A Note on Anosov Homeomorphisms

Abstract

For an $\alpha$-expansive homeomorphism of a compact space we give an elementary proof of the following well-known result in topological dynamics: A sufficient condition for the homeomorphism to have the shadowing property is that it has the $\alpha$-shadowing property for one-jump pseudo orbits (known as the local product structure property). The proof relies on a reformulation of the property of expansiveness in terms of the pseudo orbits of the system.

1. Introduction

In [1] (Theorem 1.2.1) it is proved, among other things, that Anosov diffeomorphisms has the shadowing property, called pseudo orbit tracing property there. In the proof, on [1] (p. 23), the authors only uses the so-called local product structure property: if $d(x,y)<\delta$ then $W_{\epsilon }^s\left(x\right)\cap W_{\epsilon }^u\left(y\right)\ne ⌀$ if $\delta >0$ is chosen small enough for a given $\epsilon >0$, and the special (hyperbolic) properties of the metric d coming from the Riemannian structure of the manifold supporting the system [1] ((B), p. 20).

As can be easily checked the first of these two conditions is equivalent to the shadowing property for pseudo orbits with one jump, that is, for every $\epsilon >0$ there exists $\delta >0$ such that for every bi-sequence of points of the form

$$\dots ,z_{-2}=T^{-2}y,z_{-1}=T^{-1}y,z_0=x,z_1=Tx,z_2=T^{2}x,\dots$$

with $d(x,y)<\delta$, where T denotes the diffeomorphism, there exists a point z such that $d(T^{n}z,z_n)<\epsilon$ for all $n\in \mathbb{Z}$.

On the other hand, in [2] (Theorem 5.1) it is shown that for every expansive homeomorphism on a compact space there exists a compatible metric (which we call hyperbolic metric) with similar properties to those of the metric d in the case of Anosov diffeomorphisms. Then the proof of the shadowing property for Anosov diffeomorphisms given in [1] carries over the more general case of expansive systems.

In this paper, we give an alternative and elementary proof of this well-known shadowing condition (Proposition 4), not making use of Fathi’s hyperbolic metric. Instead we use a reformulation of the property of expansiveness of a system (Proposition 1) which seems interesting in its own right.

2. Terminology and Notation

In this note X denotes a compact metric space with metric d and $T:X\to X$ a homeomorphism. The orbit of a point $x\in X$ is the bi-sequence $O\left(x\right)={\left(T^{n}x\right)}_{n\in \mathbb{Z}}$.

Definition 1.

T is said to be expansive if there exists a constant $\alpha >0$, called expansivity constant, such that if $x,y\in X$ and $d(T^{n}x,T^{n}y)\le \alpha$ for all $n\in \mathbb{Z}$ then $x=y$.

Expansive homeomorphisms was introduced in [3] with the name unestable homeomorphisms.

Definition 2.

Let$\xi ={\left(x_n\right)}_{n\in \mathbb{Z}}$be a bi-sequence of elements of X. If$\delta >0$and$d(T{x}_n,x_{n+1})<\delta$for all$n\in \mathbb{Z}$then ξ is called$\delta$-pseudo orbit. We say that ξ has a jump at the n-th step if $T{x}_{n-1}\ne x_n$. Given $\epsilon >0$ a bi-sequence $\eta ={\left(y_n\right)}_{n\in \mathbb{Z}}$ is said to $\epsilon$-shadow ξ if $d(x_n,y_n)<\epsilon$ for all $n\in \mathbb{Z}$. If in the previous situation $\eta =O\left(x\right)$ is the orbit of a point $x\in X$ we simply say that x ε-shadows ξ and that ξ is ε-shadowed (or ε-shadowable).

Definition 3.

Given$\epsilon >0$we say that T has the$\epsilon$-shadowing property if for some$\delta >0$every δ-pseudo orbit is ε-shadowable. We say that T has the shadowing property if it has the ε-shadowing property for all $\epsilon >0$. If T is expansive and has the shadowing property then it is called Anosov homeomorphism.

3. Rephrasing Expansivity

The following simple result states an equivalent condition for the expansiveness of the system $(X,T)$. This alternative characterization of expansiveness will allow us to give an elementary proof of the shadowing condition in Proposition 4.

Proposition 1.

Let$\alpha >0$. The following conditions are equivalent.

(1)

T is expansive with expansivity constant α.

(2)

For every$\epsilon >0$there exists$\delta >0$such that

$$\mathit{if}d(x_n,y_n)\le \alpha \mathit{for}\mathit{all}n\in \mathbb{Z}\mathit{then}d(x_n,y_n)<\epsilon \mathit{for}\mathit{all}n\in \mathbb{Z},$$

for every pair of δ-pseudo orbits${\left(x_n\right)}_{n\in \mathbb{Z}}$and${\left(y_n\right)}_{n\in \mathbb{Z}}$of T.

Proof. 

$(1\Rightarrow 2)$ Suppose that the thesis is not true. Then, there exists $\epsilon >0$ such that for every $k\in \mathbb{N}$ one can find $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$k$}\right.$-pseudo orbits ${\left(x_n^k\right)}_{n\in \mathbb{Z}}$ and ${\left(y_n^k\right)}_{n\in \mathbb{Z}}$ of T satisfying $d(x_n^k,y_n^k)\le \alpha$ for all $n\in \mathbb{Z}$ but $d(x_{n_k}^k,y_{n_k}^k)\ge \epsilon$ for a suitable $n_k\in \mathbb{Z}$. Changing the indexing of the pseudo orbits if necessary it can be assumed that $n_k=0$ for all $k\in \mathbb{N}$. As X is compact it can be also assumed that $x_0^k\to x$ and $y_0^k\to y$ for some $x,y\in X$. It is easy to see that then $x_n^k\to T^{n}x$ and $y_n^k\to T^{n}y$ for all $n\in \mathbb{Z}$ (the pseudo orbits converge pointwise to actual orbits). However, now, as $d(x_n^k,y_n^k)\le \alpha$ for all $k\in \mathbb{N}$ and $n\in \mathbb{Z}$ we have $d(T^{n}x,T^{n}y)\le \alpha$ for all $n\in \mathbb{Z}$, and as $d(x_0^k,y_0^k)\ge \epsilon$ for al $k\in \mathbb{N}$ we get $d(x,y)\ge \epsilon$, so that $x\ne y$. This contradicts that $\alpha$ in an expansivity constant and the proof finishes.

$(2\Rightarrow 1)$ Suppose $x,y\in X$ verifies $d(T^{n}x.T^{n}y)\le \alpha$ for all $n\in \mathbb{Z}$, and note that ${\left(T^{n}x\right)}_{n\in \mathbb{Z}}$ and ${\left(T^{n}y\right)}_{n\in \mathbb{Z}}$ are $\delta$-pseudo orbits for every $\delta >0$. Therefore, by the hypothesis, for every $\epsilon >0$ we have $d(T^{n}x,T^{n}y)<\epsilon$ for all $n\in \mathbb{Z}$, that is, ${\left(T^{n}x\right)}_{n\in \mathbb{Z}}={\left(T^{n}y\right)}_{n\in \mathbb{Z}}$. Then $x=y$ and hence $\alpha$ is an expansivity constant. □

For future reference we recall from [4] (Theorem 5) the following basic property of expansive homeomorphisms on compact spaces known as uniform expansivity.

Proposition 2.

Let$\alpha >0$. The following conditions are equivalent.

(1)

T is expansive with expansivity constant α.

(2)

For every$\epsilon >0$there exists$N\in \mathbb{N}$such that for every$x,y\in X$

$$\mathit{if}d(T^{n}x,T^{n}y)\le \alpha \mathit{for}\mathit{all}\left|n\right|\le N\mathit{then}d(x,y)<\epsilon .$$

We also recall the following easy result that can be found in [5] (Lemma 8).

Proposition 3.

T has the shadowing property if and only if T has the shadowing property for pseudo orbits with a finite number of jumps.

4. The Shadowing Condition

As pointed out in the Introduction the next is a known result that can be proved with the techniques in [1] (p. 23) replacing the metric coming from the Riemannian structure in that argument by Fathi’s hyperbolic metric [2] (Theroem 5.1).

Proposition 4.

If T is expansive with expansivity constant$\alpha >0$then the following conditions are equivalent.

(1)

T has the shadowing property.

(2)

There exists$\delta >0$such that every one-jump δ-pseudo orbit is α-shadowed.

Proof. 

Clearly we only need to prove that the last statement implies the first one. By Proposition 3 it is enough to show that for every $\epsilon >0$ there exists $\rho >0$ such that all $\rho$-pseudo orbits with a finite number of jumps are $\epsilon$-shadowed. To do that it is sufficient to find a $\rho >0$ corresponding only to $\epsilon =\alpha$, because by Proposition 1 for any $\epsilon >0$ taking a smaller value of $\rho$, more precisely choosing $\rho \le \delta$ where $\delta$ is given by the cited proposition, we have that to $\alpha$-shadow a $\rho$-pseudo orbit is equivalent to $\epsilon$-shadow it.

To find $\rho >0$ such that every $\rho$-pseudo orbit with a finite number of jumps is $\alpha$-shadowed, let $\delta >0$ be as in the statement of this proposition, that is, such that

$$\mathit{every} \delta -\mathit{pseudo\; orbit\; with\; one\; jump\; is} \alpha -\mathit{shadowed}.$$

(1)

By Proposition 1 (with $\epsilon =\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$) we can take a smaller $\delta$ to also guarantee that

$$\mathit{if\; a} \delta -\mathit{pseudo\; orbit} \xi \alpha -\mathit{shadows\; a} \delta -\mathit{pseudo\; orbit} \eta \mathit{then} \xi \raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.-\mathit{shadows} \eta .$$

(2)

Obviously we can also require that $\delta \le \alpha$. For this $\delta$ there exists $N\in \mathbb{N}$ such that

$$\mathit{if} d(T^{n}x,T^{n}y)\le \alpha \mathit{for\; all} \left|n\right|\le N\mathrm{then}d(x,y)<\delta ,$$

(3)

for all $x,y\in X$, according to Proposition 2. Finally, as T is uniformly continuous we can take $\rho >0$, $\rho \le \delta$, such that any segment of length $2N+1$ of a $\rho$-pseudo orbit, say $x_0,\dots ,x_{2N+1}$, is $\delta$-shadowed by its first element $x_0$, that is,

$$\mathit{if} d(T{x}_n,x_{n+1})<\rho ,0\le n\le 2N,\mathit{then} d(T^n{x}_0,x_n)<\delta ,0\le n\le 2N+1.$$

(4)

We will prove that this $\rho$ works by induction in the number of jumps in the $\rho$-pseudo orbits. If a $\rho$-pseudo orbit has only one jump, as $\rho \le \delta$ we know by condition (1) that it can be $\alpha$-shadowed. Assume now that $\xi ={\left(x_n\right)}_{n\in \mathbb{Z}}$ is a $\rho$-pseudo orbit with $k\ge 2$ jumps. Indices can be arranged so that the last jump takes place in the step from $x_{2N}$ to $x_{2N+1}$, so that ${\left(x_n\right)}_{n>2N}$ is a segment of a true orbit. By condition (4) we can replace ${\left(x_n\right)}_{n=0}^{2N}$ by ${\left(T^n{x}_0\right)}_{n=0}^{2N}$ in $\xi$ getting a $\delta$-pseudo orbit

$${\xi }^{\prime }={\left(x_n\right)}_{n<0}\bigsqcup {\left(T^n{x}_0\right)}_{0\le n\le 2N}\bigsqcup {\left(x_n\right)}_{n>2N},$$

where we denote ${\left(x_n\right)}_{n\in I}\bigsqcup {\left(x_n\right)}_{n\in J}={\left(x_n\right)}_{n\in I\cup J}$ if $I,J\subseteq \mathbb{Z}$ are disjoint sets of indices. We have that ${\xi }^{\prime }$ $\alpha$-shadows $\xi$ because $\delta \le \alpha$. Note that on one hand the bi-sequence given by

$$\eta ={\left(x_n\right)}_{n<0}\bigsqcup {\left(T^n{x}_0\right)}_{n\ge 0}$$

is a $\rho$-pseudo orbit with less than k jumps, then by the inductive hypothesis there exists $y\in X$ that $\alpha$-shadows $\eta$. By condition (2) we know that in fact $\eta$ is $\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$-shadowed by y. On the other hand consider

$$\zeta ={\left(T^n{x}_0\right)}_{n\le 2N}\bigsqcup {\left(x_n\right)}_{n>2N}$$

which is a $\delta$-pseudo orbit with one jump. Then by condition (1) there exists $z\in X$ that $\alpha$-shadows $\zeta$. Again condition (2) implies that $\zeta$ is $\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$-shadowed by z.

Now, as the segment of orbit ${\left(T^n{x}_0\right)}_{0\le n\le 2N}$ is in both sequences $\eta$ and $\zeta$ we have that the corresponding segments of the orbits of y and z verifies $d(T^{n}y,T^{n}z)<\alpha$ for $0\le n\le 2N$. Hence, by condition (3) we have that $d(T^{N}y,T^{N}z)<\delta$. Consequently

$$\tau ={\left(T^{n}y\right)}_{n<N}\bigsqcup {\left(T^{n}z\right)}_{n\ge N}$$

is a one-jump $\delta$-pseudo orbit that $\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$-shadows ${\xi }^{\prime }$. A new application of condition (1) gives an element $w\in X$ that $\alpha$-shadows $\tau$. Finally, as w $\alpha$-shadows $\tau$, $\tau$$\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$-shadows ${\xi }^{\prime }$ and ${\xi }^{\prime }$ $\alpha$-shadows $\xi$, we obtain by repeated application of condition (2) that w $\alpha$-shadows $\xi$, and we are done. □