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Topology and its Applications 139 (2004) 17–22

www.elsevier.com/locate/topo

Is the composite of two expansive maps expansiv✩

Mario Roy

Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West,Montreal, Canada H4B 1R6

Received 18 February 2003; received in revised form 30 July 2003

Abstract

We study the question whether the composite of two expansive maps is itself expansivanswer to this question is a resounding no. We will actually establish a stronger fact. We wian example of two dynamical systems which are topologically exact and Ruelle expandinrespect to two different, equivalent and compatible metrics, but whose composite is far fromexpansive, for it is locally eventually contracting towards a fixed point a completely invariant,open subspace. 2003 Elsevier B.V. All rights reserved.

MSC:37B99; 37B10

Keywords:(Positively) expansive maps; Ruelle expanding maps; Composite; Symbolic dynamics

1. Introduction

The notion of expansiveness was introduced by Utz [6] in 1950. Until now, two mproblems have been further investigated. On the one hand, some people (for insee [1,2]) have been working on the following problem: What kind of space doesan expansive map? Some others (see [5], for example), on the other hand, havconsidering the following problem: Given a space, what kinds of maps have an expbehavior? In both cases, one and only one map was considered at a time.

In this short note, we raise a different question. We ask whether the compositetwo expansive maps, defined on the same space, is expansive.

Obviously, the answer to this question is negative if the two maps are expahomeomorphisms. Indeed, iff :X → X is an expansive homeomorphism of a metric sp

✩ This research was supported by the Fonds québécois pour la Formation de Chercheurs et l’ARecherche du Québec (FCAR).

E-mail address:marioroy@mathstat.concordia.ca (M. Roy).

0166-8641/$ – see front matter 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.topol.2003.07.017

18 M. Roy / Topology and its Applications 139 (2004) 17–22

(X,d), then so is its inversef −1, and their composite, the identity map, is not expansivem

all

ce.

pactdposite

s it is

ase,hth”tuallyry samepletely

o-e

rt

whenever the underlying spaceX is infinite. Recall thatf is an expansive homeomorphisif there exists a constantε > 0 such thatd(f n(x), f n(y)) � ε for all n ∈ Z implies thatx = y.

However, the answer to the question whether the compositef ◦ g of two positivelyexpansive mapsf :X → X andg :X → X is positively expansive is more nebulous. Recthat a mapf is said to be positively expansive if there exists a constantε > 0 such thatd(f n(x), f n(y)) � ε for all n ∈ Z+ implies thatx = y.

Indeed, observe that if bothf andg are expanding with respect to the same metrid

on X, then their composite is also expanding with respect tod , and thereby expansivRecall that a mapf is called expanding if there exist constantsa > 0 andλ > 1 such thatd(f (x), f (y)) � λd(x, y) wheneverd(x, y) � a. Recall also thatf is said to be Ruelleexpanding (cf. [4]) if it is both open and expanding.

We will first give a simple example of two maps, defined on the same comtopological space, that are Ruelle expanding withrespect to two different equivalent ancompatible metrics, but whose composite is not expansive. More precisely, the comproves to be contracting “half” of the space towards a single fixed point (whereaexpanding on the other “half”).

Another similar, though slightly more complicated, example is provided. In this cbothf andg are additionally topologically exact. Their composite is contracting a “eigof the space towards a single fixed point as before, but it is further locally evencontracting a completely invariant, dense open subset of the space towards that vepoint (whereas it is expanding on the complement of this latter set, that is, on a cominvariant, nowhere dense, closed subspace).

2. A first example

Let X = {0,1}Z+ � {0,1}Z+ consist of two disjoint copies of the one-sided full twshift. In order to distinguish these copies, we denote the first one byΣ1 and the second onby Σ2, and byσ1 andσ2 their respective shift maps. We further use the notation(x0x1 . . .)tto denote the point(x0x1 . . .) of Σt , and[x0x1 · · ·xs−1]t (s ∈ N), to represent the cylindeof points ofΣt whose firsts coordinates are successivelyx0, x1, . . . , xs−1. We assume thaX is endowed with the metric

d(x, y) ={

2−l if x, y ∈ Σt for the samet,1 otherwise,

wherel is the smallest non-negative integer for whichxl �= yl . (In casex = y, we adopt theusual conventionl = ∞ and 2−∞ = 0.)

There is a family of metrics that will be of particular interest to us:

Definition 1. Let µ > 1. Givenn ∈ N andh :X → X a map, define

dhn(x, y) =n−1∑m=0

1

µmd(hm(x),hm(y)

).

M. Roy / Topology and its Applications 139 (2004) 17–22 19

It is well known that if hn is expanding with respect to the metricd with constants

t

t

everyt

se

s

a andµn, thenh is expanding with respect todhn with constantsa andµ. This is justMather’s change of metric (cf. [3]). It is also easy to see that ifh is Lipschitz with respecto d , thendhn is equivalent tod for everyn.

We further denote byπ :Σ1 → Σ2 the natural identification map betweenΣ1 andΣ2,that is,π(x0x1 . . .)1 = (x0x1 . . .)2. Now, we definef,g :X → X as

f (x) ={

π(σ 2

1 x)

if x ∈ Σ1,

π−1(0x) if x ∈ Σ2,

and

g(x) ={

π(0x) if x ∈ Σ1,

π−1(σ 2

2 x)

if x ∈ Σ2.

Lemma 2. The mapsf andg are Ruelle expanding with respect to the metricsdf 2 anddg2 respectively, and these latter two metrics are equivalent to the original metricd . (Theequivalence of two metricsρ and ρ means that there exists a constantC � 1 such thatC−1 � ρ(x, y)/ρ(x, y) � C wheneverx, y ∈ X, x �= y.)

However, neitherf ◦g norg ◦f is expansive. More precisely, thoughf ◦g is expandingon the “half-space”Σ2, it is contracting the “half-space”Σ1 towards the fixed poin(0∞)1. The compositeg ◦ f has similar properties.

Proof. Bothf andg are open and continuous since the image and inverse image ofcylinder under each of these maps are (finite) unions of cylinders. Now, observe tha

f 2(x) ={

0σ 21x if x ∈ Σ1,

σ2x if x ∈ Σ2,

and

g2(x) ={

σ1x if x ∈ Σ1,

0σ 22x if x ∈ Σ2.

Hence bothf 2 and g2 are expanding maps with respect tod (with common constanta = 1/4 andλ = 2). It follows from this thatf andg are expanding with respect to thmetricsdf 2 anddg2, respectively (with common constantsa andµ = √

λ ). These latter twometrics are further equivalent tod , for f andg are Lipschitz with respect tod . However,

(f ◦ g)(x) ={

00x if x ∈ Σ1,

σ 42x if x ∈ Σ2,

and

(g ◦ f )(x) ={

σ 41x if x ∈ Σ1,

00x if x ∈ Σ2.

In particular, note that(f ◦ g)(Σt ) ⊂ Σt for eacht = 1,2. Moreover, notice thatf ◦ g iscontractingΣ1 towards the fixed point(0∞)1, since every application off ◦ g adds two0’s to the beginning of every point ofΣ1. On the other hand,f ◦ g is expanding onΣ2(with constantsA = 1/24 andΛ = 24), for every point inΣ2 loses its first 4 coordinateunderf ◦ g. �

20 M. Roy / Topology and its Applications 139 (2004) 17–22

3. A more interesting example

e

ect

ge ofo that.

In the following,i, j, k, l ∈ {0,1} are arbitrary as far as they satisfyi �= j andk �= l. Thespace upon which the dynamics take effect is the same as before, but we now defin

f (x) =

π(σ 2

1 x)

if x ∈ Σ1, x0x1 = ij,

σ 21 x if x ∈ Σ1, x0x1 = ii,

π−1(σ 2

2 x)

if x ∈ Σ2, x0x1 = ij,

σ 22 x if x ∈ Σ2, x0x1 = 11,

π−1(0x) if x ∈ Σ2, x0x1 = 00,

and, similarly,

g(x) =

π(σ 2

1 x)

if x ∈ Σ1, x0x1 = ij,

σ 21 x if x ∈ Σ1, x0x1 = 11,

π(0x) if x ∈ Σ1, x0x1 = 00,

π−1(σ 2

2 x)

if x ∈ Σ2, x0x1 = ij,

σ 22 x if x ∈ Σ2, x0x1 = ii.

Lemma 3. The mapsf andg are topologically exact, and Ruelle expanding with respto the metricsdf 2 anddg2, respectively.

Proof. Both f andg are open, continuous maps since the image and inverse imaany cylinder under any of these maps are (finite) unions of cylinders. Observe alsf (Σt) = X andg(Σt ) = X for everyt = 1,2, and thereforef andg are surjective mapsMoreover,

f 2(x) =

σ 41x if x ∈ Σ1, x0x1x2x3 = ijkl,

π(σ 4

1 x)

if x ∈ Σ1, x0x1x2x3 = ij11,

0σ 21x if x ∈ Σ1, x0x1x2x3 = ij00,

π(σ 4

1 x)

if x ∈ Σ1, x0x1x2x3 = iikl,

σ 41x if x ∈ Σ1, x0x1x2x3 = iikk,

σ 42x if x ∈ Σ2, x0x1x2x3 = ijkl,

π−1(σ 4

2 x)

if x ∈ Σ2, x0x1x2x3 = ijkk,

π−1(σ 4

2 x)

if x ∈ Σ2, x0x1x2x3 = 11kl,

σ 42x if x ∈ Σ2, x0x1x2x3 = 1111,

π−1(0σ 2

2x)

if x ∈ Σ2, x0x1x2x3 = 1100,

π−1(σ2x) if x ∈ Σ2, x0x1 = 00,

andg2 has a similar expression.

M. Roy / Topology and its Applications 139 (2004) 17–22 21

Observe that, given a cylinderc = [c0c1 · · ·cs−1]t , at least one and at most four of theyst

ct

r

oft or,

leftmost coordinates ofc are lost after every iteration off 2. Observe further that ancylinder of length four or less is mapped ontoX by (f 2)2. Therefore, there is a smallen = n(c) ∈ N such that(f 2)n(c) = X for any given cylinderc. This shows thatf istopologically exact. Evidently, a similar argument applies tog.

The same observation shows that bothf 2 and g2 are expanding maps with respeto d (with common constantsa = 1/24 andλ = 2). It follows from this thatf andg areexpanding with respect to the metricsdf 2 anddg2 (with common constantsa andµ = √

λ ).Notice once again that these latter two metrics are equivalent tod , since bothf andg areLipschitz with respect tod . �

Now let us have a look atf ◦ g:

(f ◦ g)(x) =

σ 41 x if x ∈ Σ1, x0x1x2x3 = ijkl,

π(σ 4

1x)

if x ∈ Σ1, x0x1x2x3 = ij11,

0σ 21x if x ∈ Σ1, x0x1x2x3 = ij00,

π(σ 4

1x)

if x ∈ Σ1, x0x1x2x3 = 11kl,

σ 41 x if x ∈ Σ1, x0x1x2x3 = 11kk,

00x if x ∈ Σ1, x0x1 = 00,

σ 42 x if x ∈ Σ2, x0x1x2x3 = ijkl,

π−1(σ 4

2x)

if x ∈ Σ2, x0x1x2x3 = ijkk,

π−1(σ 4

2x)

if x ∈ Σ2, x0x1x2x3 = iikl,

σ 42 x if x ∈ Σ2, x0x1x2x3 = ii11,

π−1(0σ 2

2x)

if x ∈ Σ2, x0x1x2x3 = ii00.

Lemma 4. The backward orbit of the cylinder[00]1 underf ◦ g, namelyC := ⋃n∈Z+(f ◦

g)−n([00]1), is a dense, open, completely invariant subset underf ◦ g.

Proof. The openness ofC is an immediate consequence of the openness of[00]1 and thecontinuity off ◦ g.

Furthermore, observe that(f ◦ g)−1(C) = C, for (f ◦ g)−1([00]1) ⊃ [00]1 and henceC = ⋃

n∈N(f ◦ g)−n([00]1). So (f ◦ g)−1(C) = C. Sincef ◦ g is surjective, we furthe

deduce that(f ◦ g)(C) = C. HenceC is completely invariant underf ◦ g.Finally, givenx ∈ X, there is a uniquet ∈ {1,2} for whichx ∈ Σt . Denote as(y(n))n∈Z+

the sequence defined byy(n) = (x0x1 · · ·x4n−10∞)t . It is easy to see that the applicationf ◦ g to any point ofX results in either the loss of the first four coordinates of the poinin the point being mapped into[00]1. Hence(f ◦ g)n+1(y(n)) ∈ [00]1 and, consequently(y(n))n∈Z+ is a sequence inC which converges tox. SoC is dense inX. �Lemma 5. The compositef ◦ g is expanding onX\C. However, it is globally contracting[00]1 and locally eventually contracting the setC towards the fixed point(0∞)1. Inparticular, f ◦ g is not expansive. Nor isg ◦ f .

22 M. Roy / Topology and its Applications 139 (2004) 17–22

Proof. First, note that(f ◦ g)([00]1) = [0000]1. By induction, we obtain that(f ◦

e

ch

tion.ke to

y, Proc.

3–

)

,

g)n([00]1) = [02n]1. Sof ◦ g is contracting[00]1 towards the fixed point(0∞)1.Recall now that each iteration of a point underf ◦ g results in either the loss of th

first four coordinates of the point or in the point being mapped into[00]1. So, givenx ∈ C,t ∈ {1,2} with x ∈ Σt , andn = n(x) ∈ Z+ the smallest non-negative integer for whi(f ◦ g)n(x) ∈ [00]1, we have(f ◦ g)n([x0x1 · · ·x4n+3]t ) ⊆ [00]1. This means thatf ◦ g

is locally eventually contracting the dense, open, completely invariant setC towards thefixed point(0∞)1.

On the other hand,f ◦ g is expanding onX\C (with constantsA = 1/24 andΛ = 24),since every point in this set loses its first four coordinates upon application off ◦ g. �

Acknowledgements

I am really grateful to Mariusz Urbanski for bringing up this question to my attenMany thanks to Peter Walters for sharing his opinion on the subject. Finally, I would lithank the University of North Texas for its warm hospitalityduring my visit in 2001–2002,on which occasion this paper was drafted.

References

[1] K. Hiraide, Nonexistence of positively expansive maps on compact connected manifolds with boundarAmer. Math. Soc. 110 (1990) 565–568.

[2] R. Mané, Expansive homeomorphisms and topologicaldimension, Trans. Amer. Math. Soc. 252 (1979) 31319.

[3] J. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Indag. Math. 30 (1968479–483.

[4] D. Ruelle, Thermodynamic Formalism, in: Encyclopedia Math. Appl., vol. 5, Addison-Wesley, Reading, MA1978.

[5] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747–817.[6] W.R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950) 769–774.