Symbolic Dynamics and Hyperbolic Groups (Lecture Notes in Mathematics, 1539) 1409916510, 9783540564997, 3540564993

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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen Subseries: Institut de Mathematiques, Universite de Strasbourg Adviser:P.A. Meyer

1539

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen Subseries: Institut de Mathematiques, Universite de Strasbourg Adviser:P.A. Meyer

1539

Michel Coornaert Athanase Papadopoulos

Symbolic Dynamcis and Hyperbolic Groups

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Authors Michel Coornaert Athanase Papadopoulos Institut de Recherche Mathematique Avancee Universite Louis Pasteur et CNRS 7, rue Rene Descartes F-67084 Strasbourg, France

Mathematics Subject Classification (1991): 53C23, 34C35, 54H20, 58F03, 20F30 ISBN 3-540-56499-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56499-3 Springer-Verlag New York Berlin Heidelberg

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, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany

Typesetting: Camera ready by author 46/3140-543210 - Printed on acid-free paper

A Martine,

a Marie

Pascale.

Table of contents

Introduction

1

Cbapter 1. - A quick review of Gromov hyperbolic spaces. §1. - Hyperbolic metric spaces. §2. - Hyperbolic groups. §3. - The boundary of a hyperbolic space. §4. - The visual metric on the boundary. §5. - Approximation by trees. §6. - Quasi-geodesics and quasi-isornetries. §7. - Classification of isometries. §8. - The polyhedron Pd(X). Bibliography for Chapter 1.

5 6 8

Chapter 2. - Symbolic dynamics. §1. - Bernoulli shifts. §2. - Expansive systems. §3. - Subshifts of finite type. §4. - Systems of finite type and finitely presented systems. §5. - Symbolic dynamics on N and on 7l . §6. - Sofic systems. Notes and comments on Chapter 2. Bibliography for Chapter 2.

9

12 13 13 16 17 18 19

20 22

26 29 31 36

40 41

Chapter 3. - The boundary of a hyperbolic space as a finitely presented dynamical system. 43 §1. - The cocycles X is a geodesic ray, then there exists a point in ax such that r(t n) converges to for every sequence (tn) of real numbers 0 such that t n -> 00. We shall write r( 00). In the same manner, every geodesic '"'I: lR -> X defines two distinct points '"'I ( -00) and '"'I ( 00) of ax .

=

We extend the definition of a geodesic polygon, given in §1, by allowing certain vertices of the polygon to belong to ax. An n-sided geodesic polygon IT = [Xl, ..., x n] is therefore given by n points z j , ... , X n E X U ax (the vertices of IT) and n geodesics

9

Chapter 1. -

Review of hyperbolic spaces

/1, ... , /n (the sides of II) with /i joining Xi and Xi+l for each i mod n, Let us recall

that the geodesic polygon II is said to be e-narroui if every side of II is contained in the e-neighborhood of the union of the other sides. The following proposition generalizes Corollary 1.4. Proposition 3.2. In a 8-hyperbolic geodesic space X, every n-sided geodesic 3) is 4( n + p - 2)8polygon with p vertices in oX and n - p vertices in X (n + p narrow. PROOF. Let II = [Xl,X2, ... ,X n ] be such a polygon. \Ve begin by "truncating" II at each vertex situated on the boundary in the following manner: If Xi E oX, we place a point Yi (resp. Zi) on the side which goes from Xi-l to Xi (resp. from Xi to xi+d, the indices being taken modulo n. We then replace, in the polygon II, fYi, Xi] U [Xi, z;] by a geodesic segment fYi, Zi]' \Ve thus obtain an (n + p )-sided polygon, with has all its vertices in X. Let X be a point situated on one of the sides / of II, and let us take x as a basepoint. \Ve can then place the points Yi and z; sufficiently close to the point Xi so that (Yi.Zi) > 4(n + p - 2)8, for every i such that Xi E oX. Corollary 1.4 and Proposition 1.5 show then that the point x is at a dist ance v, 4( n + p - 2)8 from the union of the sides of II other than /. • Corollary 3.3. - Let /1, /2 : lR -> X be geodesics such that /1 ( -(0) = /2( -(0) and /1 (00) = /2 (00). Then every point on /1 is at distance::::: 88 from /2. • Corollary 3.4. Let 1'1,1'2 : [0,00[-> X be geodesic rays such. that 1'1 (0) and 1'1(00) = 1'2(00). Then every point on 1'1 is at distance::::: 48 from 1'2.

Remark. In fact, we can see, using the fact that the triangle h(0),Tl(T),T2(T)] is 48-thin and taking T large enough, that the folowing inequality is satisfied, for every t 0:

Corollary 3.5. - Let 1'1,1'2: [0,00[-> X be geodesic rays such that 1'](00) = 1'2(00). Then every point on 1'1 is at distance :::::1 1'1(0) - 1'2(0) I +88 from 1'2. Furih ermore, there exists a real number T 0 such that Tl(t) is at distance c: 88 from 1'2, for eveTy

t

T.

•

Recall that a metric space is said to be propel' if all its bounded subsets are relatively compact. Recall also that a geodesic space is proper if and only if it is complete and locally compact (see for instance [GLP], Theorem 1.10). In the rest of this paragraph, we suppose that the space X -is geodesic, 8hyperbolic and proper. Under these hypotheses, Ascoli's theorem shows that given a point in X and a point in oX (resp. two distinct points in oX), there exists a geodesic ray (resp. a geodesic) joining these points. The topology on X U oX is defined in the following manner. 10

Chapter 1. -

Review of hyperbolic spaces

Let us take a basepoint Xo for X. Let R be the set of geodesics a : I ---+ X such that I is an interval of type [0, T] (where T is an arbitrary nonnegative real number) or the interval [0, oo] with a(O) = xo. We extend every a E R, defined on an interval [0, T], to the interval [0, oo] by taking a(t) = a(T) for every t ;::: T. The set R appears then as an equicontinuous set of maps from [0, oo] to X. We equip R with the topology of uniform convergence on compact sets. The compactness of R is a consequence of Ascoli's theorem. There is a natural surjective map 71" : R ---+ X u ax defined by 71"( a) = a( oo) for a E R. We equip X u ax with the quotient topology. It can be shown that this topology does not depend on the choice of the basepoint Xo and that the topology induced on X is compatible with the metric. X U ax is a compact set in which X is open and dense. Every isometry of X can be extended into a homeomorphism of X U ax. Examples. 1) If X is a real tree which is proper, the map which associates to r the point r( oc] is a homeomorphism from the set of geodesic rays starting at the basepoint Xo, equipped with the topology of uniform convergence on the compact sets of [0, oo], to ax. 2) Let X be a complete and simply connected Riemannian manifold whose sectional curvature is bounded above by a negative constant. Let S(xo) be the unit tangent sphere of X at xo. For every v E S( xo), let r v be the geodesic ray in X such that = v. Then the map which associates to each such v the point r v (= ) is a homeomorphism from S(xo) to ax.

Let us recall that the Gromov product ( . ) : X x X ---+ R can be naturally extended to the whole space (X U aX) x (X u ax). This extension is defined in the following way:

If a

=

and b =

are two sequences of points in X, we let

(a.b) = lim inf (ai.bi) when i

---+ oo.

If x and y are two points of X U ax, we let

(x.y) = inf(a.b), with a = (ai) converging to x and b = (bi) converging to y. (Note that the formula is correct if the points x and y belong to X). We then have (x.y) limits)

= += if and only if x = y

E ax, and we still have (by taking

(x.y) ;::: min ((x.z), (y.z)) - 8, for every x, y, z E X U ax. 11

Chapter 1. -

Review of hyperbolic spaces

§4 - The visual metric on the boundary In this section, X is a 8-hyperbolic space which is geodesic and proper and equipped with a basepoint xo. Let a be a real number which is > 1. For a sufficiently close to 1, Gromov defines (§§7.2.K, 7.2.L and 7.2.M of [Gro 3]), a visual metric I la on oX in the following manner. Let us define the a-length £a (0") of a continuous path 0" : [iI, i2] --+ X (t 1, t z real numbers) as the integral (defined by taking limits of "Riemann sums") of the function f : X --+ lR defined by f(x) = a-lxi, along the path 0". Let us then define the a-distance I x - y la between two arbitrary points x and y of X by

Ix

- y

la=

inf £a(O"),

the lower bound being taken on the set of continuous paths 0" joining the points x and y. One can show that there exists a constant ao = ao(8) (i.e. depending only on 8), with 1 < ao ::; oo , such that for every real number a which is strictly contained between 1 and ao, the following properties are satisfied:

(PI) The identity map of X extends to a homeomorphism from X U oX to the completion of X with respect to the metric I la. In particular, the metric I la induces a metric on oX. (P2) For all distinct points and TJ of oX and for every geodesic, : lR --+ X joining and TJ, we have, denoting by d the distance from the basepoint Xo to (the image of)

,:

A-la- d

::;1 -

TJ

la::;

Aa- d

where A = A(8, a) 2:: 1 is a constant which depends only on 8 and on a. For a E]1,ao(8)[, we call the metric I \a on oX the visual metric with parameter a and basepoint Xo. Property (P2) emphasizes the "visual" character of such a metric. Example. Suppose that X is a tree (i.e. 8 = 0). Let us note that the extension of the Gromov product, ( . ) : (X U x (X U --+ [0, oo] which we have recalled in §3 above admits in that case a very simple geometric interpretation: for and TJ in oX, TJ) is the length of the common path of the two geodesic rays starting at Xo and ending respectively at and TJ.

ex,

Proposition 4.1. -

for all points

ex,

If X is a tree, we can take ao = oo and we have:

and TJ in oX and for every a

> 1.

PROOF. Let x and y be points in X. Let 0"0 be the geodesic segment joining x and a(O") 2:: a(O"O) for every continuous path 0" joining x and

y. It is clear that we have y. Hence, we have

12

Chapter 1. -

Review of hyperbolic spaces

By integrating, we find that

• §5 - Approximation by trees In this section, X is a geodesic S-hyperbolic space, equipped with a basepoint xo. The following theorem (Approximation by trees) says that the union of a finite number of geodesics starting at Xo (that is, a "star" centered at xo) is, from a metric point of view, uniformly close to a tree (the "uniform" constant depending only on 15 and on the number of geodesics involved). This theorem is useful in general hyperbolic spaces for proving inequalities which involve distances between a finite number of points. Theorem 5.1. - (Approximation by trees). Let 1'1, ... "[n. be geodesics in X. Suppose that, for all i = 1, ..., n, 1'; is defined on an interval of the form [0, T] or [0, CXJ[ and satisfies 1';(0) = xo. Let Z be the union of (the images of) the 1';. Then there exists a real tree with a natural basepoint, (T, to), and a map f: (Z, xo) -> (T, to) satisfying the following two properties: (1)

For every z and z' in Z, we have

Iz where en = 2(1

(2)

z'

I -enS

f(z) - f(z')

z - z'

I,

+ log2n).

The restriction of f to (the image of) 1'; is distance preserving, for all i.

•

We shall apply sometimes Theorem 5.1 with a finite set of points F c X u ax instead of the finite set of geodesics. This means that, among these points, there is a basepoint of X which we shall specify, and that we apply the theorem to a set of geodesics 1'; joining the basepoint to each of these points in F.

§6 - Quasi-geodesics and quasi-isometries Let X and Y be two metric spaces. Given two real numbers A ;:::: 1 and k ;:::: 0, we say that a map f : X -> Y is a (A, k)-quasi-isometry if

for every

Xl

and

X2

in X. (Note that such a map 13

f

is not necessarily continuous.)

Chapter 1.

Review of hyperbolic spaces

In a metric space X, a (A, k)-quasi-geodesic is a (A, k)-quasi-isometry a: I X, where I is an interval oflR or of 7l. In the case where 1= N, we say that we have a (A, k)-quasi-geodesic sequence. When is not necessary to specify the values of A and k, we can simply say quasiisometry (resp. quasi-geodesic), instead of (A, k )-quasi-isometry (resp. (A, k )-quasigeodesic). The following theorem is fundamental in the theory of hyperbolic spaces. Theorem 6.1. (Stability of quasi-geodesics). Let X be a geodesic space which is 5-hyperbolic. Let I be a segment in lR or in'll, whose endpoints are a and b, and a : I X a (..\, k)-quasi-geodesic. Let I be a geodesic segment in X joining the endpoints a(a) and a(b) of a. Then the (images of) a and I are at a Hausdorff distance S J( from each other, where J( = J{( 5, A, k) is a constant which depends only on 0, A and k. •

(Let us recall that we say that two subsets of a metric space are at Hausdorff distance S from each other if each of these subsets is contained in the e-neighborhood of the other.) This Theorem gives immediately the following corollaries: Corollary 6.2. - Let a : [0, X {resp . a : N X) be a (A, k)-q1lasi-geodesic. Then aCt) has a limit a(oo) E ax when t 00. Furthermore, if r : X is a geodesic ray such that r(O) = a(O) and r( (0) = a( (0), then (the images oj) a and r are at a Hausdorff distance::; J{' from each other, where J{' = J{' (0, A, k) is a constant which depends only on 0, A and k. • Corollary 6.3. Let a : lR X {resp . a : 'll X) be a CA, k)-quasi-geodesic. Then a(t) admits a limit a( (0) E ax [resp . a( -(0) E ax) when t 00 (resp. t -(0). We have a(oo) i- a(-oo). Furthermore, if I : lR --+ X is a geodesic such that a( -(0) = 1(-00) and a( (0) = 1(00), then (the images oj) a and I are at Hausdorff distance ::; J{" from each other, where J{" = J{1f (0, A, k) is a constant which depends only on 0, A and k. • Corollary 6.4. - Let X and Y be two geodesic spaces. Suppose that Y is hyperbolic and let f : X Y be a quasi-isometry. Then

(1)

X is hyperbolic.

(2) For every sequence (x n ) of points in X which converges to a point in ax, the and which we sequence (J(x n ) ) converges to a point of ay which depends only on denote by af(O.

of : ax

ay

(3)

The map

is injective.

(4)

If

(5)

If X and Yare proper, then

f(X) is e-dens e in Y, then the map of: ax

ay

is surjective.

of is a topological imbedding from ax into ay.• 14

Chapter 1. -

Review of hyperbolic spaces

Application to hyperbolic groups.

Let T be a group and G c f a finite generating set of f. Recall that the Cayley graph K(f, G) of I' relatively to G is the simplicial graph defined in the following manner. The vertices of K(f, G) are the elements of I' and two distinct elements I and I' in T are related by an edge if and only if there exists an element g in G such that I = I' g. It follows from the fact that G generates I' that K (I', G) is connected. K(f, G) is equipped with its canonical metric which will be denoted by I IG. (The canonical metric of a connected simplicial graph K is the maximal metric on K for which every edge is isometric to the interval [0,1].) One should note that the restriction of I IG to I' is the word metric relative to G, defined in §2 above. It is easy to verify that K(f,G) is geodesic and proper. The space (K(f,G), IIG) is hyperbolic if and only if (f, I IG) is hyperbolic. If G' is another generating set of f, on'e can show that the identity map of I' extends to a quasi-isometry K(f, G) -. K(f, G'). Corollary 6.4 implies therefore that the hyperbolicity of (T', I IG) depends only on f (a result which we have already announced in §2). Suppose now that I' is hyperbolic'. Corollary 6.4 shows that there is a canonical homeomorphism oK(f,G) -. oK(f,G') which is induced by the identity map of f. Let us define of = oK(f, G). As a topological space, of is compact and metrizable. Of course, it is called the boundary of the hyperbolic group f. Let us also note that the action of T on K(f, G) by left translations is an isometric action which induces a (continuous) action of f on of. This action on of does not depend on the choice of the finite generating set G of f. The dynamical system (of, f) is therefore canonically associated to the hyperbolic group f. The study of this dynamical system is the main subject matter of these notes. Examples.

1) A hyperbolic group has empty boundary if and only if it is a finite group. 2) If I' = 7l, then of = {-oo, oo}. 3) If I' is a free group of rank n, n 2, then of is a Cantor set, that is, of is homeomorphic to {a, l}N. The next statement, which is very useful for proving the hyperbolicity of certain groups and for studying their boundary, is an easy consequence of Corollary 6.4. Theorem 6.5. - Let X be a geodesic space which is proper. Let I' be a group of isometries of X which acts properly discontinuously on this space, and suppose that this action is cocompact. Then X is hyperbolic as a metric space if and only if F is hyperbolic. Furthermore, if X is hyperbolic, we have a canonical homeomorphism of -; oX which is T -equivariant. •

(Let us recall that a group facts cocompactly on a topological space Xjf is compact. Recall also that we say that a group tinuously on a locally compact space X if for every compact of elements I in I' such that IK n K #- 0 is finite. Finally, 15

space X if the quotient I' acts properly disconsubset K eX, the set recall that if E and F

Chapter 1. -

Review of hyperbolic spaces

are sets equipped with an action of I', a map f: E - t F is said to be f-equivariant if fbx) = If(x), for every I E I' and for every x E E). Example. Consider the space En or, more generally, a complete simply connected Riemannian manifold of dimension n whose sectional curvature is bounded above by a negative constant. The previous theorem shows that any discrete and co compact subgroup of I som(X) is hyperbolic, and that its boundary is homeomorphic to

s-:',

§7 - Classification of isometries In this section, X is again a geodesic metric space, which is 8-hyperbolic and proper. The isometries of X are classified into three distinct types, according to the behaviour of a point in X under iteration by the given isometry. This classification generalizes the well-known classification of isometries of En.

Definition 7.1. Let I be an isometry of X and let x be an arbitrary point of X. We say that I is elliptic if the sequence bnx)nE71 is bounded. We say that I is hyperbolic if the sequence bnx )nE7l is a quasi-geodesic. Finally, we say that I is parabolic if I is neither elliptic nor hyperbolic. It can easily be shown that this definition does not depend on the choice of the point x EX. If I is a hyperbolic isometry and x a point in X, the sequence In X admits a limit 1+ E ax (resp. 1- E ax) when n tends to oo (resp. -"-Lipschitz if we have I f(x) - f(y) >"1 x - y I for every x and y in Xl. We have the following

Proposition 7.2. -

Let I be a hyperbolic isometry of X, with 1+ (resp. 1-) its attracting (resp, repelling) fixed point. E ax. Then

(1) For every point E ax - {r-} [resp, ax - (r+}), the sequence In(O tends to 1+ [resp . 1-) when n tends to oo (resp. -" > 1, there exists an integer no such that In (resp. I- n) sends ax - U- {resp, ax - U+) into U+ (resp, U-) and is A-Lipschitz on ax - U- (resp. ax - U+ )for all n ? no. •

16

Chapter 1. -

ax.

Review of hyperbolic spaces

Let I be a parabolic isometry of X. One can show that I fixes a unique point on Furthermore, we have the following

Proposition 7.3. Then

Let I be a parabolic isometry of X, with a E

ax

For every point E ax, the sequence ,n(O tends to a when 1n more, the convergence is uniform on compact sets of ax - {a}.

(1) (2)

sends

as fixed point.

1-+ 00.

Further-

Given a neighborhood U of a in ax, there exists an integer no such that ax - U into U for every n such that 1n no.

,n •

(Note that, in the preceding two propositions, (1) is implied by (2).) Remarks. 1) A real tree does not have parabolic isometries. 2) Let r be a hyperbolic group and X its Cayley graph for a finite generating set Ger. Let us recall that r acts isometrically on X. If I is a torsion element of r, it is clear that I defines an elliptic isometry of X. If I is a non-torsion element of r, it can be shown that it defines a hyperbolic isometry of X. (In particular, the type of the isometry of X which is defined by an element of the group r does not depend on the choice of the generating set G.)

§8 - The polyhedron Pd(X) Let X be a metric space and let d be a positive real number. We construct a simplicial complex, denoted by Pd(X), in the following manner. The vertices of Pd(X) are the points of X. The simplices of Pd(X) have as a set of vertices the finite subsets of X whose diameter is ::; d. Examples. 1) If X is bounded, then Pd(X) is the standard simplex inlRx , provided d diam(X). 2) Let r be a group, and G a generating set of r. We equip r with the word metric relative to G. Then PI (r) is the Cayley graph K(r, G), as it has been defined in §6.

The complex Pd(X) is a sort of "regularizing space" for the metric space X. An important phenomenon, when X is hyperbolic, is the stable contractibility of Pd(X). More precisely, we have the following theorem (due to Rips): Theorem 8.1. Let X be a geodesic 8­hyperbolic space, and let d be a positive real number. Then the simplicial complex Pd(X) is contractible for all d 48. •

17

Chapter 1. -

Review of hyperbolic spaces

Bibliography for Chapter 1

[Bow] B. Bowditch, "Notes on Gromov's hyperbolicity criterion for path-metric spaces", in Group Theory from a geometrical viewpoint, ICTP, 'World Scientific, 1991, pp. 64-167. [CDP] M. Coornaert, T. Delzant and A. Papadopoulos, "Ceometrie et theorie des groupes: Les groupes hyperboliques de Gromov", Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, 1990. [E] D. B. A. Epstein (with J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, and W. P. Thurston), "Word processing and groups", Jones and Barnett Publishers, 1992. [GH] E. Ghys, P. de la Harpe (ed), "Sur les groupes hyperboliques dapres Mikhael Gromov", Progress in Mathematics, vol. 83, Birkhauser, 1990. [Ghy] E. Ghys, "Les groupes hyperboliques", Seminaire N. Bourbaki, expose No. 772, mars 1990. Asterisque 189-190, SMF, 1990. [GLP] M. Gromov, "Structures metriques pour les varietes riemanniennes", notes written by J. Lafontaine and P. Pansu, Fernand Nathan, Paris, 1981. [Gro 1] - - , "Hyperbolic manifolds, groups and actions", in Riemann surfaces and related topics, Ann. of Math. studies 97, Princeton University Press, 1980, pp. 213. [Gro 2] - - , "Infinite groups as geometric objects", Proc. ICM Warszawa, 1983, pp. 385-392. [Gro 3] - - , "Hyperbolic groups", in Essays in group theory, MSRI publ. 8, Springer Verlag, 1987, pp. 75-263. [Sho] H. Short (ed.), "Notes on word hyperbolic groups", in Group Theory from a geometrical viewpoint, ICTP, World Scientific, 1991, pp. 3-63.

18

Chapter 2 Symbolic dynamics

Consider a finite set 5, which will be called the set of symbols, and let r be a countable semigroup. (Recall that a semigroup is a set equipped with an internal law which is associative and with an identity element.) Symbolic dynamics studies the action of r on the set of maps a : r 5. The action of r on is given by I'ab') = ab'l') (a E and 1',1" E I"). We equip r and 5 with the discrete topology and with the product topology. The dynamical system r) is called the Bernoulli shift associated to rand 5.

Using methods which are essentially of combinatorial nature, symbolic dynamics is useful for understanding the behaviour of certain dynamical systems (n, T}, where n is a space on which I' acts. The general idea consists in "coding" in an equivariant way the elements of n by elements of some Bernoulli shift (with an appropriate set of symbols 5). This symbolic approach has been adopted (with T = 7l) in the study of Anosov diffeomorphisms (Sinai) and, more generally, in the study of Axiom A diffeomorphisms (Smale, Bowen, Manning, ... ). One of the results which have been obtained is the rationality of the (­function for Axiom A diffeomorphisms, which has been proved by Manning ([Man]). A paper of Hadamard ([Had]), published in 1898, inaugurates a long series of works concerning the application of symbolic dynamics to the study of geodesic flows in negative curvature. This approach has been adopted by several mathematicians (Koebe, Nielsen, Morse, Hedlund, Gromov, Series, ... ). In this chapter, we present, following Gromov ([Gro 1] and [Gro 3], see also Fried [Fri]), a few basic notions of symbolic dynamics. In particular, we introduce the notions of dynamical systems of finite type and of finitely presented dynamical systems. (The reader will notice a certain analogy between the terminology used for the theory of symbolic dynamics and the theory of groups. This analogy, which is stressed by Gromov, will be apparent as the theory will go on (c f. [Fri]).)

19

Chapter 2. -

Symbolic dynamics

§1 - Bernoulli shifts Let r be a countable semigroup and S a finite set. The elements of S are called the symbols. We denote by = S) the set Sf' of maps (J : I' S. We equip r and S with the discrete topology and with the product topology. Thus, a sequence ((J n) of elements of converges to an element (J E if and only if for every , E r, there exists an integer no = noel) such that (JnC!) = (JC!) for every n 2:: no·

Proposition 1.1. The space is metrizable, compact and totally disconnected. Furthermore, if r is infinite and if card( S) 2:: 2, then is a perfect set. (Let us recall that a topological space is said to be perfect if it has no isolated points, and totally disconnected if each of its connected components is reduced to a point.) PROOF. If I Is is an arbitrary metric on Sand (O,,)-YEf' a family of positive real numbers such that L: < 00, it is clear that the metric I I on defined by

0,

(1.1.1) is compatible with the topology The compactness of is a consequence of Tychonoff's theorem (by a diagonal process, we extract from each sequence in a convergent subsequence). To show that is totally disconnected, consider the map p, : S which associates to (J E the element (JC!). This map is continuous and therefore p,( C) is reduced to a point for every connected non-empty set C C I;. Suppose now that r is infinite and let us show that in this case is perfect if card(S) 2:: 2. Given (J E we construct a sequence ((In) of distinct elements of which converges to (J, in the following manner. Let C!n) be a sequence of distinct elements of r. Define (In : r S by the formula (JnC!) = (JC!) if, i=- ,n and (In(,n) = an arbitrary element of S which is distinct from (JC!). It is clear that this sequence (In does the job. •

Corollary 1.2. -

If

r

is infinite and card( S) ::::: 2, then

is a Cantor set.

(Recall that a Cantor set is a topological space which is homeomorphic to the triadic Cantor set, that is, homeomorphic to the space {O,l}N equipped with the product topology. ) From a classical theorem of general topology (see, for instance, [MoiJ, Chapter 12, Theorem 8) any compact, metrizable, perfect and totally disconnected space is a Cantor set. •

PROOF.

Vve define now a continuous left action of r on E. For, E rand (J E defined by the following rule:

20

,(J is

Chapter 2. -

Symbolic dynamics

The space equipped with this action of r, is called the Bernoulli shift on r (and with associated set of symbols 5) . Exercise. Let r be an infinite countable semigroup, 5 a finite set and Assume that the following condition holds:

=

5).

(*) For any finite subset Fer, there exists an element, E r such that F, n F = 0. (N ote that (*) is satisfied if r is a group or, more generally, if r can be imbedded in some group.) 1) Show that the action of r on is topologically transitive (i.e. admits a dense orbit). 2) Show that, for any integer n ;::: 1, the diagonal action of r on is topologically transitive. (Hint: use = 5 n ).)

Definition 1.3. -

A subshift of

is closed and r-invariant subset of

Exercise. Show that the intersection of an arbitrary family of subshifts of a subshift Show also that the union of a finite family of subshifts of subshift of

is again is also a

Let now 51 and 52 be two finite sets and let us consider the two Bernoulli shifts = 5d and = 52)' Given a finite subset F e r and a map u : 5f -+ 52, we construct a continuous and I' 0 such that if the elements 171,172 E Phil are at distance less than E (as defined by formula (1.1.1)), then T(17d and T(172) take the same value at the identity element IdE r. Hence, there is a finite set F e r such that T( 17d = T( (72) whenever 171 and 172 have the same restriction on F.

Thus, we can construct a map u : 5f -+ 52 such that u( 17 IF) = T( (7)(1d) for every 17 E !P 1. We have, for every a E !PI and for every, E r,

u oo ( (7)h) = u( ,17IF) = Th(7) (1 d)

21

Chapter 2. -

Symbolic dynamics

= IT(a)(Id) = T(a)(r),

and therefore U oo

Corollary 1.5. countable.

= T.

The set of continuous and f-equivariant maps T : with

°

E

Is - s' Is.

= O'Idin!soFsf

3) If the dynamical systems (0 1 , f d and (0 2,f 2 ) are expansive, then the system (0 1 x O 2 , f 1 x f 2 ) is also expansive. 4) Suppose that there exists a metric I I on 0, compatible with the topology, such that I' acts isometrically on 0. Then (0, f) is not expansive, unless 0 is finite. This implies that any expansive system has only finitely many fixed points. The following proposition can be useful in showing that some systems are expanSIVe.

Proposition 2.4. Let I' act on a compact metric space (0,1 I). Suppose that (Ui)iEI is an open covering of 0 such that for every i E I there exists some ,i E I' and Ai > 1 satisfying: l,i X -,iY k Ai I x - Y I \Ix, y E Ui. Then the system (0, I") is expansive. PROOF. By compactness, we can assume I to be finite. We can Lebesgue number of the covering) such that, for every point x of centered at x and of radius E is contained in some Ui. Such an E constant for (0, f). To see this, note that if x and yare two distinct are less than E apart, then x and yare in some U, and we have

l,i X

-

,iY

k

Ai I x - y

°

find an E > (a 0, the open ball is an expansivity points of 0 which

I.

If the distance l,iX - ,iY I is again < E, then we can a.pply the same reasoning to the points ,iX and 'iY. Suppose that we can iterate this construction for n steps. Then, we would have Ai E f such that

I,x -,Y I;:::

An I X

-

y

I,

where A = min Ai. As A > 1, there is an integer n for which this blowing up process must stop. Therefore, we ha.ve I,x -,y I;::: E. •

23

Chapter 2. -

Symbolic dynamics

Corollary 2.5. - Let M be a closed Riemannian manifold (with or without boundary) and let r be a semigroup of c l maps of M. Assume that for every x E M there • is a, E r such that Ild,(x)11 > 1. Then the system (D,r) is expansive. The next statement shows that any expansive system admits a "coding" by a subshift.

Proposition 2.6. Suppose that the system (D, r) is expansive. Then there exists c 5) and a continuous, surjective and r ­equivariant a finite set 5, a subshift map 7r : ---+ D. PROOF. Let 1 I be a metric on D which is compatible with the topology, and I'. > 0 an expansivity constant for (D, II, T'). By compactness of D, we can find a finite covering (B.).ES of D with closed sets of diameter < e. Let us denote by 5) the set of all maps a : r ---+ 5 and define, for every a E 5),

Let C 5) be the set of maps a : r ---+ 5 such that 1( a) is nonempty. For every a E , the set 1(a) is reduced to a point. Indeed, if x and y belong to 1(a), then

,x -

,x

and ,yare in B(7(­y)' and therefore I ,y 1< I'. for every, E r. As I'. is an expansivity constant for (D, I I, I'}, we have x = y. Let 7r : ---+ D be the map which to every a E associates the unique element in 1(a). For every, E r and for every a : r ---+ 5, it is clear that 1ha) = ,1(a). Consequently, is a r -invariant subset of 5) and the map 7r is r -equivariant. Let now (an) be a sequence of elements in which converges to a E 5). For every, E r, we can find an integer nh) such that anh) = 0-(,) for every n 2: nh)· By extracting a subsequence, we can suppose that the sequence X n = 7r(a n ) converges to a point xED. For every, E r, we have X n E B(7(­y) for n 2: nh). Therefore, x E B(7(­y)' Hence x E 1(a), which shows that is closed in 5) and that the map 7r is continuous. Finally, the surjeetivity of 7r is a consequence of the fact that the sets (B.) form a covering of D. Indeed, given the elements xED and a : r ---+ 5 such that ,x E B(7(­y) for every, E r, the definition of 7r shows that x = 7r(a). •

,-I

,-I

Definition 2.7. We say that the system (D, r) is a quotient of the system (D', r) if there exists a continuous, surjective and r -equivariant map D' ---+ D . 'With this definition, Proposition 2.6 can be formulated in the following way: Any expansive system is the quotient of some subshift.

Remarks.

1) A quotient of an expansive system is not necessarily expansive. For example, consider the N -action on the circle 51 in the complex plane {z E c II z I = I} generated by the map z l-+ z2, and consider also the N -action on the interval [-1, 1] generated by the polynomial x l-+ 2x 2 - 1. By using the vertical projection, the dynamical system ([-1,1 ],N) appears as a quotient of the system (51 ,N). As the differential of z l-+ Z2 is of norm 2 on 51, the system (5 1,N ) is expansive by Corollary 2.5. But

24

Chapter 2. -

Symbolic dynamics

the polynomial P(x) = 2x 2 -1 generates a dynamical system on [-1,1] which is not expansive, as opposite points have the same image by P. Here is now an example with r = 7l (cf. Parry and Walters, [Wall, p. 175). Let F be the homeomorphism of the torus T 2 = R 2 / 7l 2 which is induced by the linear automorphism L of R 2 defined by the matrix

). The action of 7l on

° T2R? 2

T generated by F is expansive (to see this, one can look at L in the neighborhood of E equipped with a basis of eigenvectors). Let us now consider the involution I of defined by the symmetry with respect to the origin. The quotient 52 = T 2/ I is a sphere and the projection T 2 --+ 52 is a two-sheeted branched covering. F induces a homeomorphism f of 52 which generates a dynamical system (52, 'll), which is the quotient of (T2, 'll). But the system (52, 7l) is not expansive. Indeed, for every E > 0, we can find distinct points x and y in 52 such that I r( x) - r(y) I:S E for all n E 7l (one can take x and y to be the images of two points in R 2 which are symmetric with respect to one of the two eigenspaces of L). 2) If the system (n, 7l ) is the quotient of a subshift ll> C = 5), then the topological entropy hen, 7l ) is finite. Indeed, we have

hen, 7l) :S h(ll>, 7l) :S

7l)

=

log(card(5)).

For example, this implies that the system ([0, 1]'ll ,'ll ), where 'll acts as the shift on the Hilbert cube [0,1] 7l , being of infinite topological entropy, is not the quotient of a subshift (see [Wall for the definition and the properties of the topological entropy). 3) The topological space n cannot be in itself an obstruction to the fact that the system (n, I') is the quotient of a subshift. Indeed, any compact metrizable space is the image by a continuous map of a Cantor set (see for example [Kur] , p. 214, Theorem 4). Recall that a topological space is said to have topological dimension zero if it has a basis whose elements are open and closed sets, and that a compact set has topological dimension zero if and only if it is totally disconnected (see for example [HW]). The next statement gives a characterization of the systems which are topologically conjugate to a subshift. (Recall that two systems (n 1 , I") and (n 2 , I") are said to be topologically conjugate if there exists a r -equivariant homeomorphism n1 --+ n2 . ) Proposition 2.8. equivalent:

-

For every system

(n, I"),

the following two conditions are

(i) There exists a finite set 5 and a subshift ll> C topologically conjugate to (ll>, I').

5) such that (n, I') is

(ii) The space n has topological dimension zero (i. e. zs totally disconnected) and the system (n, r) is expansive. PROOF. The first implication is immediate. Let us prove the second one.

25

Chapter 2. -

Suppose that

Symbolic dynamics

n has topological dimension zero and that (n, r)

is expansive. Let

II be a metric on n which is compatible with the topology, and let E be an expansivity constant for (n, I I, T'). As n has topological dimension 0, we can cover it by sets

which are open and closed, and whose diameter is < E. By compactness of n, we can extract a finite covering. Thus, we have a partition of n by a finite family (BS)SES of closed sets of diameter < E. By looking again at the construction which was done for the proof of Proposition 2.6, we see that the map which associates to every x E n the unique a E S) defined by iX E Bu(-y) is a r-equivariant homeomorphism from n to a subshift C S). •

§3 - Subshifts of finite type Following the terminology which is most commonly used in a cartesian product, we shall say that a subset C of is a cylinder if there exists a finite subset F of r and a set A of maps from F into S such that

I alF

C = {a E

E

A}.

Remarks. 1) The complement of a cylinder is also a cylinder. 2) The union (resp. intersection) of two cylinders is again a cylinder. 3) Every cylinder is closed and open in 4) The set of all cylinders forms a basis for the topology 5) The cartesian product of two cylinders is again a cylinder. More precisely, if Sl and S2 are finite sets, if C 1 C Sd and C2 C S2) are cylinders, then C1 x C2 C Sl) x S2) = Sl x S2) is also a cylinder.

Proposition 3.1. are equivalent:

Let be a subset of L

Then, the following three statements

(i) There exists a cylinder C such that = nI'Hi-1c.

(ii) There exists a continuous and set of fixed points. (iii) is closed in

r-

equivariant map T :

and there exists an open set U in

having as its such that

= nI'Hi-1u.

PROOF.

(i) =} (ii) : We shall suppose that card(S) 2 (otherwise, the result is trivial). Let be a map satisfying (i). Then there exists a finite subset F of r and a set A of maps from F to S such that = {a E

I Vi E r,ialF E A}.

26

Chapter 2. -

Symbolic dynamics

We can suppose that the identity element Id of r belongs to F. Let 13 be the set of all maps from F to 5. It is clear that we can construct a map u : 13 S satisfying the following property:

(3.1.1)

Vf E 13, u(J) = f(Id)

f EA.

(to define such a map u, we take u(J) = f(Id) if f E A, and u(J) = an element in 5 which is distinct from f(Id) if f ¢: A.) Let us now construct the map T : 1j 1j. Given a E 1j, we define T(a) : r 5 by (3.1.2)

V, E

r,

T(a)(,) = u(,alF)'

Let us verify that T possesses the required properties. If a E , we have ,alF E A for all, E r, and then, using (3.1.1) and (3.1.2),

T(a)(,) = ,a(Id) = a(,). Hence T( a) = a. Suppose now that a ¢: . Then there exists , E r such that ,aF ¢: A. We then have, using (3.1.1) and (3.1.2),

Hence, T( a) i- a. It follows that is the set of fixed points of T. Let us show now that Tis r-equivariant. Let 'f) E r. We have

V,

E

r, 'f)T(a)(,)

=

T(a)(,'f))

= u(,rWIF) = T('f)a)(,), which implies that T]T(a) = T(T]a). If (an) is a sequence of elements of 1j which converges to a E 1j, then, given, E r, we have, for all n large enough, ,anlF = ,alF, and therefore, using (3.1.2), T(a n ) ( , ) = T(a)(,). This proves the continuity of T.

(ii)

(iii) : Let T :

=?

1j

1j

be a continuous r-equivariant map and let E 5,

= Fix(T) (the set of fixed points of T). Define also, for every s

We have a E if and only if, for every, E r, T( a)(,) = a(,). On the other hand, we have T(a)(,) = ,T(a)(Id) = T(,a)(Id) and a(,) = ,a(Id). Therefore: (3.1.3)

= n-yH,-1 U,

where U = UsES Vs n Us. As Us is a cylinder, the sets Us and Vs are open and closed in 1j. Equality (3.1.3) implies then that satisfies (iii).

(iii)

=?

(i) : Let be a closed set and U an open set in

1j

such that =

n-yH,-lu. The set U is the union of the cylinders that it contains. By compactness of , there exists a finite family of cylinders (Ci)iEI contained in U which cover the

27

Chapter 2. -

Symbolic dynamics

set q). Let C be the union of the C j. (Note that C is therefore a cylinder.) We have q) = n-yEr'Y-1c, which implies (i). • Exercise. Show directly that (ii)

=}

(i), by using Proposition 1.4.

Let us note that any subset q) of satisfying any of the equivalent conditions of the preceding proposition is a subshift, as q) is closed in and is r-invariant.

Definition 3.2. We say that the subset q) of satisfies the equivalent conditions of Proposition 3.1.

is a subshift of finite type if it

A subshift of finite type is called also a Markovian subshift.

Remark.

Exercise. Show that every subshift is the intersection of a countable family of subshifts of finite type. Exercise. Show that the intersection of two subshifts of finite type is also a subshift of finite type. Exercise. Let r = 'll and 5 = {O,1,2}; we have therefore = {O,1,2}71. Let q)l = {a, 1}'ll and q)2 = {1,2}'ll. Show that q)1 and q)2 are subshifts of finite type, but that q)1 U q)2 is not a subshift of finite type.

Let q) c

Then, the diagonal

of q) x q), defined as

= ((o-,o-) I 0-

E q)},

is a subset of 5) x

5)

=

5 x 5).

Proposition 3.3. If q) is subshift of finite type, then the same is true for the diagonal of q) x q). PROOF. Let T : be a continuous r -equivariant map such that q) = Fix(T). It is sufficient to remark that = Fix(T 1 ) , where T 1 : x X is defined 1 , • by the formula T' (( 0-,0-/)) = (0- T( 0-)). Consider now two finite sets 51 and 52, and two subsets q)1 C 51) and q)2 C 52)' Then, q)1 x q)2 is a subset of 5r) x 52) = 51 x 52)'

Proposition 3.4. for q)1 X q)2'

If q)l and q)2 are subshijis of finite type, then the same is true

to PROOF. For i = 1,2, let T; be a continuous and r-equivariant map from itself, such that q)j = Fix(Tj ) . The proposition follows from the fact that q)1 x q)2 =

Fix(Tl

X

T2 ) .

•

28

Chapter 2. -

Symbolic dynamics

§4 - Systems of finite type and finitely presented systems Let again n be a compact metrizable space on which I' acts continuously.

Definition 4.1. We say that (n, r) is a system of finite type if there exists a finite set S, a subshift of finite type cf> C S) and a continuous, surjective and T -equivariant map 7l' : cf> ---+ n. Examples. 1) If cf> is a subshift of finite type, then (cf>, f) is a system of finite type (take, as a map tt , the identity map). 2) Given the systems (n, f) and (n', T}, suppose there exists a continuous, surjective and I' -equivariant map f : n ---+ n' (i .e. suppose that (n', I') is a quotient of (n, P)') and that (n, r) is of finite type. Then, (n', f) is of finite type. 3) If the systems (n, I') and (n', I") are of finite type, then the same is true for the system (n X n', I') (use Proposition 3.4). Consider two systems (cf>, r) and (n, r) and a map 7l' : cf> ---+ n which is continuous, surjective and f -equivariant. Let us denote by R( 7l') the (graph of the) equivalence relation associated to 7l',

This is a closed and f-invariant subset of cf> x cf>, and homeomorphism from the quotient space cf>/ R( 7l') to n.

7l'

induces a f-equivariant

S), we note that R(7l') is a subshift of

In the case where cf> is a subshift of S) x S) = S x S)

Definition 4.2. We say that (n, f) is a finitely presented system if there exists S) and a continuous, surjective and a finite set S, a subshift of finite type cf> C I'(W) = {a E 2: I Vw E W, w does not appear in a}. It is clear that 1>(W) is a closed r-invariant subset of 2:, and is therefore a subshift. Conversely, let 1> be a subshift. Let W be the set of words which do not appear in any element of 1>. It is clear that 1> C 1>(W). Consider now an element a E 1>(W). For every integer n :2: 0, we can find an element of 1> in which the word (a(O), a(I), ...,a(n)) appears (resp., if r = 7l, the word (a( -n), a( -n + 1), ... , a( n))). As 1> is r-invariant, we deduce that there exists an element an E 1> such that an (i) = a( i) for every i such that I i I::; n. The sequence (an) converges to a. It follows from the fact that 1> is • closed that a E 1>, and this finishes the proof. Given a set liV C S" of words of length n, we define 2:w = {a E 2: I Vi E

Proposition 5.2. equivalent:

r, (a( i), ... , a( i + n -

1)) E W}.

Let 1> be a subset of 2:. The following three statements are

(i) 1> is a subshift of finite type. (ii) There exists an integer n and a subset liV C S" such that 1> = 2:w. (iii) There exists an integer n and a subset W' C S" such that 1> is the set of a E 2: in which no word of lV ' appears. The equivalence of (ii) and (iii) can be seen by taking W' = S" - W. Given W C S"; let C be the cylinder defined as the set of elements a E 2: such that (a(O), a(I), ..., a(n - 1)) E W. We have 2:w = n"HI-1c, which shows that 2:w is a subshift of finite type and therefore that (ii) implies (i). Conversely, if 1> is a subshift of finite type, there exists a finite set F e r and a set A of maps from F to S such that 1> is the set of a E 2: such that the restriction of la to F belongs to A for every 1 E r. It is clear that we can suppose that F = {O,1,2, ... , n -I} for a certain n. We remark then that 1> = 2:w, where W is the set of words of the form (a(O), a(I), ..., a( n - 1)), a E A, which finishes the proof. •

PROOF.

Example. Let us take S = {O,I} and r = 7l. Let 1> be the set of sequences 7l 1-+ S having at most one zero term. It is clear that 1> is a subshift. But 1> is not a subshift of finite type. Indeed, suppose that 1> = 2:w for W C S", Let a : 7l --t S be the sequence whose terms are all zero except a(O) and a(n), which are equal to 1. We have a 1>, and nevertheless every subword of length n which appears in a appears also in an element of 1>, which is a contradiction. Note that the system (1), 7l) is topologically conjugate to (71 U {oo}, 7l), where the action of 7l on 7l U {oo} is generated by the translation z --t z + 1.

32

Chapter 2. -

Symbolic dynamics

Definition 5.3. We say that a subshift cI> C L; is of order n if there exists a subset We S" such that cI> = L;w. The following proposition allows us to reduce the study of subshifts of finite type to that of subshifts of order two.

Proposition 5.4. -

Let cI> C L;(r, S) be a subshift of finite type. Then, there exists a finite set S' and a subshift of order two cI>' c L;(r, S') such that (cI>, T') is topologically conjugate to (cI>', I'),

PROOF. Suppose that cI> is of order n. Let We S" be such that cI> = L;w. We take as a new set of symbols S' = W, and we define L;' = L;(r, S'). Consider the map u : cI> --* L;' which associates to every a E cI> the sequence u( a) : r --* S' defined by:

u(a)(i) = (a(i),a(i

+ 1), ...,a(i + n

-1)), for every i E r.

Let W' C S,2 be the set of words of length two on S' which are of the form (u(a)(O), u(a)(l))

with a describing any element of cI>. It is clear that u is a I' to the order-two subshift cI>' = •

A subshift of order two can be described by a square matrix with coefficients in

{a, I}. To see this, we begin by noting that a subset We S2 of words of length two can be defined by its characteristic function M : S2

--*

{a, I} given by

M(s,s') = 1 if (s,s') E W, M(s, s') = a if not. One can see M as a square matrix having S as a set of indices for its rows and columns, and with coefficients in {O, I}. Let us denote by L;M the subshift of order two cI> = L;M which is thus associated to the matrix M. We have therefore L;M

=

{a EL;I Vi E r,M(a(i),a(i

+ 1)) = I}.

We say that M is a transition matrix for cI> = L;M. Examples. In all the examples which follow, we take S

l)For M

=

or

= {I, 2}

=

r = 7l

.

L;M is empty.

2) For M = rduced to a point. 3) Forr M

and

or

L; M is

L;M is reduced to two points (which are fixed by the action of

7l ). 33

Chapter 2. 4) For M =

Symbolic dynamics

is, as in the preceding example, reduced to two points, but

the action of 7l interchanges the two points. 5) For M

=

presents a North - South dynamics:

or

is topologically conjugate to (71 U {6) For M

=

7) For M =

),

ex:>,

+oo}, 7l

7l )

).

= or

the two systems

7l ) which are obtained are

topologically conjugate (this is done by interchanging the two elements of S). is the union of a Cantor set. The dynamics of the action is chaotic. The same as on the periodic orbits is dense in Note that 7l ) is not topologically conjugate to 7l ) has two fixed points, whereas has only one). The set We S2 can also be seen as an oriented graph whose set of vertices is S and whose edges are the pairs of elements (s, s') E W. We say that this graph is a transition graph for the order-two subshift

Go 1

Examples. Fo, S

{1,2,3} and M

) , the associated graph is given ;n

1

Figure 1.

n

/0" o· 0 Figure 1

Let M be a square matrix having S as a set of indices for its rows and columns, and whose coefficients are in {O, I}. We can easily "read" on M certain dynamical properties of the order-two subshift As an example, let us show how to calculate the (-function of For the rest of this section, we shall restrict to 7l -systems. 34

Chapter 2. -

Symbolic dynamics

Recall (see [Sma]) that the (-function of a 'll-system (n, 'll ) is the formal powersenes ((t) = Nnt n) where N n is the number of elements in n whose period is n (i.e. the fixed points under the action of the group n'll C 'll of multiples of n). We note that the (-function is related to the formal power-series

by the formula N(t) = i('(t)j((t).

Proposition 5.5. by the formula

The (-function of the subshift of order two

'll ) is given

((t) = det(I - tM)-1 where I : Sx S if and only if s

=

{O, I} is the identity matrix, that is, the matrix defined by I( s, s') = 1 s':

To prove this proposition, we use the following lemma which follows immediately from the definition of the product of two matrices. Lemma 5.6. Let s, s' E Sand n an integer which is 1. The number of words = (S1' ... , sn) E S" such that M(s;, S;+1) = 1, for every i = 1,2, ... , n - 1, S1 = s and Sn = s' is equal to Mn(s,s'). •

W

Proof of Proposition 5.5. From the preceding lemma, we know that the number of elements (7 E I;M which are fixed under the action of n'll and such that (7(0) = s is equal to Mn(s,s). We therefore have N; = Tr u», This gives

((i) = M n))

=

= det(I -

tM)-1.

•

From the fact that ( is invariant under topological conjugacy, the preceding proposition, together with Proposition 5.4, gives the following Corollary 5.7. form

The (-function of every subshift of finite type on 'll is of the ((t) = det(I - tM)-1

where M is a square matrix with coefficients in {O, I}.

Corollary 5.8. function.

•

The (-function of every subshift of finite type on 'll is a rational

35

Chapter 2. -

Symbolic dynamics

The preceding corollary is a particular case of the celebrated rationality theorem of Manning, which can be stated in the following way: Theorem 5.9. Let (n, 7l) be a finitely presented 7l -ssrsiem, and suppose that there is a finite presentation 1r : n which is finite-to-one, that is, satisfies the following condition (F): (F)

There exists an integer N such that eard(1r- 1(x))

:s N

for all x E

n.

•

Then, the (-function of (n, 7l ) is rational.

For a proof of the theorem of Manning, the reader is referred to [Man] and to [Fri] (see also section 8.5.U of [Gro 3]).

In fact, D. Fried has proved in [Fri] that any finitely presented system (n, 7l ) admits a finite presentation 1r : f-+ n which satisfies (F). Hence, any finitely presented 7l -system has a rational (-function.

§6 - Sofie systems In this section, r is again an arbitrary countable semigroup. Let 5 be a finite set and let C 5) be a subshift. We say that is a sofie subshift if the dynamical system ( . Indeed, Proposition 4.3 shows that such a map 1r is a finite presentation of (, I'). Example. Let us look again at the subshift which we gave as an example in the preceding section. We take r = 7l , 5 = {O,1} and we let be the subshift C = 5) which is defined as the set of sequences a E which have at most one nonzero term. The system (, 7l ) is topologically conjugate to (7l U {oo}, 7l ). Consider now the subshift of order two o C which consists of the sequences s E such that (a(i),a(i + 1)) of- (1,0) for every iE'll. The system (o, 7l) is topologically conjugate to (7l U {-oo,oo}, 7l). The map 1r:

7l U {oo} =

defined by 1r(-(0) = 1r((0) = 00 and 1r( z) = z for all z E 7l , is a finite presentation of (, 7l). As a consequence, is a sofic subshift. The map 1r can be read on the labelled graph of Figure 2, in the following way. This graph is a transition graph for 5,

36

Chapter 2. -

Symbolic dynamics 1

Figure 2

o

where E stand for the set of edges of the graph. The map 7r : q,o ----+ q, is obtained by associating to each element of q,o, that is, to each 'll -path in the graph, the sequence obtained by reading successively the labelling on the edges of that path. Exercise. We use the notations of the preceding example. 1) Explicit the relation R( 7r) = {(a, a') E q,o X q,o I 7r( a) = 7r( a')}, and show directly that R( 7r) is a subshift of finite type of 'E x 'E. 2) Show that q, is not topologically conjugate to any subshift of finite type, that is, that we cannot find a finite set S' and a subshift of finite type q,' C 'E( 'll ,S') such that (q" 'll ) is topologically conjugate to (q,', 'll ).

The dynamical system (Sl, r) is said to be sofie if there exists a finite set Sand a sofie subshift q, C 'E(r,S) such that (Sl,r) is topologically conjugate to (q"r). We have the following characterization of sofic systems, which is an immediate consequence of Proposition 2.8. Proposition 6.1. For a system (Sl, r) to be sofie, it is necessary and sufficient that the space Sl is of topological dimension 0 (i.e. that this space is totally disconnected) and that the system (Sl, r) is finitely presented. • Sofic systems on 'll and on N.

In the rest of this section, we take T = 'll or N. We shall describe the sofic subshifts with the help of graphs with labelled vertices. A graph with labelled vertices (G, fl) is, by definition, an oriented finite graph G, that is, a finite set of vertices V and a set of vertices E C V X V, together with a map fl : V ----+ S from the set of vertices to a finite set S. To such a labelled graph (G, fl), we associate a sofic subshift q, c 'E(r, S) and a finite presentation 7r ; q,o ----+ q, in the following manner: Let q,o c 'E(r, V) be the subshift of order two having G as a transition graph. We associate to every element a of q,o the sequence 7r( a) : I' ----+ S defined by 7r( a)( i) = fl(a(i)) for i E T. It is clear that the set q, of 7r(a),a E q,o, is a subshift of and that the map 7r : q,o ----+ q, is continuous, surjective and r-equivariant. The sofic subshift q, is called the subshift associated to (G, fl). In fact, we have the following Proposition 6.2. - A subshift q, c S) is sofic if and only if there exists a graph with labelled vertices (G, fl) such that q, is the subshift associated to (G, fl). PROOF.

It remains to show that the condition is necessary. Let q,

37

c

S) be a

Chapter 2. -

Symbolic dynamics

sofie subshift. There exists therefore a finite set S1, a subshift of finite type -1>1 C I;(r, Sd and a map 1r : -1>1 ----; -I> which is continuous, surjective and r-equivariant. By Proposition 1.4, there exists a finite set Fer and a map u : sf ----; S2 such that 1r = U oo , i.e. such that 1r(cr)({) = u( I'crlF) for every a E -1>1 and for every I' E r. At the expense of taking F larger and of composing 1r, to the right, by an element of r, we can suppose that F = {O,1, ... , n - I} and that the subshift -1>1 is of order n. The set sf is then identified with the set of words of length ri on S 1. Consider now, as a new set of symbols, the set So of words of length n on S1 which appear in at least one element of -1>1, and let J1 be the restriction of u to So. The map which associates to every element a of -1>1 the sequence r : r ----; So defined by r( i) = (cr(O), cr(l), ..., cr( n - 1)) is a topological conjugacy between -1>1 and a subshift of order two -1>0 C I;(r, So) (proof of Proposition 5.4). If G is a transition graph for -1>0, it is clear that -I> is the subshift associated to (G, J1). •

Example. Let again r = sz , S = {O,I} and -I> C I;(r, S) the subshift consisting of the sequences having at most one non-zero term. -I> is the sofic subshift associated to the graph with labelled vertices of Figure 3.

cO-+O-+O n Figure 3

o

.0

Figure 4

38

Chapter 2. -

Symbolic dynamics

Ezercis e. We take r = 'll and S = {O,I}. Let

. Let r be a geodesic ray in X, that is to say, a map r : [0,00[-+ X such that ret') 1=1 t - t' 1for every t, t' 2: 0. The Busemann function (or horofunetion) associated to r is the function h ; : X -+ R. defined as

1ret) -

Thus, a Busemann function is a limit of normalized distance functions. This limit exists and is finite. Indeed, the triangle inequality shows that I x - ret) I -t is a non-inecreasing function of t which is bounded below by - 1 x - reO) I .

46

Chapter 3. -

The boundary as a finitely presented dynamical system The function h; is I-Lipschitz, that is to say, we have

Proposition 3.1. -

Vx,y E X, I hr(x) - hr(y) 1:::;1 x - Y

PROOF.

I·

The triangle inequality gives

1(I x - ret) I -t) - (I

Y - ret)

I -t) 1=

1(I x - ret) I) - (I Y - ret) J) 1:::;1 x - y 1. Making t tend to infinity proves the proposition.

•

Recall that a function f : X -> JR is said to be E- convex (E 2:: 0) if, for all points Yo, YI in X, for every geodesic segment [Yo, YI] joining these points and for every 0' E [0, 1], we have

f(yo) :::; O'f(yo)

+ (1 -

0' )f(yI)

+ E,

where Yo is the unique point on [Yo, yd satisfying I Yo - Yo 1= Lemma 3.2. -

For every x EX, the function f : X

->

0'

1Yo - YI \ .

JR defined by f(y) = I x - y

1

is 48-convex. PROOF. When X is a tree (8 = 0), the result follows from the convexity of the function g(O') = f(yo), 0' E [0,1]. In fact, it is easy to see in this case that 9 is a piecewise linear function whose derivative is -Ion [0,0'0] and 1 on [0'0,1], where YOo is the projection of x on [yO, yd. Let us pass now to the general case (8 arbitrary). Let Z = [Yo, YI] u {x}. By the theorem of approximation by trees (cf. Chapter 1, §5), there exists a tree T and a map F : Z -> T such that

(3.2.1) Vz, z' E

Z,I

z - z' 1-48 :::;1 F(z) - F(z') 1:::;1 z - z'

I,

and

(3.2.2) Vz, z' E [Yo, YI], I F(z) - F(z') 1=1 z - z' I . Therefore, we have: 1

x - Yo 1:::;1 F(x) - F(yo) I +48, from (3.2.1),

:::; 0' 1 F(x) - F(yo) 1+(1 - 0') I F(x) - F(yd 1+48, from (3.2.2) and the O-convexity of the function "distance to F( x)" in the tree T, and :::; 0' 1x - yo 1+(1 - 0') \ x - YI 1+48

•

using (3.2.1).

47

Chapter 3. -

Proposition 3.3. PROOF. For every t

The boundary as a finitely presented dynamical system The function h r is 48-convex.

0, we can write, using the preceding lemma:

1Yo - r(t) I::::: 0 I Yo - r(t) I +(1 - 0) I Yl - r(t) I +48. Therefore, we have: I Yo - r(t) I -t ::::: 0(1 Yo - r(t) 1 -t)

+ (1 -

0)(1 Yl - r(t) I -t)

+ 48.

By letting t tend to infinity, we obtain:

Proposition 3.4. Let A be a real number, and x a point in X such that h r( x) Then there exists a point p in X such that

1x -

p

1= hAx) -

A and hr(p)

•

A.

= A.

PROOF. For every real number t, consider a geodesic segment [x,r(t)] between x and r(t). For t large enough, let p(t) be the unique point on [x,r(t)] such that [ x - p(t) 1= hr(x) - A. We have: (3.4.1)

Ix

- r(t)

I -t = hr(x) -

A + (I p(t) - r(t) 1-t).

By hypothesis, X is proper. Consequently, the closed ball centered at x and radius h r ( x) - A is compact. Therefore, we can find a sequence t; which tends infinity and such that the sequence (p(t;)) converges to a point p of X. By passing the limit, we obtain 1x - p 1= hr(x) - A and, using (3.4.1), hr(p) = A.

of to to •

Corollary 3.5. we have:

A,

For every real number A and for every x E X such that h r( x)

PROOF. If Y is a point in h;l(] -

00,

AD, we have, by

Hence On the other hand, Proposition 3.4 gives

48

Proposition 3.1,

Chapter 9. -

The boundary as a finitely presented dynamical system

•

and this proves the corollary. Let us prove now the following Proposition 3.6. -

The function cp(x,y), defined by

cp(x, y) = hr(x) - hr(y) for all x, y E X such that I x - y belongs to

I::; 3d

Furthermore, h.; is a primitive of cp.

PROOF. We must show that cp satisfies properties (i), (ii) and (iii) of §1. The verification of (i) is immediate. (ii) results from Proposition 3.3. Let us prove (iii) (which is not an immediate consequence of Corollary 3.5 !). Let x E X and let 'f'x be the function on B 3d(X) defined by

Let y E Bd(x), and t E [-d,d] be such that 'f'x(Y)

t, or, in other words,

From Proposition 3.4, there exists a point p of X such that

and Using the fact that the function h; is I-Lipschitz (Proposition 3.1), we have

which implies that

I y - pi::; 2d.

Using the triangle inequality, we have therefore

I x - pi::; I x As p E B 3d(X),'f'x(p)

=t

and \ y - p

- y

I + I y - pi::;

\= 'f'x(Y) -

3d.

t, we obtain

On the other hand, if z E 'f';l(] - 00, t]), we have

This implies

'f'x(Y) - t::; dist(y,'f';l(] - oo,t]). 49

Chapter 3. -

The boundary as a finitely presented dynamical system

Thus, we have established that

•

that is to say, property (iii).

§4 - The gradient lines defined by 'P Let 'P be an element of .

Definition 4.1. A gradient line for 'P, or a 'P-gradient line is a rectifiable map 9 from an interval I C R to X parametrized by arclength and satisfying V t, t' E I, 'P(g(t), g(t'))

= t' - t.

Remark that a local gradient line is also a global gradient line, as can be seen from the cocycle property (I) of 'P given in Proposition 2.2. This cocycle property shows also that if we concatenate two gradient lines (which have the property that the endpoint of the first one equal to the initial point of the second), then the resulting path (parametrized by arclength) is also a gradient line. Let us begin by proving the following

Proposition 4.2. z. e.

If g : I ---. X is a gradient line for 'P E , then g is geodesic,

v t,t' E I, I get) -

get')

1=1 t -

t'

I·

PROOF. Given t and i' in I with t ::::: t', let g[t,t'] be the subpath in g comprised between get) and get'). Let us show that this path is geodesic. As the path g is parametrized by arclength, we have length(g[t,t']) = t' - t. On the other hand, we have

t' - t = 'P(g(t),g(t')) :::;1 get') - get) 1 (by property (V1) of the functions 'P). Therefore, we have

length(g[t,t '])

=1

get') - get) I,

•

and therefore the path g is geodesic.

Proposition 4.3. Let 'P E and x E X, let y E X be a point satisfying 'P(x, y) =1 x - y I, and let 9 : [to,t l ] ---. X be a geodesic segment between x and y. Then, 9 is a gradient line for 'P.

50

Chapter 3. -

The boundary as a finitely presented dynamical system

PROOF. By property (V 1) of the functions cp, we have, for every t' '2: t,

cp(g(t),g(t'))

:::;1

get) - get')

1= t'

< t' - t. We would have then

Suppose that there exists t' '2: t such that cp(g(t),g(t')) (again by application of property (V I)) tl

-

to = cp(x,y)

- t.

= cp(g(to),g(t)) + cp(g(t),g(t')) + cp(g(t'),g(td)

< (t - to) + (t' - t) + (t l

-

t') = t l

-

which is a contradiction. Therefore, we have cp(g(t),g(t'))

to,

= t'-t

for every t,t' E

[to, tl]'

•

Let cp be an element of 1> and x a point in X. Then, there exists a cp-gradient line 9 : [0,00[--+ X starting at x, that is, such that g(O) = x.

Proposition 4.4. -

PROOF. From property (vii), we can construct, by induction, a sequence of points Xi, i = 0,1, ... , such that Xo = x and such that, for every i, cp(Xi, Xi+l)

=1 Xi -

Xi+l

1= d.

Consider, for every i, a geodesic segment [Xi, xi+d between Xi and Xi+l' Let 9 : [0,00[--+ X be the infinite path obtained by concatenating the paths [Xi, Xi+l], parametrized by arclength, starting at X = Xo = g(O). By Proposition 4.3, each of the paths [Xi, Xi+l] is a gradient line. By the cocycle condition (I), the path 9 is a cp-gradient line (ef. the remark following Definition 4.1). • The preceding proposition provides, for the cocycles cp E 1>, "global" versions of properties (iii) and (vii) of §1:

Let cp be an element of 1>. Then:

Corollary 4.5. -

(V I I)

For every

X

E X and for every real number t

that cp(x,y)

=1 x -

y

'2:

a,

there exists y E X such

1= t.

PROOF. Let 9 : [0,00[--+ X be a cp-gradient line starting at x. The point y does the job. Corollary 4.6. Let cp E 1> and rp a primitive of cp. Then, for every for every real number t such that rp( x) '2: t, we have

(III)

rp(x)

X

= get) •

E X and

= t + dist(x, rp-l(] - 00, t])).

PROOF. This is an immediate consequence of the preceding corollary and of property (V I) of cp. •

51

Chapter 3. -

The boundary as a finitely presented dynamical system

§5 - The point at infinity associated to a cocycle Given an element cp E 4>, let us choose a point x E X and a gradient line 9 starting at x. As this line is geodesic, it converges to a well-defined point of ax . We will show that this point is canonically associated to cp (that is, it does not depend on x or on the chosen gradient line). We shall denote this point by a(cp), and we call it the point at infinity associated to cp. Proposition 5.1. Let cp E 4> , and let g, g' : [0,00[-' X be two cp-gradient lines starting respectively at the points x and Y of X. Then, 9 and g' stay at a uniformly bounded distance from each other. In other words, there exists a constant C such that

Vt

PROOF.

0,1 get) - g'(t) I:::: C.

The proof will be done in three steps:

°

First step. Let us show that the statement is true with the assumption that cp( X, y) = and 1x - y I:::: 268 + 1. Let (Xi) (resp. (Yi)) be the sequence of points on 9 (resp. on g') defined by Xi = g(di) (resp. Yi = g'(di)), i = 0,1,2, ... It suffices to show that the two sequences Xi and Yi stay at a uniformly bounded distance from each other. In fact, we shall prove by induction that for every i, we have

I Xi -Yi I:::: 268 + 1. This is true, by hypothesis, for i = 0. Suppose that 1Xi - Yi I:::: 268 + 1 for a given i. Let [Xi+l' Yi+l] be a geodesic segment joining the points Xi+l and Yi+l. Consider the set Z = [Xi+l' Yi+l] U {Xi, y;}. From the theorem of approximation by trees, there exists a tree T and a map f : Z ---. T with the following properties: (5.1.1) and

Vz1,zz E Z,

1Zl

-

Zz

1-68::::1

f(zd - f(zz)

1::::1 Zl

-

Zz I,

°

On B 3 d (Xi ) n B 3 d (Yi ), we have 'Px; = 'Py;, because cp(Xi,Yi) = by the co cycle condition. The point Xi+l (resp. Yi+l) is a projection of Xi (resp. Yi) on the set 'P;"/(] - 00, -d]). By 'Px;(xi+d = 'Px;(Yi+d = -d, we have, using property (ii) of the functions cp (quasi-convexity): . Vz E [Xi+l' Yi+l], 'Px;(z) :::: -d + 48. (Remark that we have Xi+I, Yi+l E B 2d(Xi) because 1Xi - Xi+l 1= d and 1Xi - Yi+l 1 :::: I Xi - v. 1+ 1Yi - Yi+l I:::: 268 + 1 + d:::: 2d.) As -d+48 :::: 0, property (iii) of the functions cp implies, for every z E [Xi+l' Yi+l],

1z -

Xi

d - 48 and

52

1z -

Yi

d - 48.

Chapter 3. -

The boundary as a finitely presented dynamical system

Therefore, using (5.1.1), we have (5.1.3)

IP -

f(Xi)

12: d -

lOb and

IP -

f(Yi)

12: d -

108,

for every point P on the segment [f(Xi+l),f(Yi+l)] which is the image of [Xi+l,Yi+d by I, as follows from (5.1.2). Consider now, in the tree T, the projections Pi and qi respectively of the points f(Xi) and f(Yi) on the segment [f(xi+d,f(Yi+l)]' If p, and qi are distinct (Figure 1), we have

f(Yi)

Figure 1

u. \2:1 f(Xi) - f(Yi) I from (5.1.1), 2:\ f(Xi) - Pi I + I f(y;) - qi I 2: 2d - 2015, using (5.1.3), which contradicts the induction hypotesis I Xi - Yi I:S 268 + 1. Therefore, 1

Xi -

f(Xi) and f(Yi) have the same projection P = f(z) on the segment [f(Xi+l),f(Yi+dJ as shown in Figure 2. P

f(Xi)

Figure 2

f(Yi)

Again by (5.1.3), we have 1

f(Xi) - Pi

12: d -

lOb.

On the other hand, we know that

1f(Xi)

- f(Xi+d

I:SI Xi -

This implies that

53

Xi+!

1= d.

Chapter 3. -

The b01tndary as a finitely presented dynamical system

In the same manner, we can show that

Finally, we have

and therefore, using (5.1.1),

1Xi+1

- Yi+1

I:::; 26 85 and the ball B, has nonempty intersection with 91. In the same way, B, has nonempty intersection with 92 for every i large enough. Let us choose an integer i such that B, has nonempty intersection with both 91 and 92. Let PI be a point in B, n gi and P2 a point in B, n 92. We have

by the cocycle property (1) satisfied by 'P2. Using property (V 1), we have

and 'P2(P2,Y)

= - I P2

- Y

'PI(P2,y).

On the other hand, we have 'P2(PI,P2) = 'PI(PI,P2), as 'PI and 'P2 coincide on B, x Bi, It follows that

In the same way, 'Pl(X, y) proposition.

'P2(X, y). Hence 'PI (x, y)

§6 - Properties of the map a : A < I x - y

(2.1.2)

If xRy and yRz then (x.z)y

(2.1.3)

If I x - x'

(2.1.4) 1908.

If xRy, yRz, and it:«, and if (x.Y)t < 1508, then (t.u)y < 1708 and (y.z)u
' C

I;

x

I;

is a subshift of finite type.

It is clear, from the definition of 1>', that it is a subshift of order two.

Let now W' = P'('lJ o ) C I; x I; be the image of the map P'. The definition of shows immediately that W' is the kernel R( 71") of 71". Proposition. PROOF.

'lJ'

•

'lJri

is a semi-Markovian subset of I; x I;.

Let us show that W' is equal to the intersection of 1>' with the cylinder

C' = {(sn'

c

I; X I;

with So

= sri =

the N-type of the basepoint xo}.

Let be an element of I; x I; which is in the kernel of 71". The two sequences (sn)nEN and are then the images by the map P of two discrete geodesic rays g( n) and g' (n) in the tree T geo which, by definition, converge to the same point of oTg eo • These two rays start at the basepoint Xo, and we therefore have So = sri = the N-type of Xo. Hence (sn' E C'. On the other hand, by the inequality (3.4.1) of Chapter 1, each of the two geodesic rays stays uniformly at a distance'S: 48 from the other one, and we therefore have (sn, E 1>'.

128

Chapter 7. -

The boundary as a semi-Markovian space

Conversely, let us take an arbitrary element (Sn, in q>' n C ', and let us show that it is in the kernel of Jr. For that, we shall construct two discrete geodesic rays g(n) and g'(n) such that I g(n) - g'(n) 48 for every n EN, and such that = pl(g(n),g'(n))nEN' The construction of g(n) and g'(n) is made by induction. We begin by defining g(O) = g'(O) = xo, which is possible since So = = the N-type of xo. For every n 2: 0, we define the pair (g(n+l),g'(n+l)) out ofthe pair (g(n),g'(n)) in the following manner. From the definition of q>', we can find in the Cayley graph K two ordered pairs of vertices, (x, x') and (y, V') satisfying the following four properties:

I x 1=1 x'I, -

The N-types ofthese ordered pairs of vertices are respectively the pairs (sn,

and (Sn+l' y follows x and y' follows x'.

Ix -

x'

48, and

Iy -

y'

48.

We have, as an induction hypothesis, I g(n) - g'(n) 48. Let us consider the left-translation T of the group which sends x on g(n), and let x" be the image of x' by this translation. We have I x" 1=1 g(n) 1=1 gl(n) I . Indeed, this is a consequence of the fact that x' is in the ball of radius N centered at x (because we have supposed N 2: 48), and that the points x and g(n) have the same N-type. Let us note now that the translation T, which preserves the N -type at x, preserves the (N - 48)-type at x'. Thus, the points x" and g'(n) have the same (N - 48)-type. By Proposition 3.1, we deduce that T(X') = x" = g'(n). As T preserves the (N - 48)-type at the points x and x', and as we have taken N large enough (so that T preserves the isometry types of the pointed trees Tgeo,x and Tgeo,xl), we conclude that T(Y) follows x and that T(y') follows x'. Applying now Proposition 2.4, we see that T(Y) has the same N-type than y and that T(Y') has the same N-type than y'. We then take g(n + 1) = T(Y) and g'(n + 1) = T(Y'), which completes the induction argument, and the proof of Proposition 3.5. • We can now state the Theorem 3.6.- The map

Jr:

\II

--+

of is a semi-Markovian presentation of of.

PROOF. This is a consequence of Propositions 2.7 and 3.5. Corollary 3.7. Markovian space.

• •

The boundary of a torsion-free hyperbolic group zs a semi-

Corollary 3.8. Let X be a hyperbolic space which is geodesic and proper. Suppose there exists a torsion-free group which acts isometrically and properly discontinuously on X, with X/f compact. Then the boundary of X is a semi-Markovian space.

PROOF. We have

ax

=

of, by Proposition 6.5 of Chapter 1. 129

•

Chapter 7. -

The boundary as a semi-Markovian space

Corollary 3.9. Let r be a hyperbolic group which has a finite-index subgroup I" c r which is torsion-free. Then, the boundary of r is a semi-Markovian space. PROOF. We know that I" is also hyperbolic and that

or = or'

(cf. Chapter 1, §2) .•

Corollary 3.10. -

Let k be a field of zero characteristic, n a natural integer and a hyperbolic group isomorphic to a subgroup of GLn(k). Then, the boundary of r is a semi-Markovian space.

r

PROOF. Selberg's lemma (see [Sell or [Cas] for a proof) asserts that any finitely generated subgroup of GLn(k) contains a finite-index subgroup which is torsion-free.•

§4 - The boundary of T p a r t as a semi-Markovian subset To see the space oTp ar t as a semi-Markovian subset of a certain one-sided Bernoulli shift, we shall follow the same outline than in §2, concerning the space oTg eo • We can adapt, step by step, the arguments that we used for the vertices of T g eo into arguments for the vertices of T par t. Let N 1 = 608 + 2, let x be an arbitrary vertex of the tree T par t and let Y1 and Y2 be two distinct vertices of this tree which are situated beyond x and at distance 1 from x. Then, there exists an integer No N 1 (which deepends only on the group r equipped with its given set of generators) such that for every N No, the vertices Y1 and Y2 are not (N - Nd-equivalent.

Proposition 4.1. -

PROOF. Let us identify the vertices Y1 and Y2 with their images as finite subsets of If card(Y1) =I- card(Y2), these two vertices are not N-equivalent, for any N O. If card(Yd = card(Y2), we can find two elements Zl and Z2, with Zl E Y1 - Y2 and Z2 E Y2 - Y1· The points Zl and Z2 are situated on the same sphere centered at the identity element. As the diameters of the three sets x, Y1 and Y2 of r are bounded above by 208, we see that the distance I Zl - Z2 I is bounded above by 3 x 208 + 2 = 608 + 2. In the same manner as for the proof of Proposition 3.1, we can see now the existence of the integer No which we are looking for. •

r.

We then proceed in the same way as we did in §2, concerning the tree T g eo : we define the set of symbols 5 as the set of N -equivalence classes of vertices of the tree T p ar t, and if 81 and 82 are two elements of 5, we say that 82 follows 81 in T p ar t if these two elements can be represented by vertices x' and y' of that tree, with v' E Tpart,x' and I x' - y ' 1= 1. We have then the following proposition, which is analogous to Proposition 2.4.

Proposition 4.2. -

Let 81 and 82 be two elements of 5 such that 82 follows 81 in T pa r t' If x E 81 is an arbitrary vertex of T par t in the class 81, then there exists a vertex Y which follows x and which is in the class 82. In fact, if x' and y' are as above and if we identify the four points x, y, x' and y' with their image sets in r, then the

130

Chapter 7. -

The boundary as a semi-Markovian space

image of y' by the left-translation of I' which sends z' on x is a subset of I' which defines the vertex y of T par t.

PROOF. (sketch). The proof is of the same type as that of Proposition 2.4. We use Proposition 1.9 which insures that the pointed trees Tpart,x and Tpart,x' are Nequivalent. • The rest of the construction is exactly the same as for the tree T g e o : We define the set llIo of discrete geodesic rays in T par t which start at the basepoint, and a map P : llIo E = E(N, S) which, thanks to Proposition 4.1, will be injective. The image III C E(N, S) of llIo by P is a semi-Markovian subset of E(N, S). There is a natural bijection map:

D :

llIo

oTpar t,

and if we call ?To the composed

we have the following

Theorem 4.3. The map ?To is a homeomorphism between the semi-Markovian subset III C E(N, S) and oTpar t. In particular, ?To is a semi- Markovian presentation of oTpar t. • §5 - A finite-to-one semi-Markovian presentation of

or

From the semi-Markovian presentation of oTpar t decribed in §4 above, we can deduce a semi-Markovian presentation of following the same type of reasoning as for the presenattion give in §3. The set of symbols S will be here the set of N-equivalence classes of vertices of T par t, with N greater than the integer No of Proposition 4.1 and the integer N I of Proposition 5.1 below which will be useful for the construction of the presentation, and which is the analogue of Proposition 3.1.

of

Proposition 5.1. Let YI and yz be two distinct vertices of TpartJ such that \ YI 1=1 yz I and such that if we consider these vertices as finite subsets of I', we have dist(YI, yz) :::; 45. Then there exists an integer N I such that for all N :::: N I J the vertices YI and yz are not (N - 45)-equivalent. PROOF. The proof is of the same style as that of Propositions 4.1 and 3.1 (in fact, • we can see this proposition as a corollary of Proposition 3.1). We then proceed in the construction of the semi-Markovian presentation of of. Following the same notations as in §3, we define the set p' C E x E as the set of sequences of ordered pairs (Si, sDiEN satisfying the following condition: 131

Chapter 7. -

The boundary as a semi-Markovian space

\In ;::: 0, two consecutive ordered pairs (sn, and (Sn+l, can be represented by ordered pairs of vertices of the tree T p a r t , (x, x') and (y, y') such that I x 1=1 x'I, y following x and y' following x', and such that if these four vertices are considered as finite subsets of r, we have dist(x, x') ::::: 48 and dist(y, y') ::::: 48. The other definitions and the rest of the construction remain inchanged with respect to §3, and we thus obtain the following Theorem 5.2. The map Jr : 'lJ - t ar which is defined in this way is a semimarkovian presentation of ar which is finite-to-one, that is, there exists an integer M such that, for every EEar, card(Jr- 1 (O) ::::: M.

132

Chapter 7. -

The boundary as a semi-Markovian space

Notes and comments on Chapter 7

Proposition 8.5.K of [Gro 3} affirms that the boundary of every hyperbolic group (without the hypothesis on torsion) is semi-Markovian. Gromov gives indications for the proof in sections 8.5.1 and 8.5.J of [Gro 3].

133

Chapter 7. -

The boundary as a semi-Markovian space

Bibliography for Chapter 7 [CanJ J. Cannon, "The combinatorial structure of co-compact discrete hyperbolic groups", Geometriae Dedicata 16, (1984), pp. 123-148. [CasJ J. W. S. Cassels, "An embedding theorem for fields", Bull. Australian Math. Soc. 14, (1976), pp. 193-198 and 479-480. [CDPJ M. Coornaert, T. Delzant, A. Papadopoulos, "Geometrie et theorie des groupes: Les Groupes hyperboliquers de Gromov", Lecture Notes in Mathematics, vol. 1441, Springer Verlag, 1990. [Gro 1J M. Gromov, "Hyperbolic manifolds, groups and actions", Ann. Studies 97, Princeton university Press (1982), pp. 183-215.

of Math.

[Gro 3] - - , "Hyperbolic groups", in Essays in Group Theory, MSRI publ. Springer, 1987, pp. 75-263.

8,

[SelJ A. Selberg, "On discontinuous groups in higher dimensional spaces", in "Contributions to Function Theory", Bombay 1960, pp. 147-164.

134

Index

Bernoulli shift one-sided two-sided bounded geometry (simplicial graph with - - ) boundary - of a hyperbolic group - of a hyperbolic space Busemann function

20 31 31 60

canonical metric on a graph Cantor set Cayley graph cocompact action cocycle cp convergent quasi-geodesic field curve Menger Sierpinski cylinder

15 20 15 15 44

diagonal dimension (topological - - ) dynamical system

15 9 46

74

113 112

26 22

104 22 16

elliptic isometry expansive system expansivity constant

22

22 29

finite presentation finite type subshift of - -

28 135

system of function

29

((- - )

35 46

(Busemann - - ) geodesic - polygon -ray - segment - space gradient line graph - with labelled vertices simplicial - - with bounded geometry Cayley - transition - Gromov product group - acting cocompactly - acting properly discontinuously hyperbolic - -

7 6 6 6

50 36 60 15 34

6

15 15 8 14

Hausdorff distance hyperbolic - group - isometry - metric space

8

16 6

7

internal points of a triangle isometry elliptic - hyperbolic - parabolic - -

16 16 16

line (gradient - - ) Lipschitz map

50 16

Markovian subshift Menger - curve - sponge matrix (transition - - ) metric canonical on a simplicial complex word - visual - - on the boundary

28

136

113 113

32 15 8

12

metric space geodesic - hyperbolic - .proper - -

6 6 10

narrow (8- ) geodesic polygon N-type N-equivalence

7 119 119

parabolic isometry 16 perfect set 20 point at infinity of a cocycle 'P 50 pointed subtree 120 polygon (geodesic - - ) 6 polyhedron Pd(X) 17 presentation (fini te - - ) of a dynamical system 28 presentation (semi-Markovian - - ) of a compact setl08 primitive of a cocycle 'P 45 product (Gromov - - ) 6 projection 96 projective sequence of sets 92 properly discontinuous action 15 proper metric space 10 13 13 24

quasi-geodesic quasi-isometry quotient system real tree

6

segment geodesic topological - semigroup semi-Markovian - presentation - space - subset sequence convergent - - at infinity projective - - of connected sets set of symbols Sierpinski - carpet - set - sponge 137

6 6 19 108 108 108

9 92 20 112 112 112

sofic - subshift - system subshift - of finite type - of order n Markovian - sofic - symbols (set of - - ) system - of finite type dynamical - expanSIve - finitely presented quotient - sofie - -

36 37 28 31 28 36 20 29 22 22 29 24

36

theorem - of approximation by trees - of stability of quasi-geodesics Manning's rationality - thin ( 5- - - triangle) totally disconnected space topological dimension 0 topological dimension n topologically conjugate systems transition - graph - matrix transitive action

13 14 36 7 20 25 104 25 34

33 21 12

visual metric

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138