2}, and Corollary 18.5 and Theorem 20.8 together imply that a 9Jt ' does not have completely positive entropy. c::J REMARK 20.13. Example 20.12 indicates that prime filtrations are an unreliable guide to completely positive entropy. However, Corollary 20.9 does allow us to conclude that a 9Jt has completely positive entropy if 9)1 is a Noetherian 9'td-module which has a prime filtration 9)1 = 918 => ... => 91 0 = {O} in which every quotient 91j /91 j - 1 ~ 9'td/qj for some positive prime ideal qj C 9't d . We return to the connection between completely positive entropy and mixing first mentioned in Lemma 20.10. A measure preserving Zd-action T on a probability space (Y, 'I, p,) is mixing of order n (or n-mixing) if, for all sets B 1, ... , Bn in 'I, miEZ d

lim P,(T-ml (Bd n··· n T_ mn (Bn)) mi-mr-+()O

= p,(Bd'"

P,(Bn).

and

for all ',J=l, ... ,n, '#J

(20.9)

For n = 2, (20.9) is equivalent to the usual definition of mixing. THEOREM 20.14. Let T be a measure preserving Zd-action on a probability space (Y, 'I, p,). If T has completely positive entropy, then T is n-mixing for every n 2': 2. PROOF. We have to prove (20.9) for every n 2': 1. For n = I, (20.9) is obvious, and we assume that (20.9) holds for all n < k with k 2': 2. Let B 1, ... , Bk be sets in 'I, and let P C 'I be the finite partition generated by B 2 , ... , Bk. Suppose that ((m(r, 1), ... , m(r, k)), r 2': 1) is a sequence in (Zd)k with limr->oo m(r, i) - m(r, j) = 00 whenever 1 :::; i < j :::; k, and that

li~~flP,(T-m(r'l)(Bd n··· n T-m(r,k) (Bk)) -

k

}] p,(Bi) I > O.

After permuting the indices 1, ... , k, replacing m(r, i) by Am(r, i) for some matrix A E GL(d, Z) and all r 2': I, 1 :::; i :::; k (cf. Lemma 11.3), and after replacing the sequence ((Am(r, 1), ... , Am(r, k)), r 2': 1) by a subsequence, if necessary, we may assume that limr->oo m(r, 1)(1) = 00, and that m(r, i)(1) 2': m(r, 2)(1) 2': m(r, 1)(1) and limr-> 00 (m(r, i)(l) -m(r, 1)(1») = 00 for i = 2, ... , k, with m(r, i) = (m(r, i)(1), ... , m(r, i)(d») E Zd for every r, i. Then Ar = T-(m(r,2)-m(r,1» (B2) n ... n T-(m(r,k)-m(r,l» (Bk)

170

VI. POSITIVE ENTROPY

E T -(m(r,2)(1) -m(r,l)(1) -l)e(1) (Pi),

where e(l) = (1,0, ... ,0) E 7l,d, and where Pi was defined at the beginning of this section. It follows that

1J-l(T-Ul(r,l) (B1) n··· n T-Ul(r,k) (Bk)) - D,J-l(Bi)1

= 1J-l(B1 nAr) - D,J-l(Bi)1 k

= lir EJ.!(lB1IT_(m(r,2)(1)-m(r,1)(1)-1)e(1)(Pi))dJ-l- gJ-l(Bi)1 :::; lIEJ.!(lB1IT-(m(r,2)(1) -m(r,l)(1) -l)e(l) (Pi)) - J-l(Bl )IdJ-l Y

g k

+ J-l(BdlJ-l(Ar ) -

J-l(Bi )

I·

By Proposition 20.2, T-me(l)(Pi) \.. {0, Y} (mod J-l) as m / decreasing martingale theorem shows that

as r

-+ 00.

00,

and the

Our induction hypothesis implies that

as r -+ 00, so that limr->oo J-l(T-Ul(r,l) (Bl ) n··· nT_Ul(r,k)(Bk)) = TI7=1 J-l(Bi ), contrary to our choice of ((m(r, 1), ... , m(r, k)), r :::: 1) and B l , ... , Bk. It follows that (20.9) holds for n = k and, by induction, for every n :::: 1, which proves the theorem. D We conclude this section with another consequence of completely positive entropy. If a is a 7l,d- action by automorphisms of a compact group X, Proposition 13.5 shows that h(a) = hAX(a) :::: hJ.!(a) for every J-l E Ml(X)'>, the set of a-invariant probability measures on IB x. If a has completely positive entropy and h(a) < 00, then this inequality is strict. THEOREM 20.15. Let a be a 7l,d- action by automorphisms of a compact group X with h(a) < 00. The following conditions are equivalent.

(1) a has completely positive entropy; (2) hJ.!(a) < h>-.x(a) whenever Ax 1= J-l E Ml(X)"'.

20.

COMPLETELY POSITIVE ENTROPY

171

For d = 1, Theorem 20.15 was established in [10], and we follow Berg's proof with the modifications described in [63]. Let T: m 1--+ T m be a measure preserving Zd-action on a probability space (Y, 'I, J-l). LEMMA

HJ1o(P)

20.16. Assume that P, Q are countable partitions in 'I such that that T has completely positive entropy on PT, and that

+ HJ1o(Q) < 00,

(20.10)

Then the sigma-algebras PT and QT are independent. PROOF. Equation (20.1) shows that (20.10) is equivalent to (20.11) We set Qk = {O, ... ,k - l}d, define Pk and Qk as in (20.6), and consider the Zd-action T(k): n 1--+ T~k) = T kn . According to (20.7),

h J1o (T(k) , Pk

V

Qk)

= k d hJ1o(T, P V Q) = k d (hJ1o(T, P) + hJ1o(T, Q» = h J1o (T(k) , Pk) + h J1o (T(k) , Qk),

and the equivalence of (20.10) and (20.11) yields that H J1o (Qkl(Qk);;(k» H J1o (QkIPT (k) V(Qk);;(k» = H J1o (Qkl(Pkh(k) V(Qk);;(k». By applying the equivalence of (20.10) and (20.11) twice more we see that h J1o (T(k) , P V Qk) = hJ10 (T(k) , P) + h J1o (T(k) , Qk) and HJ10 (PIP:;'(k» = HJ10 (PIP:;'(k) V (Qkh(k». Since k >1 P:;'(k) c P:;'oo = {0, X} by Proposition 20.2 and the assumption of completely positive entropy of T on PT,

n

HJ1o(PIQ) ~ H J1o (PIP:;'(k)

V

Q)

~ H J1o (PIP:;'(k) V (Qkh(k»

= H J1o (PIP:;'(k»

-7

HJ1o(P)

00, so that P and Q are independent. If we replace P and Q by Pk and Qk, then hJ1o(T, Pk VQk) = hJ1o(T, pvQ) = hJ1o(T, P) + hJ1o(T, Q) = hJ1o(T, Pk) + hJ1o(T, Qk), and the first part of this proof shows that Pk and Qk are independent for every k ~ 1. Hence PT and QT are independent, as claimed. 0

as k

-7

Now assume that a is a Zd-action by automorphisms of a compact group X, and that J-l is an a-invariant probability measure on \Bx. The sigma-algebras 6,6 1 C ~xxx are defined as in the proof of Proposition 13.5. LEMMA 20.17. The sigma-algebras 6 1 and 6 are independent with respect to the measure AX x J-l if and only if J-l = AX·

VI. POSITIVE ENTROPY

172

PROOF. We use the same notation as in the proof of Proposition 13.5. If JL = Ax, then

(Ax x Ax )(71"11 (E) n 7I"-1(F)) = (Ax x Ax)({(x,y) E X x X: x E E, xy E F}) =

JJ

1E1 x- I F dAxdAx

= Ax (E) Ax (F)

for all E, FE 6, so that 6 and 61 are independent. Conversely, if 61 and 6 are independent with respect to Ax x JL, then

(Ax x JL)( 71"1 1(E) n 71"-1 (F)) = (Ax X JL)(7I"1 1(E) )(Ax x JL)(7I"-1 (F)) = Ax(E)Ax * JL(F) = Ax(E)Ax(F), so that Ax x JL and Ax x Ax coincide on the sigma-algebra 61 V 6 = 93 x x 93 x (cf. (13.15)). Hence JL(F) = (Ax x JL)(7I"2 1(F)) = (Ax x AX) (71"2 1(F)) = Ax(F) for every F E ~x. 0 PROOF OF THEOREM 20.15. Suppose that we have proved the implication (1):::}(2) in Theorem 20.15 under the additional condition that JL is ergodic under a. If l/ E M1 (X) is non-ergodic and satisfies that hy(a) = h(a) (for notation we refer to the proof of Proposition 13.5), then we can write l/ as an integral of the form l/ = Ix JLx dp(x), where p is a probability measure on ~x, JLx E M1(X) is ergodic for every x E X, and the map x 1-+ I f dJLx is an a-invariant Borel map from X to lR for every continuous function f: X ~ R Then hy(a) = Ix hJ1-.,(a) dp(x) = h>-.x (a), which implies that JLx = Ax p-a.e., so that l/ = Ax. Assume therefore that a has completely positive entropy, and that JL E M1(X) is ergodic. We claim that there exists a countable partition P c ~x such that hJ1-(P) < 00 and P = ~x (mod JL), so that hJ1-(a, P) = hJ1-(a) (cf. (13.10)). If d = 1 this assertion is proved in [84]; if d > 1 it follows from Theoreme 3.2 in [17], provided that

JL({x EX: an(x)

= x}) = 0

(20.12)

for every non-zero n E 7L. d • In order to verify (20.12) we argue by contradiction and assume that (20.12) is not satisfied for some 0 =F n E 7L. d • Then Y = {x E X: an(x) = x} is a closed, a-invariant subgroup with JL(Y) > 0, and hence, by ergodicity, with JL(Y) = 1, and we conclude that hJ1-(a n ) = O. Propositions 13.2 and 13.5 imply that hJ1-(a) :::; h>-.y(aY ) = h(a Y ) = 0, which is absurd. Hence (20.12) is satisfied, and there exists a partition P with the required properties. Since a has completely positive entropy with respect to Ax, a is mixing by Lemma 20.10, hence a satisfies (20.12) with JL replaced by Ax, and we conclude as in the last paragraph that there exists a countable partition Q c ~ x with Q = ~x (mod Ax), h>-.x(Q) < 00, and h>-'x(a) = h>-'x(a, Q).

173

20. COMPLETELY POSITIVE ENTROPY

We set Q' = h>.x (a, Q)

1

7T 1 (Q)

C 6

1

and pI =

7T- 1 (Q)

c

6, and note that

+ hJL(a, P) = h>.x (a) + hJL(a) = h>.x XJL(a

x a)

< h>'xxJL(a x a, 6 1 V 6) -

~ h>'xxJL(a x a,(1) +h>'xxJL(a x a,6)

= h>'x(a)

+ h>'x*JL(a) =

(20.13)

2h>'x(a).

Here hv(T,1.1) denotes the entropy of a measure preserving Zd-action T on a probability space (Z, 3, v), restricted to a T-invariant sigma-algebra 1.1 C 3, i.e. hv(T,1.1) = sUPu hv(T,U), where U varies over all finite partitions of Z in 1.1. If hJL(a) = h>'x(a), then (20.13) shows that h>'xXJL(a x a, pI V Q') = h>'xxJL(a x a, 6 1 V 6) = h>'xxJL(a

x a, 6t} + h>'xxJL(a x a, 6)

= h>'xxJL(a x a, Q')

+ h>'xxJL(a x a, pI).

According to Lemma 20.16, the sigma-algebras p~xo: = 6 (mod AX x J-L) and Q~xo: = 6 1 (mod AX x J-L) are independent with respect to AX x J-L, so that J-L = AX by Lemma 20.17. Conversely, if (2) is satisfied, then a must be mixing by Lemma 20.10; otherwise h>'?b(a) = h>.y(a), but AX =f. Ay. According to Corollary 4.12, aXe and aXe /C(X ) are again mixing, and h(a X o/C(XO)) < h(a X o) < h(a) < 00. Propositions 2.13 and 15.3 imply that xo jG(XO) is trivial, i.e. that XO is abelian. If a does not have completely positive entropy, then Corollary 20.7 shows that here exists a closed, normal, a-invariant subgroup Y S;; X such that h(a x / Y ) = 0 and h>'x(a) = h>.y(a), contrary to (2). This shows that (2)=?(1). D CONCLUDING REMARKS 20.18. (1) The material on completely positive entropy is due to [85] and [111], and the modifications needed for the transition from Z-actions to Zd-actions are taken from [17]: Lemma 20.1 and Proposition 20.2 follow [17], Corollary 20.3 [85], and Lemma 20.4, Proposition 20.5, and Corollary 20.6 [111]. Theorem 20.8 and Corollary 20.9 were proved in [63]. Theorem 20.14 and its proof are taken from [33]; we shall return to the higher order mixing properties of Zd-actions by automorphisms of compact groups in Sections 27-28. Example 20.12 is due to Paul Smith (cf. [63]). For Theorem 20.15 and its proof we follow [10] with the modifications outlined in [63], where X was assumed to be abelian.

(2) Following [84] and [17] we call a measure preserving Zd-action T on a probability space (Y, 'I,J-L)) aperiodic if J-L({Y E Y : Tn(Y) = y}) = 0 whenever o =f. n E Zd. From the proof of Theorem 20.15 it is clear that an ergodic Zd_ action a by automorphisms of a compact group X is aperiodic if and only if it is faithful, i.e. if and only if an =f. idx for all non-zero n E Zd. If a is ergodic,

174

faithful, and h(o:) < by [17].

VI. POSITIVE ENTROPY 00,

then

0:

has a countable generator with finite entropy

21. Entropy and periodic points

Let 0: be a JEd-action by automorphisms of a compact group X. For every subgroup A c JEd of finite index we set (A)

= min{lml

:0

-I- mEA}

(21.1 )

where Iml, mE JEd, is given by (16.3), and define the fixed point set FixA(O:) as in Theorem 6.5 (3). We wish to investigate the growth rate of FixA(O:) as (A) ---> 00, and to relate this growth rate to the topological entropy of 0:. If 0: is an expansive automorphism of a finite-dimensional torus X = ']['n, then it is well known that 1

lim -log IFixnz(o:)1 n

n . . . . . . oo

= h(o:).

(21.2)

0: in (21.2) is only ergodic, but not expansive, then (21.2) remains true, but its proof becomes much more subtle, and requires a deep diophantine result on rational approximation of logarithms of algebraic numbers due to Gelfond (cf. [24], [60], and Lemma 21.8). If 0: is an expansive JEd-action by automorphisms of a compact group X, then FixA(O:) is obviously finite for every subgroup A c JEd of finite index, and it is natural to ask whether

If the toral automorphism

d)i~oo IJE}/Al log IFixA(O:) I = h(o:).

(21.3)

We shall prove in this section that (21.3) does indeed hold if 0: is expansive. However, if 0: is non-expansive, the cardinality of FixA(O:) may be equal to 1 for every A C JEd (Examples 5.6), or it may be infinite for every A (Examples 10.1 (1)-(2), and Example 6.18 (3)), so that (21.3) cannot hold in complete generality. The connection between periodic points and the d.c.c. exhibited in Section 5 (Examples 5.6 and Theorem 5.7) shows that any generalization of (21.3) to non-expansive actions will-in general-require the d.c.c. The problem of having too many periodic points (Examples 10.1) can be overcome by counting not the periodic points, but the number of connected components in the set of all points with a given period. Even with these natural adaptations of (21.3) some problems remain, and we can prove (an analogue of) (21.3) only with lim(A)--->oo replaced by limsup(A)--->oo' We begin with a few definitions. Let 0: be a JEd-action by automorphisms of a compact, abelian group X, A C JEd a subgroup with finite index, and let FixA (0:) 0 be the connected component of the identity in FixA (0:). Consider the index

(21.4)

175

21. ENTROPY AND PERIODIC POINTS

of FixA{a)O in FixA{a), and note that PA{a)

= IFixA{a)1

if and only if IFixA{a)1
0 such that Ilxo II < e for every x = (x n ) E X with 8(x, Ix) < e', which proves (1) and completes the proof of the lemma for 9J1 = rytd/ Q. If 9J1 is a general Noetherian rytd-module, we can find an integer l :::: 1 and a submodule ,c C ryt~ such that 9J1 ~ ryt~/,c. Choose a set of generators c h f(j) -- (f(j) {f (l) , ... , f(k)} lor "-', were I , ... , f(j)) I E (Y1.1 :.ltd C lor J• -- 1 , ... , k . S'lnce (>

~ ~ (']['l)zd, X can be regarded as a closed, shift-invariant subgroup of (']['I)Zd, and 0: as the shift-action of 7f.,d on X C (']['I)Zd (Example 5.2 (4)). We put IItll = maxi=l, ... ,llitill for all t = (tl, •.. ,tl) in ']['1, denote by B = {ioo(7f.,d, 0 for i = 1, ... ,d, and prime if each coordinate PI, ... ,Pd is a rational prime. For every positive element p = (Pl, ... ,Pd) E Zd we set

jp

=

Ap = {(nIPI, ... , ndPd) : (nl, ... , nd) E Zd}, Op = O(Ap), (21.14) bp = b(Ap), (¢Pl (UI), ... '¢Pd (Ud)) = ¢Pl (UI)9ld + ... + ¢pAUd)9ld,

where ¢q denotes the q-th cyclotomic polynomial for every q 2: 1. Then

and

(Ap) = . min Pi .=I, ... ,d

for every prime element p E Zd. In particular, if (p(n) = (Pln ), ... ,p~n»), n 2: 1) is a sequence of prime elements of Zd, and if p(n) -+ 00 indicates that limn--+ oo p~n) = 00 for i = 1, ... ,d, then (21.16) LEMMA 21.4. For every subgroup A C Zd of finite index, the ideal b(A) in (21.13) is radical, i.e. b(A) = Jb(A) = {J E 9ld : fk E b(A) for some k 2: I} = {J E 9ld : f(c) = 0 for every c E Vc(b(A))} (cf. Example 7.5 (3)). PROOF. According to (21.13), O(A) = Vc(b(A)) is a multiplicative subgroup of §d C Cd, which is isomorphic to By duality, Zd / A S:! n(A), and this isomorphism extends to the integral group rings 9ld/b(A) S:! Z[Zd / A] S:! Z[n(A)]; note that the composition of these isomorphisms is nothing but the evaluation of a Laurent polynomial f E 9ld (or, more precisely, of f + b(A)) on O(A). In particular, a Laurent polynomial f E 9ld lies in b(A) if and only if its restriction to O(A) is equal to zero, which is what we claimed. 0

idiA.

179

21. ENTROPY AND PERIODIC POINTS

jp

LEMMA 21.5. For every prime element p C rytd defined by (21.14) is radical. ---

PROOF. Define rytdjjp

. = X 9ld !Jp

=

(Pl, ...

,Pd) E Zd, the ideal

C ']['Z by (5.9) and set Qp = U u Pi - l for i = 1, ... d

I1id=l {I, ... ,

Pi - I} C Zd. Since Pi(U) = 1 + + ... + ,d, the projection map 1fQp: X 9l d/ip f--+ ']['Q p is a continuous group isomorphism, and we conclude that X 9l d/ip ~ ']['Q p is connected. Hence rytdjjp is a torsion-free Z-module. We denote by ryt~ 0 (in this order), we obtain (21.26). D LEMMA 21.9. Let d :::: 1, and let p C lim:up

IZd~ApIIOgIY:d/PI

~d

be a non-zero prime ideal. Then

=p+(o:!Rd/P) = h(o:!Rd/P),

(21.27)

where limsup p is interpreted as in (21.25). PROOF. If P is not principal, or if p = (f) for some generalized cyclotomic polynomial f E ~d, then Lemma 21.3, Lemma 21.7, Corollary 18.5, and Theorem 19.5, show that

lim sup 1 log ly!Rd/PI = p±(o:) = h(o:) = 0 p IZd/Apl p . Assume therefore that p = (f) for some non-zero polynomial f E ~d which is not generalized cyclotomic. We set 0: = o:!Rd/(f), choose a prime element P = (Pl,'" ,Pd) E Zd satisfying (21.17) and hence (21.18), and define jp C ~d by (21.14). If I)'tp = ~djjp then IYp!Rd/PI = I~d/((f)

+ jp)1

= Il)'tp/ f ·I)'tpl·

(21.28)

In order to calculate Il)'tp/ f ·I)'tpl in (21.28) we choose and fix an element (WI, ... ,Wd) E Vc(jp). Since V = Vdjp) is equal to the orbit of W in C Cd under the action of the Galois group, and since jp is radical by Lemma 21.5, the evaluation map T/w: 9 ~ g(w), 9 E ~d, has kernel jp. Every c = (CI, ... ,Cd) E V is of the form C = (W~l (e) , ... ,w~d(e)) for a unique element k(c) = (kl(c), ... ,kd(c)) E Qp = rr~=l{I, ... ,Pi -I} c Zd, and we define a Z-module isomorphism 'l/J: ZV f---> I)'tp by setting, for every w: V f---> Z, 'l/J(w) = LeEV w(c)uk(e) + jp. Let f3f: I)'tp f---> I)'tp be multiplication by f, and let f3j = 'l/J. f3f . 'l/J-l: ZV f---> ZV. We embed ZV linearly in C V and consider

w =

«t

185

21. ENTROPY AND PERIODIC POINTS

the linear map {3'J: C V ~ C V induced by {3j. For every c E V, the indicator function l{c} E C V is an eigenvector of (3'J with eigenvalue f(c), and the set of all such eigenvectors is a basis of C V . According to (21.18), f(c) #- 0 for every c E V, so that (3'J is non-singular, and IYp!)\d/PI

II If(c)1

= Il)'tp/f·l)'tpl = I:lY/{3jZ V I= Idet{3'J1 =

by (21.28). If we combine (21.29) with (21.24) we see that

(21.29)

cEV

IZd~ApIIOgpAp(a) ~ IZd~ApllogIYp!)\d/PI 1

= IZd/A 1 P

L

log If(c)1

cEVc(jp)

(21.30)

1

for every prime element p = (P1, ... ,Pd) E Zd satisfying (21.17). We fix the rational primes PI, ... ,Pd-l and consider the polynomial Pl-l

Pd-l- 1

kl=1

kd-l=1

II '" II

F(t) =

According to Lemma 21.8, lim

m--+oo

~ m

f(W~l, ... , w!~l\ t).

~

log IF(e21fik/m)1

L...J

{k:O:::;k Mnt:; Pi for

j = 2, ... , d - 1, where the limit in (21.32) is taken over the sequence of primes

Pd

-+ 00.

From (21.30)-(21.32) and Theorem 18.1 we conclude that

log M(f) = lim:up which proves (21.27).

IZd~Apllog IYp!1td/PI ~ p+(a) ~ h(a) =

log M(f),

0

In order to extend (21.27) to an arbitrary Noetherian torsion ~d-module 91t we have to investigate how the quantity 191tfjp ·91t1 is related to the corresponding quantity for the modules ~d/qj occurring in a prime filtration of 91t. We shall derive some of the necessary results in a slightly more general form than is currently required in order to avoid duplication at a later stage. 21.10. Assume that 0 i- I E ~d is an irreducible polynomial and an ideal with the lollowing properties:

LEMMA

aC

~d

Vc(a) a=

i- 0,

Vc(a)

n VC(/) = 0,

Va = {h E ~d : h(c) = 0

lor every c E Vc(a)}.

II 1)1 is a Noetherian ~d-module with a prime filtration {O} such that I)1j /l)1j -1 e! ~d/ (f) lor j = 1, ... ,s, then I)1j/(l)1j-l

1)1

(21.33)

= 1)18 ::::> ••• ::::> 1)10 = = a .l)1j and

a·1)1 n I)1j

+ a .l)1j) e! I)1j/(l)1j-l + a .1)1)

(21.34)

lor j = 1, ... ,s. PROOF. First we prove that (f) n a = I· a. Indeed, if there exists an element h E ((f) n a) '- I . a, then h = 9 I for some 9 E ~d '- a. Since a = {h E ~d : h(c) = 0 for every c E Vc(a)} we know that Vc(f) U Vc(g) ::::> Vc(a). However, Vc(f) n Vc(a) = 0 by assumption, so that Vc(g) ::::> Vc(a) and 9 E a. This contradiction implies that (f) n a = I . a, as claimed. Now suppose that there exists a j E {I, ... , s} and an a E (l)1j n a . 1)1) '- a . I)1j. We choose the smallest k > j such that a Ea· I)1k and note that there exist bI, ... , bn in I)1k and h!, ... , h n in a such that a = E~=1 hi . bi, and {bb . .. ,bn } ¢.. I)1k-l due to the minimality of k. Choose c E I)1k such that the map h 1-4 h· c + I)1k-I, h E ~d, induces an isomorphism of ~d/(f) with I)1k/l)1k-l. Then there exist gi E ~d with b~ = bi - gi . c E I)1k-l for i = 1, ... ,n. Since {b 1 , ... , bn } ¢.. I)1k-l we know that {gl, . .. , gn} ¢.. (f), whereas n

n

Lhi9i· C = Lhi · (b i i-I

-

bD E I)1k-l,

i=1

n

L higi E a n (f) = a . (f), i=1

187

21. ENTROPY AND PERIODIC POINTS

and there exist PI, . .. ,Pm in (I) and ql,· .. ,qm in a such that I:~=1 higi = I:;:1 pzqz and n

m

= LPZqz· c+ Lhi' b~

a

i=1

Z=1

E a ·l)1k-ll

which contradicts the rninimality of k. We conclude that I)1j n a·1)1 every j = 1, ... ,s, which implies (21.34). D

= a .l)1j

for

LEMMA 21.11. Let p C 9'td be a non-zero prime ideal, and let h E 9'td "p. If !JJ1,1)1 are 9't d-modules associated with p, and if h .1)1 c!JJ1 c 1)1, then

h( a'JJ1) = h( a')1), .

1

I 'JJ11' 1 I ')11 = hm:u p IZd/Apllog Yp ,

hm:u p IZd/Apllog Yp

1

lim sup

IZ d/ AI I!JJ1/b(A) ·!JJ11 =

~~rr2~

1 IZd/ AII!JJ1/b(A) . !JJ11 =

(A)->oo O(A)nvc(p )=0

O(A)nvc(p )=0

1

lim sup

IZ d/AI II)1/b(A) ·1)11,

~~rr2~

1 IZd / AI II)1/b(A) ·1)11,

(A)->oo O(A)nvc(p )=0

(21 35) .

O(A)nvc(p )=0

where limsup p is interpreted as in (21.25). Note that the last two equations in (21.35) may be vacuous, since there need not exist any subgroups A c Zd with finite index such that n(A) n Vc(p) = 0. PROOF. Lemma 21. 7 implies the existence of an integer M = M(p) 2: 1 such that IYpJtI = l.5tfjp . .5t1 < 00 for every prime element p E Zd satisfying (21.19), and every Noetherian 9'td-module .5t associated with p. Similarly, if A c Zd is a subgroup with finite index such that n(A) n Vc(p) = 0, then Theorem 6.5 (3) implies that PA(aJt) = 1.5t/b(A) ·.5t1 for every 9'td-module .5t associated with p. Let .5t,'c be Noetherian 9'td-modules associated with p such that h·.5t c ,C c .5t. For every prime element p E Zd and every subgroup A c Zd of finite index with the properties just described, we have that

l.5tfjp . .5t1

=

1.5t/ (,C + jp . .5t) I . I(,C + jp . .5t) fjp . .5t1

=

1(.5t/'c)fjp . (.5t/,C)I·I,C/(,C n jp ·.5t1

= = =

1.5t/b(A) ·.5t1

s 1(.5t/'c)fjp . (.5t/,C)I·I,Cfjp . 'c1,

IPA(aJt)1

1.5t/('c + b(A) . .5t)1 . I('c + b(A) . .5t)/b(A) ·.5t1

(21.36)

1(.5t/'c)/b(A) . (.5t/,C)I·I,C/(,C n b(A) ·.5t1

S PA(aJt/f!)PA (af!). In (21.36) we have used the fact that jp''c c ,Cnjp·j{ and b(A)·'c c 'cnb(A)·j{. Since every prime ideal associated with .5t/,C must contain p + h9'td, none of the

VI. POSITIVE ENTROPY

188

prime ideals associated with iil'c is principal. By Corollary 18.5, Proposition 18.6, and Lemma 21.3,

and (21.36) and the addition formula (14.1) yield that

~

lim:up IZ d Ap I log liifjp . iii :::; lim:u p

IZd~Apllog I'cfjp' 'c1,

1

IZdI AI log liilb(A) . iii

(A¥~oo

(21.37)

1

!!(A)nvc(p)=0

:::;

(lf~oo

IZdlAllog l'c/b(A)· 'c1,

!!(A)nvc(p)=0

h(a.!t)

= h(a£).

Now assume that ,C = f . ii. The map ii f---+ 'cfjp . ,C consisting of multiplication by f, followed by the quotient map, is surjective, and its kernel contains jp . ii. Hence I'cfjp ·,CI :::; liifjp . iii, and similarly we see that l'c/b(A) ·,CI :::; liilb(A) . iii. According to (21.36), lim:up IZ d~ Api (log liifjp . iii -log I'cfjp . ,CD (Alf~oo

=

0,

1

IZd I AI (log liilb(A) . iii -log l'c/b(A) . ,CD = 0,

!!(A)nvc(p )=0

so that

(l)i~oo

1

IZd I AI log If· iilb(A) . f . iii,

(21.38)

!!(A)nVc(p)=0

p±(a.!t) = p±(af·.!t).

,C

We apply (21.36)-(21.38) first with ,C = = f . 9J1 and ii = 1)1:, and obtain (21.35). 0

1)1:

and ii = 9J1, and then with

189

21. ENTROPY AND PERIODIC POINTS

LEMMA 21.12. Let P C 9td be a non-zero prime ideal, and let M be a Noetherian 9td-module associated with p. Then

where lim sUP p is interpreted as in (21.25). PROOF. According to Proposition 6.1 there exist integers 1 ::; t ::; sand submodules M = 91 8 J ... J 910 = {O} such that, for every i = 1, ... , s, 91i/91i - 1 ~ 9td/qi for some prime ideal P C qi C 9td , qi = P for i = 1, ... , t, and qi ~ P for i = t+ 1, ... , s. For each i = t+ 1, ... ,s, choose a gi E qi "p, and set h = gt+l· ... ·g8' Then h ¢:. p, and h·M C 91 t eM. Lemma 21.11 shows that limsup p IZd/Apllog IYp!JJl1 = limsup p IZd/Apllog Iy;ntl, p±(a!JJl) = p±(a'Ylt), and h(a!JJl) = h(a'Ylt), so that we may assume without loss in generality that s = t, and that qi = P for all i = 1, ... , s. If P is not principal, then Corollary 18.5 and Proposition 18.6 imply that h(a!JJl) = 0, and the Lemmas 21.3 and 21.7 yield that limsup p IZd/Apllog IYp!JJl1 = p+(a!JJl) = 0. If P = (1) for a non-zero irreducible polynomial f E 9td, we apply the Lemmas 21.6 and 21.10 to find a constant M :::: 1 such that, for every prime element p = (PI, ... ,Pd) E 7l,d satisfying (21.19), and for every j = 1, ... , s, 91j njp' M = jp ·91j and hence 191j / (91 j -1 +jp' M) 1 = 191j / (91 j - 1 + jp . 91 j ) I. It follows that

II 191j/(91j 8

IMfjp . MI =

1

+ jp' M)I

1

+ jp . 91j )1 = 19td/(P + jpW·

j=1 s

=

II 191j/(91j -

(21.39)

j=1

From Proposition 18.6 we know that h(a) = sh(a'Jt p / P ), and (21.39) is equivalent to the statement that IMfjp·MI = 19td/(P+jpW whenever the prime element p E 7l,d satisfies (21.19). An application of Lemma 21.9 completes the proof. D LEMMA 21.13. Let M be a Noetherian torsion 9td-module. Then

PROOF. Let {2!J 1 , ... , 2!Jm } be a reduced primary decomposition of M with associated (non-zero) primes {PI, ... ,Pm} (cf. (6.5)). Then the diagonal map e: a f-Y (a + 2!J1 , ... , a + 2!Jm ) from M to EB::1 M/2!Ji = JOt is injective, and the dual map 8: V

= XJt = EB:: 1 ~i ~

X is surjective. Let Z

= ker(8)

C V

VI. POSITIVE ENTROPY

190

be the kernel of O. The addition formula (14.1) shows that m

L h(o:!JJt/2Di) = h(o:fi.) = h(o:) i=1

+ h(o:z),

where o:Z is the restriction of o:fi. to Z C V = X fi.. We claim that h(o:Z) = O. If 1)1 is an ~d-module, and if 1)11, ... , I)1k are submodules of 1)1 such that 1)1 = 1)11 + ... + I)1k, then 1)1 is a homomorphic image of EB:=1 l)1i, where the homomorphism is given by addition, and hence

2::=1 h(o:'Jt;) = h(o:EB:=l 'Jt;) :::: h(o:'Jt). In order to apply this observation to o:Z we regard the i-th summand !JJ1jWi of.fi as a subgroup of.fi and note that Z = .fi/e0m) and .fij8(!JJ1) = (!JJ1jWl)j8(!JJ1) + ... + (!JJ1jWm)j8(!JJ1). If we can prove that h(o:(!JJt/ 2D i)/II(!JJt)) = 0 for i = 1, ... ,m, then h(o:fi./II(!JJt)) = 0 by the above argument. Fix i E {1, ... , m}, and consider an element (WI, ... , Wi-I, a + Wi, Wi+l,"" W m) + 8(!JJ1)

+ WI,"" -a + Wi-I, Wi, -a + Wi+l,"" -a + W m) + 8(!JJ1) in (!JJ1jWi)j8(!JJ1) c .fij8(!JJ1) whose annihilator q is a prime ideal. Then q J Pi + TI#i Pj and is therefore non-principal. In other words, every prime ideal = (-a

associated with (!JJ1jWi)j8(!JJ1) is non-principal, and Corollary 18.5 and Proposition 19.4 imply that h( o:(!JJt/2D;)/II(!JJt)) = O. From the Lemmas 21.3 and 21.7 we know that lim:up

IZd~ Api log 1Y~!JJt/2D;)/II(!JJt) 1= p+(o:z)

and that

= h(o:z) = 0,

m

Lh(o:!JJt/2Di) = h(o:fi.) = h(o:). i=1

(21.40)

From (21.36), applied to .fi and £ = 8(!JJ1), we see that

~ . 6 lIm sup 2=1

p

1 I (!JJt/2D;)I_ . 1 I fi.1 IZ d j A I log Yp - lIm sup IZdjA I log Yp P

P

P

(21.41 )

. 1 I £1' 1 I !JJt1 = IIm:u p IZdjApllog Yp = IIm:u p IZdjApllog Yp .

The proof is completed by combining (21.40)-(21.41) with Lemma 21.12.

0

PROOF OF THEOREM 21.1 (2). We write!JJ1 = X for the Zd-module arising from Lemma 5.1. Since 0: satisfies the d.c.c, !JJ1 is Noetherian by Proposition 5.4, and the finiteness of the entropy of 0: implies that !JJ1 is a torsion module: otherwise the prime ideal {O} is associated with !JJ1, which means that !JJ1 has a submodule 1)1 ~ ~d, and o:!JJt :::: o:'Jt = 00 by Theorem 17.1. Now apply Lemma 21.13. 0

191

21. ENTROPY AND PERIODIC POINTS

The last remaining assertion of Theorem 21.1 concerns the cases where a is expansive.

d

= 1, or where

d

= 1, or if a = a'Jtd/P

LEMMA

21.14. Let d ~ 1, and let jJ C !Rd be a non-zero prime ideal. If is expansive, then

h(a)

= p+(a) = p-(a) =

(l)i~CXl IZ}/Allog IFixA(a)l·

(21.42)

PROOF. If jJ is not principal, or if jJ = (f) for some generalized cyclotomic polynomial f E !Rd, then Lemma 21.3, Corollary 18.5, and Theorem 19.5 show that h(a) = p±(a) = a. If jJ = (f) for some irreducible polynomial a i- f E !Rd which is not generalized cyclotomic, then we set Wl = !Rd/(f) and note that h(a) = log M (f) ~ p+(a) ~ p-(a) by Lemma 21.4 and Theorem 18.1. If d = 1 then (21.42) is an immediate consequence of Lemma 21.8. Suppose therefore that d > 1 and a is expansive, and fix a subgroup A C Zd of index IZ d/ AI < 00. From Theorem 6.5 (1), (3)-(4), and (21.13), we know that f(w) ia for every wE O(A) = Vdb(A)), so that FixA(a) is finite. As in the proof of Lemma 21.9 we set 1)1 = !Rd/b(A), where b(A), and denote by f3f multiplication by f on 1)1. Choose a map c: Zd/A I----t Zd with c(z) + A = z for all z E Zd / A, and define a group isomorphism 'Ij;: Z71 d/ A f----+ 1)1 by setting, for every w: Zd/A f----+ Z in Z71 d/A,

'Ij;(w) =

L

w(z)uc(z)

+ b(A).

ZE71 d / A

Then

f3/ = 'Ij; . f3 f

. 'Ij;-l : Z71 d/ A f----+ Z71 d/ A is a homomorphism, and

PA(a) = IWl/b(A) . Wli = l!Rd/((f)

+ b(A))1

= II)1/f.1)11 = IZ71d / A/f3/(Z71d/A)I.

(21.43)

In order to calculate the last term in (21.43) we embed Z71 d / A linearly in C71 d / A and consider the linear map f3'j: C 71d / A I----t C 71d / A defined by f3/. For every wE O(A)), the indicator function Ww = l{w} E C 71d / A is an eigenvector of f3'j with non-zero eigenvalue f(w), and {ww : w E O(A))} is a C-basis of C 71d / A consisting of eigenvectors of f3'j. Hence

PA(a) = IZ71d / A/f3j(Z71 d/A)I = Idet f3'j1 = so that

II

If(w)l,

wEfl(A))

(21.44) Since a is expansive, f(8) i- 0 for every 8 E §d (Theorem 6.5 (4)), and log If I is continuous and hence Riemann-integrable. The second term in (21.44)

192

VI. POSITIVE ENTROPY

is a Riemann sum approximation to the integral of log If I on 18.1 implies that

p-(a:) = p+(a:) =

=

§d,

and Theorem

dj~oo IZd~AIIogPA(a:) = d?~oo IZ;/AI Iog IFixA(a:) I lim IZ;/AI (A)->oo

= log M (I) =

L

wEflA

h(a:).

log If(w)1

(21.45)

0

LEMMA 21.15. Let p C 9'td be a non-zero prime ideal, and let 9J1 be a Noetherian 9'td-module associated with p. If d = 1, or if a:!Rd/p is expansive, then p+(a:!m) = p-(a:!m) = h(a:!m). PROOF. If P is non-principal, or if p = (I) for a generalized cyclotomic polynomial f, then p±(a:!m) = h(a:!m) = 0 by Lemma 21.3, Proposition 18.6, and Corollary 18.5. If P = (I) for a polynomial f E 9'td which is not generalized cyclotomic, the proof is identical to that of Lemma 21.12, except that we use b(A) and Lemma 21.14 instead ofjp and Lemma 21.9 (note that f2(A)nVc(p) = o for every subgroup A c Zd of finite index. 0 PROOF OF THEOREM 21.1 (3). Proceed as in the proof of Theorem 21.1 (2), using Lemma 21.15 instead of Lemma 21.13. 0 CONCLUDING REMARKS 21.16. (1) The exposition in this section follows [63] and [62]. The role of Gelfond's theorem in the proof of Lemma 21.8 is discussed in [60]. (2) The necessity of the d.c.c. for Theorem 21.1 is clear from Example 5.6 (1)-(3). The assumption that h(a:) < 00 can also not be dropped: if a: is the shift-action of Zd on ']['Zd, then FixA (a:) is connected for every subgroup A C Zd of finite index, and hence 0 = p±(a:) < h(a:) = 00. In general, if 9J1 is a Noetherian 9'td-module, and if 9J1(t) = {a E 9J1 : f . a = 0 for some f E 9't d} is the torsion submodule of 9J1, then one would conjecture that (21.46) The proof of (21.45) may not be completely straightforward, since there exist Noetherian 9'td-modules 9J1 with 9J1(t) = {O}, for which FixA(a:!m) is disconnected for certain subgroups A C Zd of finite index. A simple example of this phenomenon is obtained by setting 9J1 = 29't 1 + (U1 -1) c 9't 1. If m ;::: 1, then the torsion submodule l)1(t) of the module l)1 = 9J1/(ul- 1) .9J1 has cardinality 2, so that PmZ(a:!m) = 2 for every m ;::: 1. However, limm->oo log PmZ(a:!m) = 0, and h(a:!m) = 00.

rk

(3) Theorem 21.1 (3) can be strengthened slightly. If d = 2, and if p = (I) with f = 1 + U2 + U2 E 9't2, then a: = a:!R2 I (f) is non-expansive, and Fix3z2 (a:) is not discrete (Example 18.16 (1)). By taking a careful look at the proof of

22. THE DISTRIBUTION OF PERIODIC POINTS

193

Lemma 21.14 we see that p-(a) = p+(a) = h(a); more generally, if d ~ 2, and if f E 9td is an irreducible polynomial with

Vc(J) n§d c

n = {w =

(Wl, ... ,Wd) E Cd:

wf = ... = W~ = 1 for some

k ~ I},

(21.47)

then (21.48) The difficulty in proving (21.47) for every irreducible polynomial f E 9td arises from the fact that Vc (J) may intersect §d in points whose coordinates are not all roots of unity, and that not enough is known about rational approximation to the logarithms of these points. A general proof of (21.47) would require a stronger version of Gelfond's result and may turn out to be quite difficult (cf. [102] for a quantitative form of Gelfond's result due to Feldman which does not, however, appear to be strong enough \ for our purposes) . Nevertheless it seems likely that (21.47) holds for all non-zero polynomials f, and that p± (a!m)

= h( a!moo Mn (a) = p(a) for every a E X. THEOREM 22.1. Let d ~ 1, and let a be a Zd-action by automorphisms of a compact, abelian group X with completely positive entropy. If d = 1, or if a is expansive, then lim(A)-->oo P,A = Ax in the topology of weak convergence. We begin the proof of Theorem 22.1 with two lemmas. LEMMA 22.2. Let p C 9td be a positive prime ideal (Definition 19.3), and let 9Jt be a Noetherian 9td-module associated with p. If d = 1, or if a~d/P is expansive, then there exists, for every non-zero element a E 9Jt, an L ~ 1 such that a (j. b(A) .9Jt for every subgroup of finite index A C Zd with (A) > L.

194

VI. POSITIVE ENTROPY

PROOF. According to Proposition 6.1 we can find submodules 9J1 = I)1s :) = {O} and an integer t E {2, ... , s} such that, for j = 1, ... , s, I)1j/l)1j-I ~ rytd/qj for some prime ideal Pj C rytd, qj = P for j = 1, ... , t, and qj ;? P for j = t + 1, ... ,s. Since P is principal and f2(A) n Vc(p) = 0 (Theorem 6.5), Lemma 21.10 implies that P n b(A) = p. b(A) and I)1j n b(A) ·l)1t = b(A) .l)1j for every subgroup A C Zd of finite index. We choose and fix polynomials gj E qj "P for j = t + 1, ... ,s, set 9 = gt+I ... gs, and note that 0 =f. 9 . a E I)1t for every non-zero element a E 9J1. Suppose that a E 9J1 is a non-zero element with a E b(An) .9J1 for some sequence (An' n 2 1) of subgroups of finite index in Zd with lim n --+ oo (An) = 00. Then b = g. a is a non-zero element in b( An) ·l)1t for every n 2 1, and we choose k E {1, ... , t} with b E I)1k" I)1k-I and note that b E I)1knb(An) ·l)1t = b(An) ·l)1k for all sufficiently large n 2 1. There exists an element c E I)1k " I)1k-I such that the map f f-+ f· c + I)1k-I induces an isomorphism of rytd/qk = rytd/P and I)1k/l)1k-I, and we choose h E rytd such that h . c E b + I)1k-I. Our choice of k implies that http, and Lemma 21.14 yields that ... :) 1)10

0< h(o:9l d / P) = h(o:'Jlk/'Jl k .

1)

=

lim 1 log PAn (o:'Jlk/'Jl k- 1 ) n--+oo !Zd / An!

1

nl~~ !Zd/An!log !l)1k/(l)1k-I

+ b(An) ·l)1k)!

nl~~ !Zd: An! log !l)1k/(l)1k-I + b(An) ·l)1k + rytd· b)! = lim

1

n--+oo !Zd / An!

log !rytd/(P

+ hrytd + b(An))! = h(o:9l

d

(22.2)

/(P+(h))),

since o:9ld/(p+(h)) is expansive by Theorem 6.5 (4) whenever o:9ld/P is expansive. As http, none of the prime ideals associated with rytd/(p+(h)) can be principal, and Corollary 18.5 and Proposition 19.4 imply that h(o:9l d /(p+(h))) = 0, which is impossible in view of (22.2). This contradiction shows that every non-zero a E 9J1 must satisfy that a b(A) .9J1 whenever (A) is sufficiently large. D

tt

LEMMA 22.3. Let 9J1 be a Noetherian rytd-module whose associated prime ideals PI, ... ,Pm are all non-zero and positive. If d = 1, or if o:9ld/pj is expansive for every j = 1, ... , m, then there exists, for every non-zero element a E 9J1, an integer L 2 1 such that a b(A) . 9J1 for every subgroup of finite index A C Zd with (A) > L.

tt

PROOF. We choose a reduced primary decomposition {WI, ... , W m } of 9J1 corresponding to the prime ideals PI, ... ,Pm (cf. (6.5)) and consider the injective homomorphism a f-+ (a+W I , ... ,a+Wm ) from 9J1 to 1)1 = 9J1/WI x··· x 9J1/Wm . If there exists a non-zero element a E 9J1 and a sequence (An' n 2 1) of subgroups of finite index in Zd such that limn--+oo(An ) = 00 and a E b(An) .9J1 for every n 2 1, then we can obviously also find a non-zero element ii =

22. THE DISTRIBUTION OF PERIODIC POINTS

(a + ®!, ... , a + ®m) E 1)1 such that a E beAn) latter is impossible by Lemma 22.2. 0

.1)1

195

for every n ~ 1. Alas, the

PROOF OF THEOREM 22.1. Lemma 5.1, Proposition 5.4, and (4.10) allow us to assume that a = a!U1 and X = X!U1 = 951 for some Noetherian rytd-module 9)1. If a is expansive, then Theorem 6.5 (4) shows that a!Rd/P is expansive for every prime ideal p C rytd associated with 9)1. The Fourier transform JkA of JkA is equal to the indicator function of the submodule b(A) . 9)1 c 9)1, and the sequence JkA converges weakly to AX if and only if there exists, for every non-zero a E 9)1, an L ~ 1 with a tJ. b(A) .9)1 whenever (A) > L. However, this is precisely what was proved in Lemma 22.3. 0 THEOREM 22.4. Let a be an expansive action of Zd by automorphisms of a compact, abelian group X, and let l/ be a weak limit point of JkA as (A) ---> 00. Then h,,(a) = h(a), i.e. l/ is a measure of maximal entropy for a (cf. Proposition 13.5). PROOF. By Lemma 5.1, Proposition 5.4, and (4.10) we may assume that a = a!U1 and X = X!U1 = 951 for some Noetherian rytd-module 9)1. Theorem 20.8 shows that there exists a unique maximal submodule 1)1 c 9)1 such that every prime ideal associated with 1)1 is null, and a!U1/oo(A n ) = 00 and a E S(~) = beAn) .9)1 for every n ~ 1, and Lemma 22.3, applied to 9)1/1)1, implies that a must lie in 1)1. Hence SeD) = z..L C 1)1, and Z :) Y = 1)1..L. By Proposition 13.5, Theorem 20.8, and by the addition formula (14.1), h(a) ~ h,,(a) = h(a Z ) = h(a z / Y ) + h(a Y ) = h(a Y ) = h(a). 0 We conclude this section with two examples which show that in the absence of completely positive entropy the measures JkA need not converge as (A) ---> 00. EXAMPLES 22.5. (1) Let a = (Ul - 2, U2 - 3) C ryt2, 9)1 = ryt2/a, and let X = X!U1 (cf. Example 5.2 (2)). Then 9)1 ~ Z[n and &(1,0) and &(0,1) correspond to multiplication by 2 and 3 on Z[iJ. We identify 9)1 with Z J and choose sequences (jn, n ~ 1) and (kn, n ~ 1) of positive integers such that limn-->oojn = limn-->oo k n = 00, and 2 jn == 1 (mod Pk), 3 kn == 1 (mod Pk) for every n ~ 1 and k = 1, ... , n, where Pk is the k-th prime exceeding 3: a

= a!U1 and

[i

196

VI. POSITIVE ENTROPY

= 5, P2 = 7, etc. If An = {(rjn, skn ) : r, s E Z} C Z2, and if a E beAn) .9Jt = (2 jn - 1)z[ij + (3 jn - l)Z[iJ, then a = for some l ~ 1 and some t E Z which is divisible by PI ..... Pn, and we conclude that every non-zero element a E 9Jt = Z[~;l can only lie in finitely many beAn) .9Jt. As we have see in the proof of Theorem 22.1, this implies that limn --. oo {.LAn = Ax. To see that other limits are possible, choose a sequence (k~, n ~ 1) such that limn --. oo k~ = 00 and 3k~ == 6 (mod 25) for every n ~ 1. Next we choose an increasing sequence (j~, n ~ 1) in N such that, for every n ~ 1, 2j~ == 1 (mod 5), and 2j~ == 2 (mod p) for every prime factor P i= 5 of 3k~ - 1. We define A~, n ~ 1 as above with j~ and k~ replacing jn and kn' and note that b(A~) .9Jt = 5Z[ij for every n ~ 1. In particular, the measures {.LA' , n ~ 1, are all equal, and distinct from Ax. n

PI

ir

(2) Let a = (2, 1 + Ul + U2) C 912, 9Jt = 912/ a, a = oo9Jl, and X = X 9Jl (cf. Examples 5.3 (3) and 6.18 (5)). By considering the realization of X c {O, ~ r~2 c ']['Z2 of Example 5.3 (3) one can easily verify that the identity Ox is the only element in X which is fixed under OO(2k ,0) for any k ~ 0 (cf. also Example 5.6 (2)). In particular, if Ak = {(r2k, s2k) : r, s E Z}, then FixAk (a) = {Ox} for every k ~ o. However, if A~ = {(r(2k -1),s(2k -1)): r,s E Z}, then FixA~ (a) consists of all points x E X c {O, z2 C ']['Z2 which have horizontal period 2k - 1, and from (5.7) it is clear that any non-zero a E 9Jt can lie in FixA~ (00).1. = beAu .9Jt for only finitely many k ~ o. As we have seen in the proof of Theorem 22.1, this guarantees that limk--.oo {.LA' = Ax. [J

n

k

CONCLUDING REMARKS 22.6. The material in this section is taken from [107]. For d = 1, Theorem 22.1 was proved in [60] (cf. [59] and [69]), and for expansive Zd-action it is due to [107]. Like Theorem 21.1 (3), Theorem 22.1 is a direct consequence of Lemma 21.14 (cf. (22.2)), and can be strengthened in exactly the same way (cf. Remark 21.16 (3)).

23. Bernoullicity Let d ~ 1. A measure-preserving Zd-action T on a probability space (X, 6, {.L) is Bernoulli if there exists a probability space (Y, 'I", v) such that T is measurably conjugate to the shift-action (J" of Zd on (yZd, 'I"Zd , v Zd ), where 'I"Zd is the product Borel field on yzd, and where (J" is defined as in (2.1) or (5.8). In particular T is Bernoulli if and only if there exists a count ably generated sigma-algebra il C 6 with the following properties: (1) il is independent under T, i.e.

whenever k ~ 1, B o, ... , Bk lie in il, and 0, nl, ... , nk are distinct elements in Zd,

23.

BERNOULLICITY

197

(2) E(UnEZd T_n(1.1)) = (5 (mod v), where E(C) is the sigma-algebra generated by a collection of sets C C (5. If a (countably generated) sigma-algebra 1.1 c (5 satisfies (1), but not necessarily (2), then SU = E(UnEZd T-n(1.1)) is called a Bernoulli factor of T. Since Bernoulli actions of Zd are measurably conjugate if and only if they have the same entropy ([76]), Bernoullicity is an important property in the study of conjugacy of Zd-actions. In this section we prove the following theorem.

THEOREM 23.1. Let d ;::: 1, and let 0: be a Zd-action by automorphisms of a compact, abelian group X. Then 0: is Bernoulli on (X, IJ3x, AX) if and only if it has completely positive entropy. Before discussing the proof of Theorem 23.1, let us consider the following examples, where Bernoullicity allows us to conclude the measurable conjugacy of topologically non-conjugate Zd-actions. EXAMPLES 23.2. (1) Let d = 2, f(i,j) = l+ui +u~ E 9\2, (i,j) E {I, _1}2 and let 9Jt(i,j) = 9\2/ f(i,j)9\2, o:(i,j) = o:9Jt(i,j) and X(i,j) = X 9Jt (i,j). Lemma 18.4 and its proof show that

o:(i,j)

= o:~,l) with A = (b ~), where o:~,l) is the Z2_

action n f--7 o:~;.l). By applying either Proposition 13.1 or an elementary change of variable in Theorem 18.1 we see that h(o:(i,j») = h(o:(l,l») for every (i,j) E {l, _1}2, and Proposition 19.7 gives the exact value of h(o:(l,l»). According to Theorem 20.8, o:(i,j) has completely positive entropy, and Theorem 23.1 implies that the Z2-actions o:(i,j) are Bernoulli with equal entropy. By [76] all these actions are measurably conjugate; however, since the prime ideals associated with the various 9\2-modules 9Jt(i,j) = 9\2/(f(i,j») are all distinct, Theorem 5.9 implies that the Z2-actions o:(i,j) and o:(i',]') are topologically non-conjugate whenever (i,j) =1= (i',]'). (2) More generally, if 0: is a Zd-action by automorphisms of a compact, abelian group X with completely positive entropy, and if A E GL(d, Z), then Proposition 13.1 implies that the Zd-action O:A: n f--7 O:An has completely positive entropy, and that h(o:) = h(O:A)' By Theorem, 0: and O:A are Bernoulli, and hence measurably conjugate by [76]. (3) If we consider, in the notation of Example (1), the prime ideals p (i,j) = = 29\2 + f(i,j)9\2 C 9\2 and set l)1(i,j) = 9\2/p(i,j) for every (i,j) E {I, _1}2, then Theorem 25.15 will imply that the Z2- actions o:')1(i,j) and o:')1(i',]') are measurably non-isomorphic whenever (i, j) =1= (i', j'). This is an indication that in the absence of completely positive entropy (and hence of Bernoullicity) the measurable conjugacy problem for Zd-actions by automorphisms of compact, abelian groups becomes considerably more complicated (cf. Chapters 7 and 9). c::J (2,j(i,j»)

198

VI. POSITIVE ENTROPY

If T is a measure preserving Zd-action on a probability space (X, 6, /-1), then a direct proof of the Bernoullicity of T amounts to the explicit construction a sigma-algebra 'I C 6 satisfying the conditions (1) and (2) at the beginning of this section, which is in general very difficult. Fortunately Bernoullicity can be shown to be equivalent to variety of apparently weaker and more easily verifiable properties (cf. Theorem 23.4 and [34]). In order to formulate some of is a Noetherian !nd-module, then Example these conditions we recall that, if 5.3 (4) shows that there exists an integer k :2': 1 such that the Zd-action am has a natural realization as the restriction of the shift-action a of Zd on (ll'k)Zd to some closed, shift-invariant subgroup X = Xm C (ll'k)Zd. The Bernoullicity of am is equivalent to the statement that the measure-preserving Zd-action a on ((ll'k)Zd, 113 (1fk)Zd ,Ax) is Bernoulli, where Ax is viewed as a shift-invariant

m

probability measure on (ll'k)Zd. For the following definitions we adopt a slightly more general point of view by considering shift-invariant probability measures on yZd, where (Y, 8) is a compact, metric space with diameter diam(Y) = maxy,y'EY 8(y, y') = 1, and where the shift-action a of Zd on yZd is defined as in (5.8). For every subset F C Zd we denote by yF the compact, metrizable space of all maps x: F f-----+ y, write Il3 YF for the Borel field of yF, and denote by Ml (yF) the weak* -compact set of probability measures on ll3 y F. In the special case where F = Zd we write Ml ( y zd) 0, an integer k :2: 0 and an e:' > 0 with -

A

dBN(p" p, ) < e: for every K,N:2: 1, and for every (k,K,e:')-cover A of B N . (3) p, is almost box independent if lim lim sup dB M(p" p,13 N) =

N->oo M->oo

o.

(4) p, is summably Vershik if there exists an increasing sequence (Nj, j :2: 1) in N and, for each j :2: 2, a partial cover AU) of BNj by translates of B Nj _ 1 with the following properties. (a) Lj~2(1 -1[A(j)]I/IBNj I) < 00;

(b) Lj~2dBNj(P"p,A(j»)
0, and (23.35)

such that (23.36)

= 1, ... , s, where s

for j

~ 1. Then ex ')1 is Bernoulli.

For the proof of Proposition 23.16 we require a relative version of one of the characterizations of Bernoullicity in Definition 23.6 and Theorem 23.7. Suppose that (Y,I5) and (Z, 15') are compact, metric spaces with diameter 1, and let u(Y), u(Z) and T = u(Y) x u(Z) be the shift-actions (5.8) of Zd on y Zd , ZZd and (Y x Z)Zd , respectively. We write a typical element in (Y x Z)Zd ~ yZd X ZZd as (y, z) with y = (Yn) E yZd and z = (zn) E ZZd and denote by 7r(Y)(y, z) = y and 7r( Z) (y, z) = z the projections of (y, z) onto its coordinates in yZd and ZZd. Let /-L E M1((y X Z)Zd V , set /-L(Z) = /-L(7r(Z»)-l E M1(ZZd),P) , and apply standard decomposition theory to obtain a Borel map z f-> /-Lz from ZZd to Ml (YZd) such that

(23.37) for every continuous map h: yZd x ZZd /-L (j~Y)

f----+

lR, and

(Y)

= /-Lzu- n

(23.38)

for every z E ZZd and n E Zd. DEFINITION 23.17. The measure /-L E Ml ((Y X Z)Zd) T is relatively almost box independent with respect to ZZd if, for /-L(ZL a.e. z E ZZd,

PROPOSITION 23.18 ([91]). Suppose that the following conditions.

(1) (2) Then

/-L

E M1((y

X

Z)Zd)T satisfies

M1(ZZd){j(Z) is Bernoulli; is relatively almost box independent with respect to ZZd.

/-L(Z) E /-L

/-L

is Bernoulli.

VI. POSITIVE ENTROPY

216

We turn to the proof of Proposition 23.16 and denote, for every l ~ 1, by the shift-action (5.8) of Zd on VI = (1[' Zd)1 ~ (1['I)Zd. If I = 1 we write (J instead of (J(l). For the next two lemmas we assume that f = EnEZd cf(n)u n E !.nd is a non-zero, irreducible polynomial with h( O/'ftd/(f)) > 0, which is nice in the sense of Definition 23.11. We regard Y = X'Ytd/(f) as the closed, shiftinvariant subgroup (5.9) of V, identify a'Ytd/(f) with the restriction of (J to Y (cf. (5.8) and (5.10)), and view Ay as an element of M 1 (V)0". For every v E V we define a probability measure A(v) E M 1 (V) by setting (J(l)

(23.39) for every B E !Bv. LEMMA 23.19. The measures pendent in the sense that

A(v),

v E V, are uniformly almost box inde-

(23.40) PROOF. Proposition 23.15 and Theorem 23.7 together imply that Ay E

M1 (V)O" is Bernoulli and thus almost box independent. Hence there exists, for every M,N ~ 1, a probability measure v(N) E C(Ay,A~N) C M1(V2) which is

invariant under the Zd-action n d(\ \BN) · 11m BM /\y, /\y = -+00

M

where 7r~~}(v(l), v(2))

= v~)

f---+

(Jg~+l)n on V 2 , and which satisfies that

1" 1v2 l:( 7r{n} , 7r{n}

1·1m -IBI -+00 M

M

u

~

nEBM

for every (v(l) , v(2)) E V 2 , i

(2))d (N)

(1)

V

= 1,2 and n

,

E Zd. For

every v E V we define a homeomorphism Rv: V 2 f--+ V2 by R v (V(l),V(2)) = (v(1) +v,v(2) +v) for every (v(1),v(2)) E V 2 . The measure v(N)Rv E M1(V2) satisfies that dB

(A(v) (A(v))BN)

M'

= _1_ IB I M

" ~ nEBM

1 V

2

8(7r(1) 7r(2)) dv(N) {n}'

{n}

D

LLV

-

B

= dBM(Ay, Ay N ) for every v E V and M, N ~ 0, and by letting first M and then N tend to infinity we obtain (23.40). 0 LEMMA 23.20. Let 1)1 be an !.nd-module which satisfies (23.35)-(23.36) for some s ~ 1, and for p = (J) = f!.nd. Then a')1 is Bernoulli. PROOF. We prove the Bernoullicity of a')1 by induction on the integer s in (23.35). If s = 1 then 1)1 = !.nd/(J), and a')1 is Bernoulli by Proposition 23.15. Assume therefore that s > 1, and that we have proved the Bernoullicity of a')1l for every !.nd-module 1)1' of the form 1)1' = 1)1~_1 J ... J 1)1~ = {O} with I)1j/l)1j_1 ~ !.nd/(J) for every j = 1, ... , s - 1.

217

23. BERNOULLICITY

Let 1)1 be an rytd-module with submodules 1)1 = I)1s :::) ... :::) 1)10 = {O} such that I)1j/l)1j-1 ~ rytd/(J) for every j = 1, ... ,8. Choose elements a1,"" as in 1)1 such that I)1j = rytd . aj + I)1j -1 for j = 1, ... , 8, and consider the corresponding surjective homomorphism{/;: rytd f----+ 1)1 with (/;(h, ... , Is) = 2:::=1 Ii . ai for every (h, ... ,Is) E rytd' The injective dual homomorphism 'IjJ: X'J1 f----+ V S = ~ satisfies that a~) ·'IjJ(x) = 'IjJ·o:,!:(x) for every x E X and n E Zd, and allows us to regard X = X'J1 as a closed, shift-invariant subgroup of V S (cf. Example 5.2 (4)). Furthermore, if Xj = I)1t eX c VS, then Xo = X and

Xj

= {x

EX: 7f(1)(x)

= ... = 7f(j)(x) = O},

Xj-d Xj ~ X'Rd/(f) , x'J1 j

= X/Xj

(23.41)

~ ry(j)(X) C vj

for every j = 1, ... , 8. We set W = ry(s-l)(X) C v s- 1 and note that 7f(s)(Xs _d = y = X'Rd/(f) C V. According to (23.41) our induction hypothesis implies that the restriction of a(s-l) to W ~ X'J1 s -1 is Bernoulli, and Proposition 23.15 guarantees that a = a(1) is Bernoulli on Y. As in (23.37)-(23.38) we obtain a family {/Lw : W E v s- 1 } E M1 (V) with

J

hdAX

=

r }vr

}VS-l

h(v(1), ... , v(s)) d/L(v(1), ... ,V(S-l»)(V(S)) dAW(V(1), ... , v(S-l)),

and (23.41) implies that there exists, for V with /Lw

AW-a.e. W E

vs- 1 , an element v(w) E

= A(v(w)).

According to Lemma 23.19 and Definition 23.17, Ax is relatively almost box independent with respect to vs- 1 , and Proposition 23.18 shows that AX = Ax 'l1 E M 1(VS)CT(S) is Bernoulli. D PROOF OF PROPOSITION 23.16. Let pC rytd be a prime ideal with h(o:'R d / P ) > 0, and let 1)1 be an rytd-module satisfying (23.35)-(23.36). Then p is principal by Corollary 18.5, and the description at the beginning of the proof of Lemma 23.19 shows that there exists an 8 ~ 1 and elements a1, ... ,as in 1)1 such that 1)1 = 2::j=lrytd' aj and {h E rytd : h· aj E I)1j-d = P for every j = 1, ... ,8, where 1)10 = {O} and I)1j = 2::i=l rytd . ai for j = 1, ... ,8. In particular, if p = {O}, then {a1,' .. , as} is linearly independent over rytd, 1)1 ~ rytd' and 0:'J1 is conjugate to the shift-action (5.8) of Zd on ('JI'S)Zd and hence Bernoulli. If p = (p) for some rational prime p > 1 we set Xj = I)1t c X = X'J1 = !Jt, j = 0, ... ,8, and observe that Xs = {O}, X s- 1 ~ X'R d/ p , and use Remark 6.19 (4) to identify the Zd-action o:X 1 induced by 0: = 0:'J1 on X s - 1 with S -

218

VI. POSITIVE ENTROPY

the shift-action a of Zd on Z~;. We claim that there exists a Haar measure preserving Borel isomorphism ¢: X I---t X/X s- 1 X (Z/pZ)Zd which carries a to the cartesian product a X / XS - 1 x a, where a X / Xs - 1 is the Zd-action induced by a on X/X s - 1' In order to construct ¢ we set W = X/ X s - 1 and choose a Borel map (: W I---t X with ((x + X s - 1) + X s - 1 = X + X s - 1 for every x E X (cf. Lemma 1.5.1 in [78]). Define a Borel isomorphism 'lj;: X I---t W X X s - 1 by setting 'lj;(x) = (x + X s- 1, X - ((x + Xs-d) for every x E X, and use the identification of X s - 1 with V = (Z/pZ)Zd to regard 'lj; as a Borel isomorphism 'lj;: X I---t W X V. The Zd-action a' on W x V defined by a~ = 'lj;. an' 'lj;-1 is of the form a~(w, v)

for every n E Zd, where c: Zd

am(c(n, w))

= (a!{ (w), an(v) + c(n, w)) X

W

I---t

(23.42)

V is a Borel map with

+ c(rn, a!{ (w)) = c(rn + n, w)

(23.43)

for all rn, n E Zd and w E W. If c is of the form c(n,·) =an''Y-'Y'a!{

(23.44)

for every n E Zd, where T W I---t V is Borel, then the map ¢(w, v) = (w, v + 'Y(w)) from W x V to W x V carries a' to the product action a W x a of Zd on W x V. In order to find a solution 'Y of (23.44) we write 1T{n} : V I---t Zip for the n-th coordinate projection and set cn(rn, w) = 1T{n} (c(rn, w)) for every n E Zd and wE W. Then (23.44) is equivalent to the solution of the equations cm(n, w)

= 'Ym+n(w) - 'Ym(a!{ (w))

(23.45)

for every rn, n E Zd and w E W in terms of Borel maps 'Ym: W I---t Zip, rn E Zd; if all these equations can be solved, then 'Y: W I---t V is obtained by setting 1T{m} . 'Y = 'Ym for every rn E Zd. In order to solve (23.45) we set, for every wE W, 'YO(w) = 0 and use (23.43) to solve (23.45) inductively for rn = 0 and for every n E Zd. This shows that a is indeed conjugate to a X / Xs - 1 x a on X/xs- 1 x V, and by replacing 1)1 = I)1s with I)1s-1, X with ~ = X/ Xs-l, and a with a'Jl s - 1 we see that a is conjugate to a X / Xs - 1 x a x a on X/X s - 2 x V 2 . By using induction we obtain after s steps that a is conjugate to a x ... x a on V x ... x V = VS, and hence Bernoulli. Finally we have to deal with the case where p = (f) for some irreducible element f = LnEZd CJ(n)un E 9td which has at least two non-zero coefficients. The same consideration as in Lemma 23.10 allows us to assume that f is nice (Definition 23.11), in which case the Bernoullicity of a = a'Jl is proved in Lemma 23.20. D

23. BERNOULLI CITY

219

PROOF OF THEOREM 23.1. Let a be a Zd-action by automorphisms of a compact, abelian group X. If a is Bernoulli, then it is clear that a has completely positive entropy ([76]). Conversely, if a has completely positive entropy, and if 9J1 = X is the ~d-module defined by Lemma 5.1, then Theorem 20.8 shows that every prime ideal P c ~d associated with 9J1 is positive, i.e. satisfies that h( aSJ'td/P) > O. If 9J1 is not Noetherian there exists an increasing sequence of finitely generatedand hence Noetherian-submodules (9J1 k , k ~ 1) of 9J1 with 9J1 = Uk>I 9J1k· Since every prime ideal associated with any of the 9J1 k is also associated with 9J1, and since every prime ideal associated with 9J1 is associated with 9J1k for some k ~ 1, Theorem 20.8 guarantees that a!Ul has completely positive entropy if and only if a!Ulk has completely positive entropy for every k ~ 1. For every k ~ 1, a!Ulk is the Zd-action induced by a on the quotient group Xj9J1t. As the a-invariant subgroups 9J1t c X decrease to {O} as k ~ 00, Lemma 23.5 shows that a is Bernoulli if and only if a!Ulk is Bernoulli for every k ~ 1. This implies that it is enough to prove Theorem23.1 under the additional assumption that the ~d-module 9J1 = X is Noetherian. We denote by {PI, ... , Pm} the set of prime ideals associated with the module 9J1 (which is now assumed to be Noetherian) and consider the Noetherian ~d-module 1)1 = 1)1(1) EB· .. EBI)1(m) ::> 9J1 constructed in Corollary 6.3. Since each Pi is positive, and since each of the modules l)1(i) satisfies (23.35)-(23.36) for some s ~ 1, and for P = Pi, Proposition 23.16 shows that al)1(i) is Bernoulli for i = 1, ... ,m. Hence al)1(l)E!l"'E!lI)1(>n) = al)1(l) x ... x al)1(>n) is Bernoulli, and a is Bernoulli by (6.7) and Lemma 23.4. 0 CONCLUDING REMARK 23.21. The exposition in this section follows [91], except that the proofs of some of the purely measure theoretic results presented there are omitted. Example 23.2 (1) is taken from [106J. For d = 1, Theorem 23.1 was established in [3J, [4], [41], [58J and [72J (note that, for d = 1, ergodicity and completely positive entropy are equivalent by Corollary 20.6 and Theorem 20.14). The idea to use valuations in the proof of Bernoullicity of certain automorphisms of solenoids already appears in some unpublished notes of Katznelson. A proof of Theorem 23.1 for d = 1, based on a suggestion due to Lind to explore the connection between the product formula (17.15) for global fields and a certain asymptotic independence property ensuring Bernoullicity, can be found in [61J. For d = 2, the Bernoullicity of aSJ't2/(l+u 1+U 2 ) was appears in [106], and of expansive Z2-actions with completely positive entropy in [108J.

CHAPTER VII

Zero entropy

24. Entropy and dimension One of the most interesting phenomena which arises in the transition from Z-actions to Zd-actions by automorphisms of compact groups is the existence of non-trivial (e.g. mixing) actions with zero entropy. Theorem 19.5 characterizes the principal prime ideals p C 9td which are null, and from Corollary 18.5 we know that every non-principal prime ideal p C 9td is null. Theorem 6.5 (2) and Proposition 19.4 yield an abundance of mixing Zd-actions with zero entropy. One class of such actions on compact, connected, abelian groups is introduced in Section 7, where we investigate actions of the form a'Jtd/ a arising from prime ideals a C 9td for which Vc(a) is finite, and other zero entropy Zd-actions are considered in the Examples 4.16 (1), 5.3 (5), 6.18 (5), and 8.5 (1). Before introducing further examples of mixing Zd with zero entropy we shall discuss briefly the restrictions a(r) of a Zd-action a by automorphisms of a compact, abelian group X to various subgroups r c Zd. If h(a) = 0, then h( a(r») may be positive (even infinite) for certain subgroups r c Zd of rank r < d. For a Zd-action of the form a = a'Jtd/p, where p C 9td is a prime ideal, this dependence of entropy on the rank of r involves the number r(p) introduced in the Propositions 8.2-8.3. We begin our investigation with prime ideals p C 9td with p(p) > 0, where p(p) is the characteristic of 9td/P defined as in (6.2). For notation we refer to Section 13; in particular,

Q(m)={-m, ... ,m-1}d C Zd for every m2:1.

c

(24.1)

PROPOSITION 24.1. Suppose that p 9td is a prime ideal such that p = and a = a'Jtd/P is ergodic, and let r = r(p) E {I, ... ,d} be the integer

p(p) >

°

221

VII. ZERO ENTROPY

222

defined in Proposition 8.2. If s is a non-negative real number, then

J~oo s~p (2~ )8 log N ( V

(L

m (U) )

mEQ(m)

= J~oo s~p

(2~)8 log h

AX (

V

CLm(P))

(24.2)

mEQ(m)

if and only if s < r(p), if and only if s > r(p),

={~

where the suprema are taken over all open covers U and all finite, measurable partitions P of X = X 9l d/p, and where Ax is the normalized Haar measure of X. In particular, the integer r( q) does not depend on the choice of the primitive subgroup r c Zd in Proposition 8.2, and is a measurable conjugacy invariant. PROOF. We realize the totally disconnected group X shift-invariant subgroup (6.19) of lF~d and assume that action of Zd on X. As in (24.3) we denote by

= X 9l d/P as the closed, 0: = o:9ld/P is the shift-

Po = {[Olo, ... , [p - 1lo}

(24.3)

the state partition of X, where film = {X = (xn) EX: Xm = i} for i = 0, ... ,p - 1 and m E Zd. Then Po generates the topology of X under 0:, and we set, for every m 2: 0,

Po(m)

=

V

O:-m(PO).

mEQ(m)

Let r, Q c Zd be the primitive subgroup and the finite set appearing in Proposition 8.2, and let r = r+Q. As explained in the proof of that proposition, we may assume without loss in generality that r = {n = (nl, ... ,nd) E Zd: nr+l = ... = nd = O}. In this case we may also take it that nl = ... = nr = for every n = (nl' ... ,nd) E Q; these additional assumptions will obviously not affect our claim. For every m 2: we put Q(m)' = Q(m)nr and denote by S(m)' :::) Q(m)' the unique maximal subset of Zd with the property that 7rS(m)1 (x) = 7rS(m)' (x') for all x,x' E X with 7rQ(m)l(x) = 7rQ(m)l(x' ), where 7rs, S c Zd, denotes, as usual, the coordinate projection. If S(m) = S(m/)nQ(m), then it is clear from the proof of Proposition 8.2 that there exists a positive constants c such that Q(cm) C S(m) C Q(m) for all m 2: 1. Since Po generates to topology and hence the Borel field 113 x of X, we know that

°

°

lim sup -1) u 2m(s lOgN(

m-->oo

V mEQ(m)

o:-rn(U)) = m-->oo lim _1) ( logN(Po(m)) 2m s

223

24. ENTROPY AND DIMENSION

and lim SU P m--->oo 'P

1 ) logh>.x( -( m2 S

V

mEQ(m)

1 ) logh>.x(Po(m)). (Lm(P)) = m--->oo lim - ( m2 S

Furthermore, .

1

hm -(2 m )S logN(Po(cm))

m-+oo

.

1

= m--->oo hm -(2m - ) logh>.x(Po(cm)) S

::; m-+oo lim 1) (S log h>.x ( 2m =

nEToo

= lim

V

(Lm(PO))

mES(m)

(2~)S logh>.x ( V

mEQ(m)'

(Lm(PO))

_1_pIQI(2m)T

m-+oo

(2m)s

12 ) logN(Po(cm)) ::; m---i'OO lim - ( m S =

.

1

hm -(2 m )S log h>.x (Po (cm))

m---i>(X)

for every sufficiently large m

~

1, which proves (24.2).

0

If the prime ideal p c 9'td satisfies that p(p) = 0, then the following examples show that we cannot expect a precise analogue of Proposition 24.1. EXAMPLES 24.2. (1) Let (PL the L-th cyclotomic polynomial for some L ~ 1, and let f(Ul,"" Ud) = ¢dUd) E 9't d. We use (5.9) to realize a = a'J{d/(f) as the shift-action of Zd on X = X'J{d/(f) C 'll'Zd. Since ¢L(Ud) divides u~ -1, every x E X satisfies that aLe(1) (x) = x, where e(j) E Zd is thej-th unit vector. We define Q(m), m ~ 1, as in (24.1) and obtain that

s~p .l~oo (2~)S logN( V

mEQ(m)

(Lm(U))

= sup lim -1) ( lOgN( u m--->oo 2m S

={oo o

V

mEQ(m)

a-m(U'))

(24.4)

ifs::;r(p)=d-1, ifs>r(p),

where the supremum is taken over all finite, open covers of U of X, and where, for each such cover U, U' = V~~~ a_ke(l) (U) is a finite, open cover of X which is invariant under a e (1). (2) Let 0 =I f E 9'td be an irreducible Laurent polynomial with M (I) > 1, and let a = a'J{d/(f) be the shift-action of Zd on the subgroup X = X'J{d/(f) c

224

VII. ZERO ENTROPY

described in (5.9). Then h(a) = logM (I) > then (24.4) changes to

']['Zd

sup lim -1) u m-+oo 2m(s log N (

V

o. If U

is an open cover of X,

a- m (U))

mEQ(m)

oo = { h(a)

o

(24.5)

ifsr(p)+1. 0

In Example 24.2 (1)-(2), s = r(p)+1 is the smallest integer such that the expressions in (24.4) and (24.5) are finite. The next proposition shows that this property characterizes r(p) for every prime ideal p C 9ld such that p(p) = 0 and a!Rd/p is ergodic. PROPOSITION 24.3. Let p C 9ld be a prime ideal such that p(p) = 0 and a = a!Rd/P is ergodic. If r = r(p) is defined as in Proposition 8.3, then s = r(p) + 1 is the smallest integer for which sup lim -1) u m-+oo 2m(s log N (

V

a- m (U))

mEQ(m)

= sup lim -1) ( logh>,x ( p m-+oo 2m s

V

(24.6)

a-m(P))
,x ( p m-+oo 2m s ={h(a) r(p) + 1.

This proves the proposition for prime ideals of the form p = jc. If P is not of this form, then Proposition 8.3 shows that r(p) ~ 1. If P = {O}, the assertion of the Proposition is obvious. If P i- {O}, let f, Q C Zd

225

24. ENTROPY AND DIMENSION

be the primitive subgroup and the finite set appearing in Proposition 8.3, put = r + Q, and assume as in the proof of Proposition 24.1 that r = {n = (nl' ... ' nd) E 7l.. d : nr+1 = ... = nd = O}, and that ni = ... = nr = 0 for every n = (nl, ... ,nd) E Q. If fr+I, ... ,fd are the Laurent polynomials appearing in the proof of Proposition 8.3, then our choice of r implies that each fJ is a function of the variables UI, ... , Uj. We write the Laurent polynomials fj in the form (5.2) and define a linear subspace S C jRZd by setting

r

S = {Z =

(zrn) C jRZd :

L nEZ

cfJ (n)zrn+n d

=0

for all m E 7l.. d and j = r

+ 1, ... , d}

(cf. (6.9)). For every m ;::: 1 we set Q(m)' = Q(m) n r and denote by S(m)' :J Q(m)' the largest subset of 7l.. d such that 1TS(m)1 (z) = 1TS(m)1 (z') for all z, z' E S with 1TQ(m)/(z) = 1TQ(m)/(z'), where 1Tp is, as usual, the coordinate projection which restricts each z: 7l.. d f----> jR in S to a set F C 7l.. d . If S(m) = S(m)' nQ(m), then it is clear that there exists a positive constant C such that Q(cm) C S(m) C Q(cm) for all m ;::: l. We realize X = X'Rd/p C T Zd and a = a'Rd/p as in (5.9) and denote by 'I9(s, t) = lis - til the metric (17.7) on T. The Propositions 13.1-13.2 and 13.7 guarantee that sup lim -1) u m->oo 2m(s log N (

= sup lim -1) ( log h>,x ( p m->oo 2m s =

V

a- rn (U))

rnEQ(m)

V

a-rn (P))

rnEQ(m)

1

lim lim -(2 ) logAx(B~(Q(m),c)), 6"--+0 m--+oo m S

where B~(c) = {x EX: Ilxoll < c} and B~(Q(m),c) = nrnEQ(m)(a_rn(B~(c)). From (5.9) and the choice of fr+I, ... , fd in the proof of Proposition 8.3 it is clear that we can find a constant C > 0 such that, for all sufficiently small c > 0 and all sufficiently large m,

AX(

n a_rn(B~(c)));:::

rnES(m)

CmAX(

n

Hence lim lim _1) ( 2m s

e->O m->oo

a_rn(B~(c)))

rnEQ(m)' = cmcIQ(m)/1 = C m c IQ1 (2m)r.

logAx(B~(Q(m),c))

226

VII. ZERO ENTROPY S; lim lim -1) ( log AX

",_Om_oo 2m s

(

n

mES(m/c)

CLm(B~(c)))

< lim lim _1_log(cm/ccIQI(2m/c+W) - ",-om-oo(2m)s

'

so that 1 lim lim -(- ) 10gAx(B~(Q(m),c)) < ",-Om-oo 2m s

if s

~

r(p)

+ 1. On the other hand, lim lim _1) ( ",-Om-oo 2m s

if s S; r(p), then

10gAx(B~(Q(m),c))

~ ",_Om_oo lim lim 1) (s log AX ( 2m =

00

n

mEQ(m)'

lim lim _1_logcIQI(2mnp) ",-om-oo(2m)s

CLm(B~(c)))

= 00

'

which completes the proof of (24.6). The remaining assertions of this proposition are immediate consequences of (24.6). 0 REMARK 24.4. The Propositions 24.1 and 24.3 introduce a notion of 'entropy dimension' for an ergodic Zd-action a by automorphisms of a compact, abelian group X: put

(24.7) where s(a) is the unique positive real number such that

s~p J~oo (2~)S 10gN( V

mEQ(m)

= s~p J~oo =

{oo

°

eLm (U))

(2~)S 10ghAx ( V

mEQ(m)

a-m(P))

(24.8)

if s < s(a), ifs>s(a).

If a = a!Rd/p for a prime ideal p C ~d, then (24.2) and (24.4)-(24.6) imply that dimh(a) = r(p) if p(p) > 0, and that dimh(a) E (r(p), r(p) + 1] if p(p) = 0. If a is an arbitrary, ergodic Zd-action by automorphisms of a compact group X satisfying the d.c.c., then dimh(a) is the maximum of the values dimh(a!Rd/P) as p runs through the set of prime ideals associated with the Noetherian module rot = X (cf. Lemma 5.1 and Proposition 5.4). Note that dimh(a) is also related to the dimensions of those subgroups r c Zd such that the restrictions a(r)

227

24. ENTROPY AND DIMENSION

of a to r have positive entropy (cf. Example 24.5 below). Other dynamical properties of the actions a(r) were discussed in Remark 8.4. We end this section with a few more examples. EXAMPLES 24.5. (1) In Example 24.2 (1), h(a) = 0, and dimh(a) = d-1. If k E {I, ... , d - I}, then every subgroup r c Zd of rank k with r n {ke(l) : k E Z} = {O} satisfies that h(a(r)) = 00. However, if r c Zd is a subgroup of rank k with r n {ke(l) : k E Z} ;2 {O}, then h(a(r») = 0. (2) In Example 24.2 (2),

°< h(a)
dimh(a) = d -1, but a is obviously non-mixing (cf. Theorem 6.5 (2)).

228

VII. ZERO ENTROPY

25. Shift-invariant subgroups of (7/.,/p7/.,)Z2 Let p > 1 be a rational prime, d > 1, and let lFpk be the field with pk elements for every k ?: 1. From Remark 6.19 (4) we know that every closed, shift-invariant subgroup X C lFf is isomorphic to X 9t d/ a ~ X 9t 9l~) l(f) is the quotient map. In particular, ker(TJ) e:! 9l~p) 1(91,92) is finite by Lemma 25.3, which implies that (f) 1(h , h) is finite. Suppose that the lemma has been proved for every ideal in 9l~p) with n generators, and assume that 0 = (h, ... ,fn+d C 9l~p). Let f' = gcd{h, .. ·, fn} and f = gcd{J', fn+1} = gcd(o). By the induction hypothesis, (f')/(h,···, f n) and (f) 1(f' , f n+ d are finite. Since

+ (fn + 1) )/((h, ... ,fn) + (fn + 1))1 1(f')/(h, .. · ,In) I < 00

l(f', fn+1)/ol = I((f') :::; and 1(f)I(f',fn+dl
O. Then h((J"n) The automorphism

(J"n

= w(n.l.,8(f)) ·logp.

(25.9)

is expansive on X if and only if

(25.10) in this case

(J"n

is algebraically conjugate to the shift on (lF~(n1- ,s(f»)z.

PROOF. Assume for the moment that n = e(l), where e(j) is the j-th unit vector in Z2, and set S( -m, n) = Z x {-m, ... , n} C Z2 and X( -m, n) = 1TS( -m,n) (X) c IF:( -m,n) for all m, n 2: o. If w(±e(2), 8(f)) = 0 and m+n+ 1 = w( e(2) , 8 (f)), then every x E X is completely determined by its projection onto the strip S(m, n). It follows that (J"e(l) is expansive, and that it is conjugate to the shift on (lF~(e(2),S(f»)z. If w(e(2), 8(f)) +w((O, -1), 8(f)) > 0 and m+n+ 1 = w(e(2), 8(f)), then the obvious projections from X( -m, n + 1) and X( -m - 1, n) onto X( -m, n) have kernels of size pw(e(2) ,S(f) and pW( _e(2) ,s(f», respectively. Hence h( (J"e(1») = w(e(2),8(f)) ·logp, and (J"e(1) is non-expansive whenever w(e(2), 8(f))

+ w( _e(2), 8(f)) > 0

(it helps to draw a picture and to study the Examples 25.2). In order to prove (25.9) for an arbitrary primitive element n E Z2 we use Lemma 11.3 to find a matrix A E GL(2, Z) with AT n = e(l). Assume that f is of the form (5.2), set fAT = L:mEZ 2 cf(m)u AT m, and consider the shift-action (J"' = a!)t~p) /(fA T) of Z2 on the closed, shift-invariant subgroup /(fAT ) Z2. . X I = X !)t(p) 2 C IFp . As we saw m Lemma 18.4, there eXIsts a continuous group isomorphism 'l/JAT: X f----> X' with 'l/JAT . (J"m = (J"~Tm . 'l/JAT for every m E Z2. The first part of this proof and (25.7) together imply that h((J"n) = h«(l») = w(e(2),8(fAT)) . logp = w(e(2),A T 8(f)) . logp = w(Ae(2),8(f)) ·logp = w(n.l.,8(f)) ·logp (since 0 = (n ..l,n) = (e(2),e(1») = (Ae(2),(A T )-le(1») = (Ae(2),n), we know that n.l. = ±Ae(2»). An analogous argument shows that (J"n is expansive if and only if (25.10) is satisfied. D Proposition 25.7 shows that, for every non-zero f E 9'\~p), the widths w(n,8(f)) are measurable conjugacy invariants of the Z2- action a = a!)td/(f) on X = X!)t~p) /(f), as n varies over the primitive elements of Z2.

25.

SHIFT-INVARIANT SUBGROUPS OF (Z/pZ)Z2

233

EXAMPLES 25.8. Let Fi C 7i}, i = 1, ... ,7, be the sets in Example 25.2, and let a Fi be the shift-action of Z2 on the subgroup XFi defined by (25.1). (1) If F = Fl = {(O,O), (1,0), (0, I)}, then w(e(1),F) = w(e(2),F) 1, w((l, 1), F) = 2, and w(( -1,1), F) = 1. Hence h(a:C2)) = h(a:Cl)) = h(ar-l,l») = log 2, and h(a(;.,l») = log 4. Furthermore,

w(( -1,0), F) = wecO, -1), F) = w((l, 1), F) = 1, F . b u t anF·lsexpanslve . £or every arenon-expanslVe, so th a t a eF(2)'a eF(1)' an d a(l,_l) primitive n rt. {±(1,0),±(0,1),±(1,-1)}.

(2) Let F = F2 = {(0,0),(1,0),(1,1)}. Since w((1,1),F2) = 1 we know from Proposition 25.7 and Example (1) that 10g2 = h(a0~l») i- h(a 1,l») = log 4, so that the Z2- actions a F1 and a F2 cannot be measurably conjugate.

0

(3) Since a F3 is not mixing, it cannot be measurably conjugate to a Fi for i = 1,2. Another way of distinguishing a F3 from a Fi , i = 1,2, is by observing that w((l,l),H) = w((-1,1),F3) = 2, which is obviously untrue for Fl and F2 . (4) By using the widths w( n, F) (or, equivalently, the directional entropies h( at:)) we can distinguish the Z2- actions a Fi , i = 4,5, from all the other a Fj in Example 25.2, but not from each other. However, a F4 is mixing, whereas a F5 is non-mixing. (5) Similarly we see that a Fi cannot be measurably conjugate to a Fj whenever i E {6,7} and j E {1,2,3,4,5}, but that h(at:6 ) = h(at:7 ) for all n E Z2. Since both a F6 and a F7 are mixing, we are unable to distinguish these actions with the tools developed so far. [J Although the widths wen, S(f)) provide a certain amount of information about the convex hull e(f) of the support of a Laurent polynomial I E I.Rr), they do not-in general-allow us to distinguish between Laurent polynomials whose supports have quite dissimilar convex hulls. EXAMPLES 25.9. (1) Let p = d = 2, and let It = 1 + uI + u~, 12 = Ul + U2 + + u~ + UrU2 + Ul u~. In the representation of Example 25.2 we have that

ur

...

= e(/l) n Z2 = :. ' F2 = e(h) n Z 2 = •• •• • , •• and w(n,S(It)) = w(n,Fd = w(n,F2) = w(n,S(h)) for every primitive eleFl

ment n E Z2.

f* =

(2) Let p = d

=

i- I E 1.R~2), and let It = P, 12 = If*, where (cf. (5.2)). Then wen, s(fd) = wen, S(h)) for every

2,0

I:mEZ2 cf(m)u- m

primitive element n E Z2. Example (1) arises by setting

I = 1 + Ul + U2.

[J

VII. ZERO ENTROPY

234

By using the notion of relative entropy for Zd-actions developed in [33], one can obtain further measurable conjugacy invariants for the Zd-actions o:'.R~p) l(f), f E ryt;t) , which will allow us to prove that the convex hull e(J) isup to translation-a measurable conjugacy invariant of o:'.R~p) l(f). In the spirit of this section we restrict ourselves to the case where d = 2 before presenting a more general picture in Section 26. Those proofs which are not significantly simplified by assuming that d = 2 will be postponed until Section 26 in order to avoid excessive duplication. If p > 1 is a rational prime, X C 1F~2 a closed, shift-invariant subgroup, and (1 the shift-action of Z2 on X, then the state partition (24.3) generates ~x under the (1. More generally, suppose that Q c ~ x is a finite partition. Put

v

Qtx(Q)-oo

=

n

(25.11) QtX(Q)k.

kEZ

IFr

If the space X is understood we suppress the subscript X in these definitions. For the following lemmas we assume that X S;; is a closed, shiftinvariant subgroup such that the shift-action (1 of Z2 is ergodic on X, and choose a (non-zero) polynomial f E ryt~) with (J) = n, where n c ryt;t) is the principal ideal (6.20) (cf. Proposition 25.5). The state partition (24.3) of X is denoted by Po. LEMMA

25.10. For every finite partition Q c Qt(Q)k

= (1_ke(2) (Qt(Q)o),

~x,

(1e(1) (Qt(Qh)

and for every k E Z,

= Qt(Q)k,

(25.12)

and Qt(Q)-l C Qt(Q)o. If Q is a generator, then lim Qt(Q)k

k->oo

(25.13)

= ~x,

and Qt(Q)-oo

whenever w(e(2), S(J))

= Qt(Q)o = Qt(Po)-oo = Qt(Po)o = ~x

(25.14)

= O.

PROOF. The relations (25.12)-(25.13) are obvious from the definition of Qt(Q)o. In order to prove (25.14) we assume that w(e(2),S(J)) = O. Fix E > 0 and use (25.13) to find an integer M :::: 0 such that (1Me(2) (Po) C Qt(Q)o (this c

means that there exists, for every P E (1(O,M) (Po), a set pI in the sigma-algebra Qt(Q)o with Ax(P6P' ) < E, where Ax is the normalized Haar measure on X). According to (25.12), (1(m"m2)(PO) C Qt(Q)o for all m2 :::: M and ml E Z. For every k

=

c

(kl, k 2 ) E Z2 we can choose an integer j :::: 1 and an element m E Z2

25. SHIFT-INVARIANT SUBGROUPS OF (Z/pZ)Z2

235

such that k E S(u rn f1?) = m+piS(f), and S(urnfPj ) '- {k} C {n = (nl,n2) E 'Z} : n2 :::; -M}. Since urn fP' E a, Xk = - EnES(f)'-.{k} cj(n)xrn+pJn for every x = (xn) E X, so that Xk is completely determined by the coordinates Xn with n E (m + piS(f» '- {k}. In particular we conclude that O"-k(PO) c 2((Q)o for BE

every k E 7i}, where s = IS(f)I. As c was arbitrary, we obtain that O"-k(PO) c 2(( Q)o for every k E Z2, so that 2(( Q)o = 23 x for every finite generator Q of 23x. It follows that 2((Qh = 0"_ke(2) (2((Q)o) = 23x for every k E Z, and that 2((Q)-oo = 23x. In particular, if Q = Po, then 2((Po)-oo = 23 x , which completes the proof of (25.14). 0 LEMMA

2((Po )-oo) (mod AX).

=

25.11. Ifw(e(2),S(f» > 0, then 2((Po )-oo f. 23x and H.xx(23xl 00. If f is in addition irreducible, then 2((Po)-oo = {0,X}

PROOF. For every k E Z we set

n(Xh =

{X

=

=0 for every n = (nl,n2)

(Xn)

EX:

xn

and write

n(x)_oo =

E 7i} with n2:::; k},

Un(X)k

(25.15)

(25.16)

kEZ

for the closure of UkEZ n(Xh. Then n(X)k and n(X)-oo are closed subgroups of X, n(Xh is invariant under O"e(l), n(X)_oo is shift-invariant, and

(25.17) Let H

= {n = (nl,n2)

E Z2: n2:::; O}, H'

= {n = (nl,n2)

E Z2: n2

< O}.

= {OlF:}, then the coordinates X(m,O), mE Z, are completely determined by the coordinates X(k,O) with 0:::; k < w(e(2), S(f»; conversely, if we choose ik E IFp arbitrarily for k = 0, ... , w( e(2), S(f» -1, then there exists a point y E X with Yn = 0 for n E H', and Y(k,O) = ik for k = 0, ... , w( e(2), S(f» - 1. Hence

If x E X is a point with 7rH'(x)

In(X)-dn(X)ol = I7rH(n(X)-l)1 = pw(e(2) ,s(f».

(25.18)

> 0 then In(X)-ool ~ In(X)-m/n(X)ol = pmw(e(2),S(f» for 1, so that n(X)-oo is infinite, 2((Po)-oo = 23 x / o (x)_oo f. 23x , and

If w(e(2),S(f»

every m

~

(25.19) The annihilators of the subgroups {OlF:/;2} c n(X)_oo c X c lF~2 satisfy p

that 9l~p) ::) (n(X)_oo)J. ::) (f) ::) {O}, and the shift-invariance of n(X)-oo

236

VII. ZERO ENTROPY

shows that b = (n(X)_oo).L is an ideal containing (f). If f is irreducible, then Lemma 25.3 implies that either b = (f) and n(X)-oo = X, or that 19t~) /bl = In(X)-ool < 00. In the first case the lemma is proved, and the second case is impossible by (25.19). D LEMMA

25.12. For every finite generator Q

c

~x,

::) ~(Po)-oo (mod AX).

~(Q)-oo

Furthermore, if Q

c

(25.20)

23 x is a finite partition satisfying (25.20), then

H>.x (~(Q)ol~(Q)-l) :S H>.x (~(Po)ol~(PO)-l)

= w(e(2), S(f)) logp :S h>.x (ae(2»).

(25.21)

PROOF. Lemma 25.11 implies that (25.20) is trivially satisfied if f is irreducible; a general proof of (25.20) will be postponed until Lemma 26.6. For every finite partition Q c 23x satisfying (25.20) we set Q(m) = V1kl:5m a_ke(1) (Q) and ~(m)(Q) = Vk:50 a- ke (2)(Q(m)). Then

H.xx(Q(m)I~(Q)_d

:S H>'x(Q(m)lae(2)(~(m)(Q))) :S h(ae(2»),

so that H>'x(~(Q)ol~(Q)-l) :S h>'x(ae(2») < 00. The decreasing martingale theorem, combined with (25.20), allows us to find, for every e: > 0, an N ~ 1 such that

H>.x (~(Q)ol~(Q)-d

~ ~

for every n

~

H>.x (~(Q)ol~(Q)-l V ~(PO)-n) H>.x (~(Q)ol~(Q)-l) - e:

N. It follows that, for every t

~

1 and n

~

N,

tH>.(~(Q)ol~(Q)-l) = H>'x(~(Q)ol~(Q)-t) ~ ~

Since

M

~

H>.x (~(Q)ol~(Q)-t V ~(PO)-n-t) tH>.(~(Q)ol~(Q)-l) - te:.

(25.22)

~(PO)m increases to 23 x as m --+ 00, there exists, for every e: > 0, an 1 with H>'x(~(Q)ol~(Q)-l V~(PO)m) < e: for every m ~ M. Hence

(25.23) for every t

~

1 and m

~

M. Finally,

tH>.x (~(Q)ol~(Q)-d = H>.x (~(Q)ol~(Q)-t) :S H>.x (~(Q)ol~(Q)-t V ~(PO)-n-t) + te: :S H>.x (~(Q)o V ~(Po)ml~(Q)-t V ~(PO)-n-t)

by (25.22)

+ te:

= H>'x(~(Q)ol~(Po)m V~(Q)-t)

+ H>.x (~(Po)ml~(Q)-t V ~(PO)-n-t) + te: :S H>.x (~(Po)ml~(Po)-n-t) + 2te: by

(25.23)

25. SHIFT-INVARIANT SUBGROUPS OF (71/p71)Z2

237

= (m + n + t)H>-.x (Ql(PO)OIQl(PO)-l) + 2tE:. If we divide by t and let t H>-.x (Ql(Po)oIQl(PO)-l). The identity

--+ 00

we obtain that H>-.x (Ql(Q)oIQl(Q)-l)

H>-.x (Ql(Po)oIQl(PO)-l) =

w(e(2),


-'x(Ql(Po)oIQl(Po)-d is constant and equal to log IO(X)-dO(X)ol = log I7fH(O(X)-dl = l ogpw(e(2) ,s(f)). D EXAMPLES 25.13. In the following examples we choose a non-zero polynomial f E 9l~p) and write (J" = o:9l~p) l(f) for the shift-action of Z2 on the closed, shift-invariant subgroup X = X9l~p) l(f) C IF!2. (1) Let p

=

2,

f = 1 + Ul + ui + U2 + Ul U2 F

E

9l~2), and put

= 8(f) = ::.

° °

(cf. Example 25.2). Then X = X9l~2) I(!) = X F C lFf (cf. (25.1)), w(e(2), F) = w( -e(1) , F) = w((l, 1), F) = 1, w( _e(2) , F) = 2, and w(n, F) = for all other primitive elements n E Z2. If a point x E X satisfies that Xn = for all n = (nl' n2) E Z2 with n2 < 0, then x(m,O) + X(m+1,O) = for

°

all m E Z, so that the choice of x(O,O) will determine x(m,O) for all m E Z. In particular, I7fH(O(X)-l)1 = IO(X)-dO(X)ol = 2 = 2w (e(2) ,F), and H>-.x (Ql(Po)oIQl(PO)-l) = J l>-.x (Ql(Po)oIQl(PO)-l)dAX = log IO(X)-dO(X)ol = log I7fH(O(X)-dl = log 2w (e(2) ,S(f)) = log 2. Since f is irreducible and w(e(2), 8(f)) > 0, Ql(Po)-oo = {0, X} by Lemma 25.1l. (2) If p

= 2, f = 1 + Ul + U2 + Ul U2 + UiU2

E 9l~2), and

F = 8(f) = : : • , then X = X F C lFf, w(e(2) , 8(f)) = 2, and I7fH(O(X)-dl = IO(X)-dO(X)ol = 4 = 2w (e(2) ,S(f)). The irreducibility of f again implies that Ql(Po)-oo =

{0,X}.

(3) If p is arbitrary and f = 1 + U2 + Ul U2, then w( e(2), 8(f)) = 1 > 0, and Lemma 25.11 implies that Ql(Po)-oo = {0, X}. The automorphism (J"e(l) is ergodic (by Proposition 25.5) and has finite entropy (by Proposition 25.7), and there exists a finite partition Q C ~x which is a generator for (J"e(l) (cf. [48]). Then VkEZ (J"-ke(l) (Q) = Ql(Q)o = Ql(Q)-oo = ~x, and H>-.x (Ql(Q)oIQl(Q)-l) = 0. Moreover, the partition Q' = Q V Po is a finite generator for (J" which refines Po, but Ql(Q')-oo = Ql(Q')o = ~x and H>-.x (Ql(Q')oIQl(Q')-d = 0. This shows that the inclusion in (25.20) may be strict, and that the conditional entropy H>-.x(Ql(Q)oIQl(Q)-d is not a monotonic function of Q. D

VII. ZERO ENTROPY

238

THEOREM 25.14. Let Pi, i = 1,2, be rational primes, and let Ii E 9l~Pi) be non-zero Laurent polynomials. If the ',f} -actions 0:(1) = o:!R~p) IUd and 0:(2) = o:!R;P) 1(12) are measurably conjugate, then P1 = P2 and e(h) = e(h) + m for some m E 71}. Furthermore, if X(i) = X!R;p) lUi), and if ¢: X(1) f---> X(2) is a measure preserving isomorphism with ¢ . 0:~1) = o:~) . ¢ for every n E 7l 2, then (25.24) where

P6i ) is the state partition (24.3) in X(i) .

PROOF. Lemma 25.12 implies that, for every non-zero f E 9l~p), the numbers w(e(2), 8(f)) and P are measurable conjugacy invariants of the 7l 2 -action 0: = o:!R;p) I(f). In particular, if h E 9lr d and 12 E 9lr 2 ), and if 0:(1) and 0:(2) are measurably conjugate, then P1 = P2 = p, say, and we may assume that X(i) C F~2, and that a(i) = o:(i) is the restriction to X(i) of the shift-action a of 71 2 on F~2. If a(1) is measurably conjugate to a(2), then the 7l 2 -action

a~21 : n

a~21n is conjugate to a~21 for every A E GL(2,71). According to the proof of Lemma 18.4, a~~l is (algebraically conjugate to) o:!R;p) IUiA ), and 1-+

by applying Lemma 25.6 and Proposition 25.7 to fiA instead of fi we see that w(A T e(2), 8(h))

= w(e(2), A8(h)) = w(e(2), 8(f~)) = w( e(2) ,S(ft-)) = w(A T e(2), 8(12))

for every A E GL(2,71). From Lemma 11.3 we conclude that w(n, 8(h))

= w(n, 8(12))

(25.25)

for every primitive element n E 7l 2 , which implies that e(h) must be a translate of e(h). The equation (25.24) is a direct consequence of Lemma 25.12. 0 THEOREM 25.15. Let P be a rational prime, and let h,h be non-zero Laurent polynomials in 9l~p) such that h is irreducible and 0:(1) = o:!R;p) I(h) is mixing on X(1) = X!R;p) I(ft). Let 0:(2) = o:!R;p) 1(12), X(2) = X!R;p) I(h), and assume that that there exists a measure preserving, surjective Borel map f---> X(2) such that ¢. 0:~1) = o:~) . ¢ AX(Wa.e., for every n E 71 2 (i.e. that 0:(2) is a measurable factor of 0:(1»). Then w(n, 8(fd) ~ w(n,8(h)) for every primitive element n E 7l 2. Furthermore there exists a finite set F C 71 2 such that e(h) + e(F) = e(h).

¢: X(1)

PROOF. Assume that X(1) and X(2) have been realized as closed, shiftinvariant subgroups of (cf. (6.19)-(6.20)), and that 0:(1) = a(1) and 0:(2) = 1) and 2) a(2) are the shift-actions of 71 2 on X(1) and X(2). We denote by

Ff

P6

P6

25. SHIFT-INVARIANT SUBGROUPS OF (Z/pZ)Z2

239

the state partitions (24.3) in X(1) and X(2), respectively. According to the Lemmas 25.10-25.11,

(p(1»)

Q{ x(1)

0

_ {{0,X(1)} -00 - lE X(l)

if w(e(2),S(/1)) > 0, if w( e(2), S(fd) = o.

(25.26)

> 0, then Q{X(2)(p~2»)_00 1:- lE X(2) by Lemma 25.11, so that {OX(2)} 1:- f2(X(2»)_00 c X(2) by (25.17). Hence y = ¢-1(f2(X(2»)_00) is a non-trivial, closed, shift-invariant subgroup of X(1). The annihilators of the subgroups {OIFZ2} eYe X(l) c lF~2 satisfy that

If w(e(2),S(fd) = 0, but w(e(2),S(h))

m:~p) ::J

P

y-L

::J

(/1)

::J {O}, and the shift-invariance of Y implies that b

=

Y -L is an ideal containing (/1). Since /1 is irreducible, we either have that b = (fd and Y = X(l), or that 19\~2) /bl = IYI < 00. In the first case ¢: x(l) I----t X(2) must be an isomorphism, and Theorem 25.14 shows that w(e(2), S(/1)) = w(e(2), S(h)) > O. In the second case f2(X(2»)_00 is finite, in violation of (25.19). These contradictions show that w(e(2), S(h)) = 0 whenever w(e(2), S(/1)) = o. If w(e(2), S(/1)) > 0 we put Q = ¢-1(p~2») and apply the Lemmas 25.1125.12 to obtain that {0,X(1)} = Q{X(l) (Po)-oo c Q{X(1) (Q)-oo and

o = w( e(2), S(/1)) logp = H>. x(1) (Q{X(1) (P~l))o IQ{X(l) (p~l) )-1) ;::: H>'x(1)

(Q{X(1) (Q)OIQ{X(l) (Q)-d =

H>'X(2)

(Q{X(2) (P~2»)01Q{x(2) (p~2»)_1)

= w(e(2), S(h)) logp, so that (25.27) For every A E GL(2, Z), the map ¢: X(l) I----t X(2) satisfies that ¢ . a A-ln = a~2ln . ¢ for every n E Z2. It follows that the Z2- action a'Y.;p) jut) is a factor of a'Y.~p) /ut), where we are using the same notation as in the proofs of Lemma 18.4 and Theorem 25.14. By replacing Ii with fl in (25.27) we obtain that w(A T e(2), S(fd) = w(e(2), S(ft)) ;::: w( e(2), S(f~)) = w(A T e(2), S(h)) for every A E GL(2, Z), and Lemma 11.3 yields that

w(n,S(/1)) ;::: w(n,S(h))

(25.28)

for every primitive element n E Z2. We set F1 = S(f) and F2 = S(f(2»). For every finite set E C Z2, the boundary of the convex hull e(E) is a closed polygon in ]R2 with edges {vn = wen, E)n-L : n E Z2 and wen, E) > O}, and we conclude that L Vn = 0, where the sum is taken over all primitive elements in Z2. From (25.28) it is clear that wen, Fd ;::: wen, F 2) for every primitive n E Z2. As one traverses the edges Vn of the polygon formed by the boundary of e(Fd, the vectors

240

VII. ZERO ENTROPY

Wn = (w(n, F 1) -wen, F2))n.1 traverse another polygon, which is the boundary of a convex set F' C ]R2 satisfying that F' + e(F2) = e(F1). The proof is completed by setting F = F' n 7i}. D

EXAMPLES 25.16. (1) Theorem 25.14 enables us to distinguish between all the Z2- actions a Fi in Example 25.2 with the exception of a Fe and a F7 : for all i,j with 1 :::; i < j :::; 7 and (i,j) i= (6,7), e(Fi) i= e(Fj), so that the Z2- actions a Fi and a Fj cannot be measurably conjugate. Theorem 25.14 does not enable us to prove that a F6 and a F7 are not measurably isomorphic. As we shall see in Example 28.10 (3), these actions can be distinguished by their higher order mixing behaviour.

(2) The sets Fi C Z2, i = 1,2, in Example 25.9 (1) satisfy that wen, F 1) = wen, F2) for every primitive element n E Z2, so that, in the notation of (25.1), h(a;:l) = h(a;:2) for all n E Z2. However, since 0 = w(e(2),Fd i= w(e(2),F2) = 1, the Z2- actions a F1 and a F2 cannot be measurably conjugate by Theorem 25.14. However, since a;:i is ergodic and hence Bernoulli for every non-zero n E Z2 ([72]), a;:l is measurably conjugate to a;:2 for every n E Z2. (3) Equation (25.24) can be used to prove that certain Z2- actions of the form a 9l ;p) lUi), i = 1,2, are non-isomorphic, even if e(h) = e(12). Let d = 2, and consider the polynomials h = 1 + U1 + + UrU2 + u~ + U1U~, 12 = 1 + U1 + + U2 + UrU2 + u~ + U1 u~ in !J\~2). In the representation of Example 25.2,

uI

uI

= X F1 ClFf, X(2) = X 9l ;2) 1(12) = X F2 c lFf, and a(1) = a9l;2) I(h) = a F1 , a(2) = a9l~2) 1(12) = a F2 are the restrictions of the shift-action X(1)

=

X 9l ;2) IUd

lFf

lFf.

to the closed, shift-invariant subgroups X(1) and X(2) in of Z2 on We write p~i) for the state partition (24.3) of X(i). Since 12 is irreducible and w(e(2), 8(12)) = w(e(2), F2) > 0, Lemma 25.11 shows that 2{X(p~2»)_oo = {0,X'}. The polynomial h is, however, reducible: h = glg2 with gl = 1 + U1 + + U2, g2 = 1 + U2 + U1 U2. Furthermore, since {O} £; (h) £; (g2) £; !J\~2), the annihilators of these groups satisfy that ;2 X ;2 Y = (g2).1 ;2 {Ox}. We write Po for the state partition of the closed, shift-invariant subgroup Y = X9l~2) /(g2) C and apply Lemma 25.11 to the irreducible polynomial g2 to obtain that 2{y(Po)-oo = {0, Y}. According to (25.16)-(25.17) this means that f2(Y)-oo = Y, and hence that {Ox} i= Y c f2(X)-oo. It follows that {0,X} £; 2{X(p~l»)_oo £; IB X (1) (cf. Lemma 25.11), and (25.24) implies that a(1) = a F1 and a(2) = a F2 cannot be measurably conjugate, although e(Fd = e(h) = e(12) = e(F2).

uI

lFf

lFf

(4) If p > 1, and if X c lF~2 is a closed, shift-invariant subgroup such that the shift-action a of Z2 on X is ergodic, then Lemma 25.12 reveals a remarkable

26. RELATIVE ENTROPIES AND RESIDUAL SIGMA-ALGEBRAS

241

(potential) asymmetry between the automorphisms (7e(2) and (7 _e(2) = (7~;). If Po is the state partition (24.3) of X, then Ql(Po)-oo measures the extent to which every point x E X is determined by its coordinates X(ml,-m2) with m2 very large. Similarly we see that the sigma-algebras

Ql(Po)k

=

V

{m=(ml,m2)EZ2 :m2::;k}

(7m(PO) , k E Z, Ql(Po):'oo =

n

Ql(PO)k'

kEZ

measure the dependence of x on its coordinates x(ml,m2) with m2 very large. If p = 2, F = Fl = {(O, 0), (1,0), (0, I)} in Example 25.2, and X = X F C IF'f, then

but

In Example 25.13 (1) we had p = 2, f = 1 + Ul + uI + U2 + UIU2 E 9t~2), F = S(f) and X = X F C IF'f, and by comparing the Examples 25.13 (1) and (2) one sees that

but

CONCLUDING REMARKS 25.17. (1) The material in this section is taken from [46] and [47]. The Examples 25.9 were communicated to me by Gruber.

(2) Let p > 1, and let X, X' be closed, shift-invariant subgroups of IF'f. We write (7 and (7' for the shift-actions of Z2 on X and X', respectively. If (7 and (7' are measurably conjugate, then Theorem 25.14 implies that {n E Z2 : (7n is expansive} = {n E Z2 : (7~ is expansive}. If (7' is a measurable factor of (7, then Theorem 25.15 shows that {n E Z2 : (7n is expansive} C {n E Z2 : (7~ is expansive}. 26. Relative entropies and residual sigma-algebras The proof of Theorem 25.14 depends on the fact that, for every closed, shift-invariant subgroup X C IF'~2, the conditional entropy H AX (Ql(Q)oIQl(Q)-d in Lemma 25.12 is finite for every finite, measurable partition Q of X. However, if d > 2, then the analogous conditional entropy will-in general-no longer be finite, and we have to use a more sophisticated approach in order to extract the necessary information from the sigma-algebras Ql( Qk The key tool for this is the notion of relative entropy in [33]. We begin with some general definitions.

242

VII. ZERO ENTROPY

Let T be a measure preserving Zd-action on a probability space (Y, 'I, p,), and let Q c 'I be a countable partition with HJ.L(Q) < 00. For every primitive element n E Zd and every t E lR we set ~T(Q,n)t

=

V V

T_m(Q),

{mEZd:(m,n):-;;t}

=

~T(Q,n)t

T_m(Q),

(26.1)

n

{mEZd:(m,n)

Q(m)

°and note that there exists an integer M 2: °such that, for all m 2: M, Ih~T(Q)O (T(r) I21T ( Q)-I)

-I~I HI" ( V T_n(Q(m)) I21 nEQ

(26.10) T

(Q)-I) I < c

244

VII. ZERO ENTROPY

and

Ih~T(Q)o (T(r) IQlT( Q)-2) - 1~IHJt( V T_n(Q(m) V Te(d)(Q(m»))

IQlT(Q)-2) 1
0, an N'

~

0 such that

for all n ~ N' and Q E Q. Furthermore, since conditional expectation is a contraction on Ll (Y, '!, /1),

26. RELATIVE ENTROPIES AND RESIDUAL SIGMA-ALGEBRAS

s

J

245

\E,.(Q\21T (P)-n) - E,.(Q\21T (P)-oo) \ dp,

for all n ~ 1, Q E Q, and for every sigma-algebra 6 c 'I. We write I,.(Q\6) for the conditional information function of the partition Q, given 6, and note that the family of functions {I,.(Q\6) : 6 C 'I} is uniformly integrable. Hence there exists an N ~ 1 such that \H,.(Q\21T (P)-n V 6) - H,.(Q\21T (P)-oo V 6)\ < c' for every sigma-algebra 6 C 'I and every n ~ N. In particular, (26.14)

for all m E Z and n ~ N, since 2lT(P)-oo C 2lT (Q)-m (mod p,). Since P is a generator for T, the increasing martingale theorem implies the existence of an integer m ~ 0 such that (26.15)

where p(m) is defined as in (26.9). We fix c > 0 and apply Lemma 26.2 to obtain that, for every t ~ 1, th~T(Q)O(T(r) \21T(Q)-d

=

If h~T(Q)o(T(r)\21T(Q)_l)

M

1 such that, for all k

~

and hence lim _l_H,. (Q)-.oo t\Q\

~

V···

... V T(t-l)e(d) (Q(k))) I21T(Q)-t

for all t > l.

If h~T(Q)O(T(r)\21T(Q)_d


K

(26.16)

> 0, then we can find an integer

Ih~T(Q)O (et(r) \21T (Q)-l) - (J}~oo \~\ H,. C'% et_n(Q(k)) I21T (Q)-l) I < c, \HJl (Q\21 T (Q)-l) - H(Q\et ed (Q(k)) V 2lT (Q)-2)\ < c. Then

h~T(Q)O(T(r)\21T(Q)_d S

=

lim _l_H,. (Q)-.oo t\Q\ ...

~h~T(Q)O(T(r)\21T(Q)_t)

(VQT_n(Q(ko)

V···

V;~_l)e(d)(Q(kt_Il)) I21T(Q)-t) +c

(26.17)

246

VII. ZERO ENTROPY

for all t ;::: 1 and k o, ... , k t - 1 ;::: M. Having chosen M ;::: 1 as in (26.16) or (26.17), we fix k ;::: M and apply (26.14), with c' = c(2tk + l)l-d, to find an integer n ;::: 0 with lim

(Q)--+oo t

(V (V p,

IQl IHp,

nE

< lim _1_H - (Q)--+oo

tlQI

Q

nEQ

T_n(Q(k) V··· V T(t-l)e(d) (Q(tk))) I2lT(Q)-t) T_ (Q(k) V···

(26.18)

n

... V T(t-l)e(d)(Q(tk))) I2lT (Q)-t V2lT (P)-n-t)

+c

for every t ;::: 1. By setting c' = c(2tk + l)l-d in (26.14) we see that there exists an m ;::: 0 such that, for the integer n in (26.18), and for every t ;::: 1, lim

(Q)--+oo

_1_Hp,(

tlQI

... V

VQ T_n(Q(k) V···

T(t:~)e(d) (Q(tk)))

I2lT (Q)-t V 2lT (P)-n-t) - c

: :; J~r tl~IHp,( V T-n(T_me(d) (p(m+k)) V··· ... V

T(t::~l)e(d) (p(m+(t+2m)k)))

I2lT(P)-n-t)

: :; Ji7r tl~1 Hp, ( VT-n(T_me(d) (p(m+k)) V··· ... V

:::;

T(t::~l)e(d) (p(m+(t+2m)k)))

(26.19)

I 2lT (P)-n-m-t)

~h:T(P)O (T(r) I2lT(P)-n-m-t) + c

= t

+ ~ + n h:T(P)o (T(r) I2lT (P)-l).

From (26.16)-(26.19) and the arbitrariness of K it is clear that h:T(P)o (T(r) I 2lT (P)-t) = 00 whenever h:T (Q)O(T(r)I2lT (Q)_t) = 00. If h:T(Q)O(T(r) I 2lT(Q)-l) < 00, we let t -+ 00 in (26.17)-(26.19); since c was arbitrary, we obtain that h:T(Q)O(T(r)I2lT(Q)_t) :::; h:T (P)O(T(r)I2lT (P)_l).

0

Lemma 25.12 corresponds to the special case of Lemma 26.3 where

r

=

{Oza}. The relative entropies h~n) (T(r)) in (26.4) are-in general-very difficult to compute. One of the reasons for this difficulty is that h:T(Q)o (T(r) , 2lT(Q)-l) is not monotonic in Q, as we saw in Example 25.13 (3). However, for certain Zd-actions T by automorphisms of compact, abelian groups X, they can be calculated explicitly, since there exist distinguished, finite generators P of X for which 2lT (P)-oo is minimal, and h~n) (T(r)) = h:T(P)o (T(r) , 2lT (P)-l). Before we can proceed with these calculations we need two definitions.

26. RELATIVE ENTROPIES AND RESIDUAL SIGMA-ALGEBRAS

247

26.4. (1) For every non-zero element f E ~r) and every primitive element n E 7l,d we define Laurent polynomials fn,., fn,. E ~r) as follows. If f = hI ..... hr is a decomposition of f into irreducible elements of ~r) such that IS(hj)1 > 1 for every j = 1, ... , k (i.e. none of the h j is a unit in ~r\ then we set DEFINITION

r,· =

IT

{j:j=I, ... ,k and

hj,

ln(S(hj »I=I}

IT

{j:j=I, ... ,k and

ln(S(hj »I>I}

,

The Laurent polynomials fn,. and fn,. are determined up to multiplication by a unit in ~r) and f = fn,. fn, •. For simplicity of notation we put f· = If a

c ~r)

fe(d),.,

f· = fe(d), •.

is a non-zero ideal we write

an,.

(fn,., f

=

E

a), an,. = (fn,., f E a)

for the ideals in ~r) generated by {tn,. : f E a} and {tn,. : f E a}, and set a. = ae(d),. and a. = ae(d), •. (2) For every non-zero element f E ~r) we define Laurent polynomials ·th 9j E :Hd-l n->(p) £or WI

f t; , fV E n->(p) nd-l by wrl·t·mg f·m the £orm f = "k+ uj=k- Udj j = k- , ... , k+ and 9k- . 9k+ i= 0, and by setting ft; If a

c ~r)

= 9k-,

fV

= 9k+.

is a non-zero ideal we write at; = (ft;, f E a), a V = (fv, f E a)

for the ideals in ~r~l generated by {tt; : f E a} and {tv : f E a}. EXAMPLES 26.5. (1) The polynomial f = 1 + Ul + u~ + U~U2 + u~ + Ul u~ in Example 25.16 (3) satisfies that f = 9192 with 91 = 1 + Ul + u~ + U2 and 92 = 1+U2+ UIU2, and f· = 91,f· = 92. Furthermore, ft; = l+Ul +u~,fv = 1 +Ul. (2) Let f = 1 +Ul +U2 +U3 E ~r). Then f· = f, = 1, ft; = 1+Ul +U2, and = l. (3) If a c ~r) is a non-zero ideal of the form a = (ft,···, fk) = ft~r) + ... + !k~~k) with non-zero generators ft, ... ,!k in ~r), then

r

r

an,.

= (fn,. 1

a·

, ... ,

jn,.) an,. k'

= (ft, ... ,ft),

a·

= (fn,. 1

, ... ,

fn,.) k '

= (fi,···,fn,

VII. ZERO ENTROPY

248

all

= (ft,···,ff), a"=(fi,···,fn· 0

26.6. Let d ;::: 2, p a rational prime, a

C IJir) an ideal, and let .. d ryt(p) I Zd a = a d 11 be the shijt-actzon of Z on X = X d 11 C Zip and Po the state partition of X (cf. (6.19)-(6.20) and (24.3)). Then 2ta (PO )-oo C 2ta (Q)-oo (mod AX) for every countable generator Q C 113 x for a with finite entropy. LEMMA ryt(p)

I

PROOF. As in (25.15)-(25.17) we set, for every m E Z,

f2(X)m

= {x = (xn) EX: Xn = 0 n

for all

= (n1, ... ,nd) E Zd with nd:::; m},

f2(X)_oo

=

U f2(X)m.

(26.20)

mEZ

Then f2(X)m is a closed, 3-invariant subgroup of X, f2(X)_oo is a-invariant, and

(26.21) Our first task is to determine the group f2(X)-oo. If a = 0, then a is the shift-action of Zd on X = F~d, f2(X)-oo = X, and 2ta (Po)-oo = {0, X} c 2ta (Q)-oo for every countable partition Q C Il3 x . Now assume that {O} =f=. a = (f) = flJir) for some non-zero elements f E IJir). We write ~± C IJir) for the subrings consisting of all Laurent polynomials involving only positive, resp. negative, powers of Ud. After multiplying f by a monomial we may take it that f E ~-, and that f has constant term 1. For all m ;::: 0, the ~- -module N m = (udm~- + flJir)) / flJir) is Noetherian, and its associated prime ideals are all principal and of the form hi~-' where f = h1 ..... hr is a decomposition of f into irreducible elements of ~-. Put 0 = Ud1~-, £.n = nk>o Ok . N m = nk>o u d k . N m , and note that £.m = £.m· By Corollary 2.5 in [5) there exists an element 'Ij; E ~- such that (l+u d 1'1j;) ·£'m = {O}. If £.m =f=. {O} this means that l+u;t1'1j; lies in at least one of the prime ideals associated with the Noetherian ~- -module N m ~ ~/ f~-, so that 1 + u d 1'1j; and f have a non-trivial common factor in ~-. We assume without loss in generality that each hi has constant term 1, and that hi E I+Ud1~- for i = 1, ... , s-l, but h j tic I+Ud1~- for j = s, ... , r. Then

o.

8-1

f& =

II hj,

j=l

l' =

r

II hj

(26.22)

j=8

(cf. Definition 26.4), and the argument in the preceding paragraph shows that f& . Lm = {O} for all m ;::: 0, since 1 + u d 1'1j; cannot be divisible by any hi with

26. RELATIVE ENTROPIES AND RESIDUAL SIGMA-ALGEBRAS

i E {S, ... ,r}. We put Hm = {n for every m ~ 0 and k ~ 0,

iT;]X) = N m , and

= (nl, ... ,nd)

(iTH",

(iTH",

249

E Z2: nd ~ m} and note that,

(O(Xh))-L = u~-mNm

c Nm ,

(O(X)_oo))-L ='cm C N m.

(26.23)

For every 'I/J E 9lr) we denote by a",: X I-----> X the homomorphism dual to multiplication by 'I/J E 9(- c 9lr) on X = 9lr) /(f). Since fA ·'cm = {O}, the last equation in (26.23) shows that iTH", (af" (x)) E iTH", (O(X)_oo) for every m ~ 1 and x E X, and we conclude that af" (X) C O(X)-oo. The annihilators of the a-invariant subgroups {Ox} C af" (X) eX C lFf satisfy that 9lr) :J a f" (X)-L :J (f) :J {O}. Furthermore, if'I/J E 9lr) lies in the annihilator of af" (X), then (af" (x), 'I/J) = (x, fA'I/J) = 1 for all x E X, so that r·'I/J E (f); as fA and J" have no common factors and f = fA J" this implies that af" (X)-L C (J"). Conversely, every 'I/J E (f~) annihilates af" (X), since (af" (x), J") = (x, fA J") = (x,f) = 1 for every x E X. We have proved that af" (X)-L = (f~), so that (f~)-L = af" (X) C O(X)-oo. Since fA E 1 +Udl 9(-, we see exactly as in the proof of Lemma 25.10 that O(X)_oo O(X)_oo ker(af") for every

--

n ker(af") = O(Y)-oo = {O}, where Y = ker(af") = 9lr) /(fA). If ;2 (f~)-L, and if x E O(X)_oo ,,(f~)-L, then Ox "# ah(x) E O(X)-oon = {Ox}, which is impossible (note that af" . af.(x) = af(x) = Ox x E X). This contradiction shows that

--

O(X)-oo = af" (X) = (f~)-L = 9lr) /(r),

(26.24)

where fA and J" are chosen as in (26.22). If a is non-principal we choose non-zero elements il, ... ,fk in 9lr) such that a = (il, ... , h), assume without loss in generality that fJ E 9(- and f(O, ... ,0) = 1 for j = 1, ... , k, and that ff and fl are chosen as in (26.22). Then (26.24) shows that k

__

k

X = n 9lr) /(fi) = n(fJ)-L C IF!d, j=l

O(X)-oo

k

__

j=l

= n 9lr) /(fn = (a~)-L

(26.25) C lFf,

i=l

where the notation is as in Definition 26.4. Having identified O(X)-oo for X = X 0, since X is infinite. Fix jo E S, and apply Proposition 8.2 to find, for p = qjo' a primitive subgroup r ~ in Zd with the properties described there. According to Proposition 8.2 (3), the r-action (a9't:t') /qjo )(r) = (a9't d/ p jo )(r) is algebraically conjugate to the

zr

shift-action of r on (lF~)r for some finite set Q, so that 0 < h( (a9't:t') / qjo )(r)) 10gplQI
'x{Bn) = for all n E F, but

>'x (

n

nEF

!

(Lkn(Bn))

= >.x( {x EX: X(O,O) = X(O,2k) = X(2k,O) = O}) = >'x({x EX: x(O,O) = X(O,2k) = O}) = t "I- k= IT >'x(Bn) nEF

for every k 2: 1. This shows that F = S(J) is non-mixing for a, although a is mixing by Theorem 6.5 (2).

(2) More generally, let p be a rational prime, d 2: 2, and let {O} "I- a c Vl.~) be an ideal. Exactly the same argument as in Example (1) shows that, for every non-zero element f E a, the support S(J) C Zd, as well as every finite set F' with S(J) c F' C Zd, is non-mixing for a = aJ{d is a prime ideal associated with 9)1, then there exists an element a E 9)1 such that I' = {f E >J{d : f .a = O}, and we set ~ = >J{d . a C 9)1. Then ~ ~ >J{d/p and Y = = X/~l., where ~l. = {x EX: (x,a) = 1 for all a E ~} is the annihilator of~. Since ~ is invariant under the Zd-action a: n f-+ an dual to 0:, ~l. is a closed, 0:invariant subgroup of X, and the Zd-action o:y induced by 0: on Y is a factor of 0: and hence r-mixing. Since the >J{d-module arising from o:y is equal to y = ~ ~ >J{d/p we conclude that 0: 9t d/P must be r-mixing. Conversely, if 0: is not r-mixing, then (27.4) shows that there exists a nonzero element (al, ... , ar) E 9)1r and a sequence (nCk) = (nik), ... , n~k)), k 2:: 1) in (Zdt such that nik) = 0 for every k 2:: 1, limk-->CXJ n;k) - n~k) = 00 for

iJ

1 :::; i < j :::; r, and Un;k) . al + ... + un~k) . a r = 0 for every k 2:: 1. There exists a Noetherian submodule 1)1 C 9)1 such that {aI, . .. ,a r } C 1)1, and (27.4) implies that the Zd-action 0:')1, which is a quotient of 0:, is not r-mixing. Since 1)1 is Noetherian, the set of prime ideals associated with 1)1 is finite and equal to {PI, ... , Pm}, say, and we choose a corresponding reduced primary decomposition WI' ... ' 217m of 1)1 (d. (6.5)). The map a f-+ (a+W I , ... , a+Wm ) from 1)1 into .it = EB~1 I)1/Wi is injective, and the dual homomorphism from X = ji to = X')1 is surjective. Hence 0:')1 is a factor of o:~, so that o:~ cannot be r-mixing. By applying (27.4) to the >J{d-module .it we see that there exists a j E {1, ... , m} such that o:')1/2lJj is not r-mixing. Put m= I)1/Wj , I' = Pj, and use Proposition 6.1 to find integers 1 :::; t :::; s and submodules m= 1)18 => ... => 1)10 = {O} such that, for every k = 1, ... ,s, I)1k/l)1k-1 ~ >J{d/qk for some prime ideal I' C qk C >J{d, qk = I' for k = 1, ... , t, and qk ;2 I' for i = t + 1, ... , s. We choose Laurent polynomials gk E qk "'I', k = t + 1, ... ,s, and set g = gt+1 ... g8. Since o:m is not r-mixing, (27.4) implies the existence of a non-zero element (aI, ... , ar ) E mr and a sequence (nCk) = (ni k), ... , n~k)), k 2:: 1) in (Zdt such that ni k) = 0 for every k 2:: 1, limk-->oo n;k) - n~k) = 00 for 1 :::; i < j :::; r, and Un;k) . al + ... +un~k) . a r = 0 for

m

27. MULTIPLE MIXING AND ADDITIVE RELATIONS IN FIELDS

265

every k 2: 1. Put bi = g·ai, and note that 0 =f. (b 1 , ... ,br ) E (I)1tY, since g·a =f. 0 for every non-zero element a E 52J. There exists a unique integer l E {I, ... , t} such that (bI, ... , br ) E (1)11Y " (1)11-1Y, and by setting b~ = bi + 1)11-1 E 1)1z!1)11-1 ~ 9'td/p we obtain that 0 =f. (bi, ... , b~) E (1)1z!1)11-1Y ~ (9'td /PY and Unik) . bi + ... + un~k) . b~ = 0 for every k 2: 1, so that a!Rd/P is not r-mixing by (27.4). Since the prime ideal P is associated with the submodule 1)1 C 9Jt, P is also associated with 9Jt, and (1) is proved. The proof of (2) is identical, except that we use (27.5) instead of (27.4). 0 Theorem 27.2 shows that a Zd-action a by automorphisms of a compact, abelian group X is mixing of order r 2: 2 if and only if the Zd-actions a!Rd/P are r-mixing for all prime ideals P c 9'td associated with the 9'td-module 9Jt = X defined by a. In order to apply this result we shall characterize those prime ideals P c 9'td for which a!Rd/p is r-mixing for every r 2: 2. For every prime ideal P C 9'td we define the characteristic p( p) of 9'td / P as in (6.2). THEOREM 27.3. Let d 2: 1, and let p C 9'td be a mixing prime ideal (cf. Definition 6.16 and Theorem 6.5 (2)).

(1) If p(p) > 0 then

a!Rd/P is r-mixing for every r 2: 2 if and only if p = (p(p)) = p(p)9'td ; (2) If p(p) = 0 then a!Rd/P is r-mixing for every r 2: 2.

We postpone the proof of Theorem 27.3 for the moment and look instead at some of its consequences. If a is a Zd-action by automorphisms of a compact, abelian group X with completely positive entropy, then it is mixing of all orders by Theorem 20.14. If the group X is zero-dimensional, the reverse implication is also true. COROLLARY 27.4. Let a be a Zd-action by automorphisms of a compact, abelian, zero-dimensional group X. The following conditions are equivalent.

(1) a has completely positive entropy; (2) a is r-mixing for every r 2: 2. PROOF. Since X is zero-dimensional, every prime ideal p associated with the 9'td-module 9Jt = X arising from a via Lemma 5.1 contains a non-zero constant, so that p(p) > O. According to Corollary 18.5 and Theorem 20.8, a has completely positive entropy if and only if p = pep )9'td for every prime ideal p associated with 9Jt, and the equivalence of (1) and (2) follows from Theorem 27.2 and Theorem 27.3 (1). 0 EXAMPLE 27.5. Let d = 2, and let J = (4,2 + 2Ul + 2U2) = 49't2 + (2 + 2Ul + 2U2)9't2 C 9't2. The 9't2-module 9Jt = 9't 2/J has two associated primes: PI = (2) = 29't2, and P2 = (2,1 + Ul + U2) = 29't2 + (1 + Ul + U2)9't2. Since 9't2/Pl ~ 9't~2), a!R 2/Pl has no non-mixing sets, and the collection of non-mixing sets of the Z2- ac tion a = aWl coincides with that of a!R 2/P2 ~ a!R~2) /(l+Ul +U2) (cf. Theorem 27.2). 0

266

VIII. MIXING

The next corollary shows that the higher order mixing behaviour of Zd_ actions by automorphisms of compact, connected, abelian groups is quite different from the zero-dimensional case, and requires no assumptions concerning entropy. COROLLARY 27.6. Let d ~ 1, and let a be a mixing Zd-action on a compact, connected, abelian group X. Then a is r-mixing for every r ~ 2. PROOF. The group X is connected if and only if the dual group X is torsionfree, i.e. if and only if na i- 0 whenever 0 i- a E X and 0 i- n E Z. We write 9Jt = X for the rytd-module defined by a via Lemma 5.1, note that the connectedness of X implies that p(p) = 0 for every prime ideal p C rytd associated with 9Jt, and apply the Theorems 27.2 and 27.3 (2). 0 COROLLARY 27.7. Let AI, ... , Ad be commuting automorphism of the ntorus Tn = ]Rn /zn with the property that the Zd -action a: (ml, ... , md) f---> a(ml, ... ,md) = Ai"l ... A;;'d is mixing. Then a is r-mixing for every r ~ 2. PROOF OF THEOREM 27.3 (1). Suppose that p = p(p) > 0, and that a Dtd / p is r-mixing for every r ~ 2. If Pi- (p), then the prime ideal q = {t/p : f E q} C

ryt~) is non-zero, where f/ p E ryt~) is obtained by reducing the coefficients of a Laurent polynomial f E rytd modulo p (cf. (6.2)). Since rytd/P 2:! ryt~) /q, we may identify the Zd-actions a Dtd / P and a Dt :!') /q (cf. Remark 6.19 (4)). We choose a non-zero element g E q and see as in Example 27.1 (1)-(2) that the support S(g) is a non-mixing set. In particular, a Dtd / P is not mixing of order IS(g)l. Conversely, if p = p(p) and p = (p), then a Dtd / P is (conjugate to) the (Bernoulli) shift-action of Zd on lF~d, and therefore mixing of all orders (cf. (6.19)). 0 The proof of Theorem 27.3 (2) depends on the following result concerning additive relations in fields of characteristic zero. PROPOSITION 27.8 ([81]). Let IF be a field of characteristic zero, a c lF x = IF''-..{O} a finitely generated, multiplicative subgroup, n ~ 1, and al, .. ·, an non-zero elements of IF. Then the equation

(27.6) has only finitely many distinct solutions (Xl' ... ' Xn) in an for which no proper subsum ail XiI + ... + aikxik vanishes. PROOF OF THEOREM 27.3 (2). Suppose that a Dtd / P is not s-mixing for some s > 2, and that s is the smallest integer with this property. According to (27.4) there exists a non-zero element (al, ... , as) E (rytd/P)S and a sequence (n(k) = (n~k), ... , n~k)), k ~ 1) in (Zd)s such that n~k) = 0 for every k ~ 1, limk->oo njk) _n~k) = 00 whenever 1 ::; i < j ::; s, and Un;k) ·al +- ..+un~k) ·a s = 0

27. MULTIPLE MIXING AND ADDITIVE RELATIONS IN FIELDS

267

n;l)

for every k 2: 1. For simplicity we assume that n;k) =f. whenever 1 ::; k < l and i E {2, ... ,s}. The minimality of s is easily seen to imply that ai =f. 0 for i = 1, ... ,s. We write If for the field of fractions of the integral domain 9\d/P, regard 9\d/P as a subring of If, denote by G the multiplicative subgroup of If X generated by {un+p : n E Zd}, and observe that G is a free abelian group on d generators, since a!)td!P is mixing (Lemma 6.6 (2)). It follows that the equation a2 al

- -X2 -

... -

as al

-Xs

=1

has infinitely many distinct solutions (X2, ... , xs) in Gs- l C (If X )s-l, and Corollary 27.8 implies that all but finitely many of these solutions have vanishing subsums. In particular there exists a subset {iI, ... ,im } ~ {2, ... ,s} such that

for infinitely many k 2: 1, so that a!)td!P is not mixing of order m < s. This contradiction to our choice of s implies that a!)td!P is mixing of every order. 0 CONCLUDING REMARKS 27.9. (1) Theorem 27.2 is taken from [47] and [98], and Theorem 27.3 from [98]. Proposition 27.8 can also be found (with a different proof) in [21]; the reference [81] was pointed out to me by Ward. In [98] Proposition 27.8 is derived in a somewhat disguised form from the main result in [92] which may yield further information about the rate of multiple mixing of Zd-actions by automorphisms of finite-dimensional tori or solenoids. In order to describe Schlickewei's result in [92] we assume that lK is an algebraic number field of degree Dover Q and denote by P(lK) the set of places and P 00 (lK) the set of infinite (or archimedean) places of lK (cf. Section 7). For every v E P(lK), I· Iv denotes the associated valuation, normalized so that lal v = modocJa) for every a E lK (cf. Section 7). Let S, Poo(lK) eSc P(lK) be a finite set with cardinality lSI. An element a E lK is called an S-unit if lal v = 1 for every v E P(lK) "- S. Then the following is true: For every n 2: 1, and for all non-zero elements all ... ,an in lK, the equation

has not more than

(27.7) solutions (Xl' ... ' xn) in S-units such that no proper subsum ai, Xi, + .. ·+aikxik vanishes. A glance at the proof of Theorem 27.3 reveals that Corollary 27.7 also

follows from the theorem of Schlickewei just quoted; however, Corollary 27.7 does not require the full strength of of the estimate (27.7), but only the finiteness assertion of Proposition 27.8. What further dynamical information about mixing Zd-actions by automorphisms of compact, connected, finite dimensional, abelian groups can be gained from (27.7)?

268

VIII. MIXING

(2) The non-existence of non-mixing sets for mixing Zd-actions by automorphisms of compact, connected, abelian groups follows from a result in [66] (cf. [94]), which can be regarded as a precursor of Proposition 27.8: Let lK be an algebraic number field of degree Dover Q, and let al, .. . ,an and Xl, ... ,X n be non-zero elements of lK. If there exist infinitely many integers o < kl < k2 < . .. such that (27.8)

for every i 2: 1, then there exist integers 1 :::; k

< 1 :::; nand m > 0 such that (27.9)

If p is a rational prime such that lai Iv = 1 for every i E {I, ... , n} and every valuation v of lK which lies above p, then Mahler's proof gives an effective bound on m in terms of p and D. By using a straightforward argument described in [93] one can extend the implication (27.8)=>(27.9) to an arbitrary field lK of characteristic O. If P C ~d is a mixing prime ideal with p(p) = 0, and if lK equal to the quotient field of ~d/P, then (27.5) and (27.8)-(27.9) together imply that every non-empty, finite set F C Zd is mixing.

28. Masser's theorem and non-mixing sets In Section 27 we saw that every mixing Zd-action by automorphisms of a compact, connected, abelian group X is mixing of all orders, so that every non-empty, finite set Fe Zd is mixing for a (cf. (27.1)). If X is not connected we denote by XO the connected component of the identity in X and write aX/xc for the Zd-action on the zero-dimensional group X/Xc induced by a. Let 9Jl = X be the ~d-module arising from a, and observe that ryt

=~=

{a E 9Jl: ma

=0

for some non-zero mE Z}.

Since every prime ideal p C ~d with p(p) > 0 associated with 9Jl is also associated with the submodule ryt c 9Jl we see from the Theorems 27.2-27.3 that a = a 9Jl and aX/xc = a'.n have the same mixing sets. This reduces the general problem of determining the mixing sets for a Zd-action a on a compact, abelian group X to the special case where X is zero-dimensional. Assume therefore from now on that a is a Zd-action by automorphisms of a compact, abelian, zero-dimensional group X, and let 9Jl = X be the ~d­ module arising from a via Lemma 5.1. In order to determine whether a nonempty, finite set F C Zd is mixing for a one has to go through the following steps. (1) Find all prime ideals associated with the ~d-module 9Jl; (2) For every prime ideal p C ~d associated with 9Jl, check whether F is mixing for al)td/P.

269

28. MASSER'S THEOREM AND NON-MIXING SETS

The first step is not really feasible unless 9J1 is Noetherian or, equivalently, unless a is expansive (otherwise 9J1 may have infinitely many distinct associated prime ideals). In order to understand what is involved in the second step we note that every prime ideal p C !Jld associated with 9J1 satisfies that p(p) > O. Assume therefore that p c !Jld is a prime ideal with p = p(p) > O. If P = (P), then we know from Theorem 27.3 (1) that a9'td/P is mixing of all orders, and that every non-empty, finite set F C Zd is mixing for a9'td/P. If p =I- (p), the prime ideal (28.1 )

(cf. (6.2)) is non-zero, !Jld/p ~ !Jl~) /q, and we identify the Zd-actions a9'td/p and 9't(p) /

.

d

a d q (cf. Remark 6.19). Accordmg to (27.5), a set F = {nl,"" n r } C Z is mixing if and only if, for every (II, ... ,fr) E (!Jl~) with fi rt. q for some

r

i E {I, ... ,r},

(28.2) for all sufficiently large k ;::: O. In order to gain insight into the meaning of (28.2) we denote by IF pk the field with pk elements and write ifp ::J IF p for the algebraic closure of the prime field IFp. We shall need the following theorem by Masser ([70], [47]). THEOREM 28.1. Let IF be a field of characteristic p =I- 0 with algebraic closure if ::J IF, r ;::: 1, and let S = {Xl, ... , Xr } C IF x. The following conditions are equivalent.

(1) There exists a non-zero element

6x~

(6, ... '~r) E IFr such that

+ ... + ~rx~ = 0

for infinitely many k ;::: 0; (2) There exist positive integers a, b, and an element (YI, ... ,Yr) E (if X such that Xi = Y'!: for i = 1, ... , r, and {y~ , ... , y~} is linearly dependent over iFp; (3) If IE = IF n iFp, then there exist positive integers a, b, and elements

r,

r,

r,

(WI, ... ,wr ) E (lEx (ZI, ... , zr) E (iFx such that Xi = WiZ'!: for i = 1, ... , r, and {zr, ... , z~} is linearly dependent over IE. In order to apply Theorem 28.1 to (28.2) we fix d ;::: 1 and denote by

(28.3) the ring of Laurent polynomials in the variables {u~t , ... ,u~t, k ;::: 1, p f k} with coefficients in iFp, where p f k indicates that p does not divide k. Every

VIII. MIXING

270

f E R(p,l) _ n>(p) :.nd - :Jtd ,

then

U~(p,k)

R(p) d

-

d,

k21

and there exists, for every non-zero such that

f

E

R 0 for some n E 'Z}, then p, = Ax.

X

= X'.R

Let d 2: 2, and let a be an expansive, mixing and almost minimal ;Zd_ action by automorphisms of a compact, connected, abelian group X. Under certain additional conditions it is shown in [40] that every a-invariant and mixing probability measure p, on X with h/L(a n ) > 0 for some n E ;Zd is equal to Ax; under even stronger assumptions on a the same conclusion can be obtained for every a-invariant and ergodic probability measure p, on X. All these results deal with special cases of the following problem motivated by the seminal paper [23].

30. COHOMOLOGICAL RIGIDITY

293

PROBLEM 29.10. Let a be an almost minimal, expansive and mixing 7L,d_ action by automorphisms of a compact, abelian group X, and let JL be a nonatomic, a-invariant and ergodic probability measure on X.

(1) If JL is ergodic under some an, n E 7L,d (and, in particular, if JL is mixing), is JL = AX?

(2) If X is connected and h!"(a n ) > 0 for some n E 7L,d, is JL = AX? Is the assumption that JL has positive entropy under some an necessary? The paper [23] raises a second and related problem concerning the nature of all closed, invariant subsets of an almost minimal 7L,d- action a by automorphisms of a compact, abelian group. In [23] it is shown that in Example 5.3 (4) every closed, a-invariant subset Y £; X is finite. Further developments concerning this problem can be found in [6], [7], [8], [9], [18] and [46]. In view of the somewhat diverse evidence emerging from these papers a general solution to this problem seems currently out of reach.

30. Cohomological rigidity Let A be a Polish (=complete, separable and metric) abelian group, d ::::: 1, and let T be a continuous 7L,d- action on a compact, metrizable space Y. A continuous map c: 7L,d x Y f----+ A is a (continuous 1-) co cycle for T if c(m, Tn(Y))

+ c(n, y)

= c(m + n, y)

(30.1 )

for every m, n E 7L,d and y E Y. A co cycle c: 7L,d x Y f----+ A for T is a homomorphism if c(n,·) is constant for every n E 7L,d, and a co boundary if there exists a Borel map b: Y f----+ A with c(n,·)

= b· Tn - b

JL-a.e.

(30.2)

for every n E 7L,d. Two co cycles Cl, C2: 7L,d x Y f----+ A are cohomologous if they differ by a coboundary, i.e. if there exists a Borel map b: Y f----+ A such that cl(n,') - c2(n,') = b· Tn - bJL -a.e.

(30.3)

for every n E 7L,d. Finally, a co cycle c: 7L,d x Y f----+ A is trivial if it is cohomologous to a homomorphism. The functions b in (30.2) and (30.3) are called the cobounding function of c and the transfer function of (Cl' C2), respectively. The set Z~(T, A) of all continuous cocycles c: 7L,d x Y f----+ A is a Polish group under point-wise addition and uniform convergence on compact subsets of 7L,d x Y, and the sets Bl (T, A) and B~ (T, A) of coboundaries, and of coboundaries with continuous cobounding function, are subgroups of Z~ (T, A). The quotient group H~(T, A) = Z~(T, A)/ B~(T, A) is called the continuous first cohomology group of T with values in A. In general, the subgroup B~ (T, A) c Z~ (T, A) is not closed, and the cohomology group H~ (T, A) has therefore no nice topological

294

IX. RIGIDITY

structure. The following well known proposition is an example of this general phenomenon. PROPOSITION 30.1. Let a be a mixing Zd-action by automorphisms of a compact, abelian group X. If A = IR or A = § ~ '][', then B~ (a, A) is not closed in Z~(a, A).

If one restricts the class of co cycles under consideration, the picture may change considerably. Consider, for example, the Z-action T: n ~ an defined by the powers of a single expansive automorphism a of X = ,][,n for some n 2 1. The continuous cocycles c: Z x X f----+ IR for T are in one-to-one correspondence with the continuous, real-valued functions f = c(l,·): X f----+ R If c E B~(T,IR), then Jc(l,·)df..l = 0 for every a-invariant probability measure f..l on X. Proposition 30.1 implies that the converse is not true: there exist co cycles c E Z~(T,IR) '-.. B~(T,IR) with Jc(l,.)df..l = 0 for every a-invariant probability measure on X. However, if c is Holder-continuous (i.e. if c(l,·) is Holder-continuous), then Livshitz' theorem ([65]) implies that c E B~ (T, IR) if and only if J c(l,·) df..l = 0 for every a-invariant probability measure f..l on X or, equivalently, if and only if c(n, x) = 0 for every n 2 1 and every x E X with an(x) = x. Furthermore, if c E B~(T,IR), then the cobounding function of c is again Holder-continuous. In particular the set of Holder-coboundaries is a closed subgroup of the group of Holder-cocycles in the usual topology on that space. The effect of Holder continuity is even more dramatic if one considers co cycles for Zd-actions by automorphisms of compact, abelian groups with d > 1 (cf. [38], [97]). As the groups carrying these actions are in general not finitedimensional tori, the classical notion of Holder continuity is not meaningful; there is, however, a natural Holder structure associated with a given continuous and expansive Zd-action T on a compact, metrizable space X, which coincides with the usual one for an expansive Zd-action on a finite-dimensional manifold (cf. [38]). Let d 2 I, and let T be a continuous Zd-action on a compact, metric space (X,8). We write I . II and (.,.) for the Euclidean norm and inner product on IRd :::) Zd, and put B(r) = {m E Zd : Ilmil ::; r} for every r 2 o. Suppose that A is a Polish abelian group, 'Y: A x A f----+ IR+ a distinguished invariant metric on A, and f: X f----+ A a continuous function. Put, for every c, r 20, w~''Y(f,T,c) =

The function

f

sup

{(x,x')EXxX:8(T".,(x),T".,(x')) 0 such that 00

w8,'Y(f,T,c) = Lw~''Y(f,T,E:) < r=l

00,

30.

and

COHOMOLOGICAL RIGIDITY

295

f is T-Holder ifthere exist constants c,w' > 0 and wE (0,1) with w~'Y(f, T, c)

< w'w r

for every r > o. These notions obviously depend on '"Y, but are independent of the metric 8 on X, and every T-Holder function has T-summable variation. If the group A is discrete, and if '"Y(a, a') = 1 for a -:J. a' and 0 otherwise, then a function f: X 1----+ A is Holder if and only if it is continuous. DEFINITION 30.2. Let a be a Zd-action by automorphisms of a compact, abelian group X, and let A be a Polish abelian group. A co cycle c: Zd x X 1----+ A is algebraic if c( n, .): X 1----+ A is a continuous homomorphism for every n E Zd, and affine if c = c' + c", where c' is algebraic and c" a homomorphism. FUrthermore, if A has a distinguished, invariant metric '"Y: A x A 1----+ jR+, and if a is expansive, then a co cycle c: Zd x X 1----+ A for a is Holder, or has summable variation, if the maps c( n, .): X 1----+ A have the respective property for every n E Zd. THEOREM 30.3 ([38]). Let d > 1, and let a be an expansive and mixing Zd-action by automorphisms of a compact, abelian group X. Every cocycle c: Zd x X 1----+ lR. with a-summable variation is continuously cohomologous to a homomorphism. If c is Holder, then the transfer function is again Holder. THEOREM 30.4 ([97]). Let d > 1, and let a be an expansive and mixing Zd-action by automorphisms of a compact, abelian group X. Every cocycle c: Zd x X 1----+ § with a-summable variation is continuously cohomologous to an affine cocycle. If c is Holder, then the transfer function is again Holder. COROLLARY 30.5 ([97]). Let d > 1, and let a be an expansive and mixing Zd-action by automorphisms of a compact, abelian group X. If A is a discrete abelian group and c: Zd x X 1----+ A a continuous cocycle, then there exists a discrete, abelian group A' :J A such that A' / A is finite and c is cohomologous to an affine cocycle c': Zd 1----+ A', with continuous transfer function. If a is an expansive and mixing Zd-action by automorphisms of a compact, abelian group X and A a Polish abelian group with a distinguished, invariant metric '"Y, then we write Z1(a,A) for the group of all Holder co cycles c: Zd X X 1----+ A, denote by B1(a,A) C Z1(a,A) the subgroup of all coboundaries with Holder cobounding function, and set H1(a, A) = Z1(a, A)/ B1(a, A) ~ Z1(a,A)/B~(a,A). As in [38] and [97] one checks easily that Z1(a,A) n Bl(a,A) = B1(a,A), so that H1(a,A) ~ Z1(a, A)/Bl(a, A). According to Theorem 30.4, every element of Z1(a,§) is cohomologous to an affine cocycle. If Z~ (a, §) and Hom(Zd, §) ~ §d are the subgroups of Z1 (a, §) consisting of all algebraic co cycles and all homomorphisms, respectively, then Hom(Zd, §)nB1(a, §) = {I}, and Z~(a, §).Hom(Zd, §).B1(a, §) = Z1(a,§). This makes it desirable to find an explicit formula for the group Z~(a,§).

296

IX. RIGIDITY

THEOREM 30.6. Let d > 1, a a mixing Zd-action by automorphisms of a compact, abelian group X, and let 9J1 = X be the !Jtd-module defined by a. An algebraic cocycle c: Zd 1---+ 9J1 is a coboundary if and only if c is an algebraic coboundary, i.e. if and only if there exists an element b E 9J1 such that

c(n) = (un - 1) . b for every n E Zd. Furthermore, if B~(a, §) algebraic co boundaries, then

and

H~(a, §) ~ 9J1* = (n IT

i=115,j5,d,#i

C Z~(a, §)

(Uj - 1) . 9J1) /

(IT

is the subgroup of

(Uj - 1) . 9J1).

i=1

EXAMPLES 30.7. (1) Let d = 2, p > 1 a rational prime, and let 9J1 = !Jtr) / j!Jt~p), where 0 -=f. f E !Jtr) (cf. (6.1)). The ideal m = (U1 -1)!Jtr) + (U21)!Jt~P) C !Jtr) is maximal, and consists of all elements 9 E !Jtr) with g(1, 1) = 0 (mod p). We claim that H~(a, §) ~ 9J1* -=f. {O} if and only if f Em. Indeed, if f E m, then there exist elements g1, g2 E !Jtr) such that f = (Ul - 1)g1 + (U2 - 1)g2' and we set a1 = g2 + 11 E 9J1, a2 = -g1 + 11 E 9J1, and observe that (U2 -1) . a1 = (U1 - 1) . a2 = a, say. From (11.5)-(11.6) it is clear that there exists an algebraic cocycle c: Zd 1---+ 9J1 with c(e(i)) = ai for i = 1,2. Since a!m is mixing, multiplication by Ui - 1 is injective on 9J1 for i = 1,2, so that a1 rJ. (U1 -1)· 9J1, a = a(c) = (U2 -1)· al E ((U1 -1)· 9J1n (U2 -1)· 9J1) "(Ul - 1)(U2 - 1) .9J1, and c is not a coboundary by Theorem 30.6. If f rJ. m we apply Hilbert's Nullstellensatz to find elements g1, g2, g3 E !Jt~p) with (U1 - 1)g1 + (U2 - 1)g2 + jg3 = 1. In particular, multiplication by Ui - 1 on 9J1/ (1 - Uj) .9J1 is invertible for i, j E {1, 2} and i -=f. j. If c: Zd 1---+ 9J1 is an algebraic co cycle , and if aj = c(e(j)) and a(c) = (Ui - 1) . aj for all i, j E {1, ... ,d} with i i j, then

a(c)

= (U1 -1)g1 . a(c) + (U2 - 1)g2 . a(c) = (U2 -1) . a1 = (U1 - 1) . a2,

so that (U1 - 1)g1 . a(c) E (U2 - 1) .9J1. As mentioned earlier, multiplication by U1 - 1 is invertible on 9J1/(U2 - 1) .9J1, and we conclude that g1 . a(c) E (U2 - 1) .9J1. Similarly we see that g2 . a(c) E (U1 - 1) .9J1 and hence that a( c) E (U1 -1) (U2 -1) .9J1. From Theorem 30.6 we conclude that every algebraic co cycle c: Zd 1---+ 9J1 is a coboundary. (2) Let d = 2, 11 = (U1 -p, U2 -q) = (Ul -p)!Jt2 + (U2 -q)!Jt 2, where p, q are positive integers, and let 9J1 = !Jt2 /11. Then 9J1 ~ Z[;q] = {k/(pq)l : k E Z,l ~ O}, and this isomorphism carries multiplication by U1 and U2 to multiplication by p and q, respectively. Then (U1 -1) . 9J1 n (U2 -1) .9J1 = lcm(p -1, q -1)9J1,

31.

ISOMORPHISM RIGIDITY

297

(Ul -I)(u2 - 1) . rot = (p - I)(q - I)rot, and H~(a, §) ~ rot* = {O} if and only if lcm(p - 1, q - 1) = (p - I)(q - 1). El

Many questions concerning the cohomology of Zd-actions by automorphisms of compact, abelian groups are unresolved. For example, the Z2-action a in Theorem 29.9 has no non-trivial, continuous cocycles with values in any compact, abelian group A (Example 30.7 (1)), but there exist non-trivial, continuous cocycles for a with values in certain non-abelian groups ([95]). Can one describe all such cocycles? Can they again be characterized in algebraic terms? The paper [37] investigates certain higher cohomology of Zd-actions by commuting toral automorphisms. In what form do these results extend to arbitrary expansive and mixing Zd-actions by automorphisms of compact, abelian groups?

31. Isomorphism rigidity This section is devoted to another striking manifestation of rigidity: certain Zd-actions by automorphisms of compact, abelian group have the property that every measurable self-conjugacy of this action is (up to modification on a null-set) a continuous group automorphism. Since the extent of this phenomenon is not yet understood we restrict ourselves to a particularly simple example. THEOREM 31.1. Let p = (2,1 + Ul + U2) c !Jh, a = a'.Jt 2 /p, X = X'.Jt 2 / p , and let ¢: X 1---4 X be a measure preserving Borel map with an . ¢ = ¢ . an for every n E Z2. Then ¢ is Ax-a.e. equal to a continuous, surjective group homomorphism 'IjJ: X 1---4 X. For the proof of Theorem 31.1 we require an elementary lemma. LEMMA 31.2. Let T be a measure-preserving and mixing tmnsformation of a probability space (Y, 'r, j.t). Then

J!"'~ for every f E Ll(y

X

J

f(y, my) dj.t(y) =

JJ

f d(j.t x j.t)

(31.1)

y, 'r QSl 'r, j.t x j.t).

PROOF. If f is of the form f(Yl, Y2) = IA(Yl)IB(Y2) for some A, BE 'r then (31.1) is an obvious consequence of mixing, and the proof for a general function f E Ll(y X y, 'r QSl 'r, j.t x j.t) follows from an approximation argument. 0 PROOF OF THEOREM 31.1. Since 1 + ur definition of X in (6.19) implies that

+ ur E p for every n

~ 0, the

(31.2)

298

IX. RIGIDITY

for every x E X and n ;::: o. For every finite subset F c Z2 we set A(F) = {x = (x n ) EX: xn = 0 for every n E F}, B(F) = ¢-l(A(F)), and conclude from (6.19) and (31.2) that

A(F) n ct(-2n,0) (A(F)) C ct(0,_2n)(A(F)), B(F) n ct(_2n,0) (B(F)) C ct(0,_2n)(B(F)) for every sufficiently large n ;::: O. Hence

1

IB(F) (x )lB(F) (ct(2n ,0) (x) )lB(F) (ct(0,2n)(x)) dAX (x) =

as n

-+ 00.

1

IB(F)(x)lB(F)(ct(2n,0)(X)) dAX(X)

-+

Ax(B(F))2

(31.3)

According to (31.2), the first term in (31.3) is equal to

and by letting n

nl~~

-+ 00

1

and applying Lemma 31.2 we conclude from (31.3) that

IB(F) (x )lB(F) (ct(2n ,0) (x)) IB(F) (ct(0,2 n)(x)) dAX (x)

=

=

11

1

IB(F)(x)lB(F)(y)lB(F)(x+y)d(AX x AX)(X,y)

IB(F) (x)

= Ax(B(F))2 =

1

IB(F)(x)

(1

IB(F)(Y)lB(F)(X+Y)dAx(y)) dAX(X)

(1

IB(F)(X+Y)dAx(y)) dAX(X).

(31.4)

Since all the functions involved are indicator functions we conclude that

Ax((B(F)

+ x),6,B(F))

=

{o

2AXB(F)

for AX-a.e. x E B(F) for AX-a.e. x EX" B(F).

The convolution IB(F) * IB(F) is thus AX-a.e. equal to IB, and a standard argument involving Fourier transform shows that B(F) differs by a null-set from a subgroup of X. As every subgroup of X with positive Haar measure is open and closed we can modify ¢ on a null-set, if necessary, and assume that ¢-l(A(F)) = B(F) is an open and closed subgroup of X for every finite set F C Z2 with Ax(A(F)) > o. We put Y = n{FCZ 2 :AX(A(F»>0} B(F) and observe that ¢ induces a continuous group isomorphism 'ljJ': X /Y f----? X and hence a continuous, surjective group homomorphism 'ljJ: X f----? X with 'ljJ(x) = ¢(x) for Ax-a.e. x E X. 0

31.

ISOMORPHISM RIGIDITY

299

COROLLARY 31.3. If 0; is the 7/,2-action on the compact, abelian group X defined in Theorem 31.1, then every measure preserving automorphism 4> of the measure space (X, ~ x, AX) with O;n . 4> = 4>. O;n Ax-a.e. is Ax-a.e. equal to O;n for some n E 7/,2. PROOF. From Theorem 31.1 we know that 4> is Ax-a.e equal to a continuous group automorphism 1/J of X which commutes with O;n for every n E 7/,2. We write the dual group X = ryt2/P as ryt~2) /q with q = (1 + U1 + U2)ryt~2) (cf. Remark 6.19 (4)) and note that the dual automorphism ¢: ryt~2) / q 1---+ ryt~2) / q satisfies that ¢(1 + q) = 9 + q for some 9 E ryt~2) and hence that ¢ is equal to multiplication by 9 on ryt~2) / q. By replacing 4>,1/J and 9 with O;m . 4>, O;m .1/J and umg for some mE7/,2, if necessary, we may assume without loss in generality that 9 is a polynomial in U1, U2, i.e. that 9 involves no negative powers in these variables. Furthermore, since 1/J is an automorphism of X, ¢ is an automorphism of ryt~2) / q; in particular, multiplication by 9 on ryt~2) / q is surjective, which is the same as saying that (31.5) Since 9 is determined only up to addition of an element in q we may also take it that 9 is a function of U1 and does not depend on U2 (just replace all the terms u~, n 2: 1, by (1 + ud n ). If the polynomial 9 = g(U1) has a zero ~ E 1F'2 ,,1F'2, then every element h E q + gryt~2) vanishes at the point (~, 1 +~) E (IF' 2 X)2, in violation of (31.5). Hence g(U1) = uf(l- ud for some k, l 2: 0, and 1/J = O;(k,l) ' as claimed. 0 REMARK 31.4. Even if the continuous, surjective group homomorphism 1/J: X 1---+ X is not bijective, its dual ¢: ryt~2) / q 1---+ ryt~2) / q is equal to multiplication by 9 for some 9 E ryt~2) " q, and 1/J = O;g (cf. (6.14)-if 9 were in q, then ¢ would not be injective and 1/J would not be surjective). By applying Hilbert's Nullstellensatz to the module ryt~2) /q we see that I ker(1/J)I = I ker(O;g)I = 21V(p)nv(g)l. EXAMPLES 31.5. (1) Let 9 = 1 +ur +u~. As is an automorphism of X.

(2) Let 9 = V (p)

and I ker(O;g)I

uI + U2.

g+p = U1U2 +p, O;g = 0;(1,1)

Then

n V (g) = {(W, 1 + w) : w E F2 and 1 + w + w2 = O},

= 4.

[]

CONCLUDING REMARKS 31.6. (1) Theorem 31.1 is originally due to Kitchens (unpublished) and to [99] (with somewhat different proofs). It is not difficult to extend Theorem 31.1 to 7/,2-actions of the form 0; = O;!R~p) /q, where

300

p

IX. RIGIDITY

> 0 is a rational prime and q c 9t~p) is an ergodic-and hence principal-

prime ideal generated by a Laurent polynomial f E 9t~) with extremal nonmixing support (cf. Proposition 25.5 and Definition 28.8 (1)). Further extensions are a little more complicated, but still possible, but the class of prime ideals p C 9t d for which the Zd-action a = a'.Rd/p satisfies (the analogue of) Theorem 31.1 is still a mystery. Does Theorem 31.1 hold for every expansive, mixing, almost minimal Zd-action a by automorphisms of a compact, abelian group X?

(2) One of the remarkable aspects of Corollary 31.3 is that each an, n i- 0 is an ergodic automorphism of X and hence Bernoulli (cf. Theorem 23.1 and 23.21), so that the set of automorphisms of (X, IB x, Ax) commuting with each individual an, n E Z2, is very large. We also note that there exist infinitely many finite-to-one, measure preserving maps ¢: X 1----+ X which commute with the Z2-action a, so that a does not have minimal self-joinings.

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Index Symbols

fq" 153

Ox, 1x,O, 1,1 1B, xvi (A), 174 (Q), 105 aT, 247 a A , 247 all, 247 at>, 247 2tT (Q, n)-oo, 242 Q(T(Q, n}t, 242 2tT (Q, n);, 242 2tx(Q)-00,234 2tx(Q)k,234 a(e) 64 a(~;, 2

lFp, 43 lFpk' 228 lFp,43

Fix(a),94 FixA(a),47 Frac(r),67 gcd(a),229 GL(n, 9

ie, 60 lKA,,62

adj(A),91

lKv, 61

ann(a),44

ker(a), 1 Ml(X),111

Aut(X),l {3A,80 'Ex, 2

Ml(X)T,l11 MI (f), 125

BN,201 C(J.Ll, J.L2)' 199 C(X),l IC, xvi e(f), 153, 270

N, xvi N*, xvi ooc, 62 Ov,62 w(n, F), 231 O(X)_oo, 235 O(X)k, 235 O(X, n)-oo, 243 O(X, n}t, 243 O(X, n);, 243

char(!J\t/p), 43 deg(f), 180 dimh(a), 226 EIJ-,108

71 c(f), 270 F(n),29 r,247 fA, 247 fn,T,247 fn,A,247 !",247 ft>,247 e(i),

Out(X), 1 p(p), 43 pOC,62 pr,62 P!,62 p+(a),175 p-(a),175 306

INDEX I.P(T) , 162 Po,222 PT,162 PT , 109, 162 Per(a), 94 IQ, xvi IQp, 61 ij,43 r(p), 71, 73 R e ,64 Rp,63 :R", 62 JR, xvi JR+, xvi !>\t, 35 9l(IK) 49 d

'

d

'

9l(p) 43 R(k),290

R