Volume preserving diffeomorphisms with orbital shadowing

Abstract

Let f be a volume-preserving diffeomorphism of a closed $C^{\mathrm{\infty }}$ Riemannian manifold M. In this paper, we show that the following are equivalent:

(a) f belongs to the $C^1$-interior of the set of volume-preserving diffeomorphisms with orbital shadowing,

(b) f is Anosov.

MSC:37C50, 34D10.

1 Introduction

A fundamental problem in differentiable dynamical systems is to understand how a robust dynamic property on the underlying manifold would influence the behavior of the tangent map on the tangent bundle. For instance, in [1], Mañé proved that any $C^1$ structurally stable diffeomorphism is an Axiom A diffeomorphism. And in [2], Palis extended this result to Ω-stable diffeomorphisms. Let M be a closed $C^{\mathrm{\infty }}$ Riemannian manifold endowed with a volume form ω. Let μ denote the Lebesgue measure associated to ω, and let d denote the metric induced on M by the Riemannian structure. Denote by ${Diff}_{\mu }(M)$ the set of diffeomorphisms which preserves the Lebesgue measure μ endowed with the $C^1$-topology. We know that every volume preserving diffeomorphism satisfying Axiom A is Anosov (for more details, see [3]).

For $\delta >0$, a sequence of points ${\{x_i\}}_{i=a}^b$ ($-\mathrm{\infty }\le a<b\le \mathrm{\infty }$) in M is called a δ-pseudo-orbit of f if $d(f(x_i),x_{i+1})<\delta$ for all $a\le i\le b-1$. Let $\mathrm{\Lambda }\subset M$ be a closed f-invariant set. We say that f has the shadowing property on Λ (or Λ is shadowable) if for every $ϵ>0$, there is $\delta >0$ such that for any δ-pseudo-orbit ${\{x_i\}}_{i=a}^b\subset \mathrm{\Lambda }$ of f ($-\mathrm{\infty }\le a<b\le \mathrm{\infty }$), there is a point $y\in M$ such that $d(f^i(y),x_i)<ϵ$ for all $a\le i\le b-1$. It is easy to see that f has the shadowing property on Λ if and only if $f^n$ has the shadowing property on Λ for $n\in \mathbb{Z}\setminus \{0\}$. The notion of pseudo-orbits often appears in several methods of the modern theory of dynamical systems. Moreover, the shadowing property plays an important role in the investigation of stability theory. In fact, Pilyugin [4] and Robinson [5] showed that if a diffeomorphism f is structurally stable, then f has the shadowing property. Moreover, Sakai [6] proved that if there is a $C^1$-neighborhood $\mathcal{U}(f)$ of f such that for any $g\in \mathcal{U}(f)$, g has the shadowing property, then f is structurally stable. For each $x\in M$, let ${\mathcal{O}}_f(x)$ be the orbit of f through x; that is,

$${\mathcal{O}}_f(x)=\{f^n(x):n\in \mathbb{Z}\}.$$

We say that f has the orbital shadowing property on Λ (or Λ is orbitally shadowable) if for any $ϵ>0$, there exists $\delta >0$ such that for any δ-pseudo-orbit $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda }$, we can find a point $y\in M$ such that

$${\mathcal{O}}_f(y)\subset B_{ϵ}(\xi )\text{and}\xi \subset B_{ϵ}({\mathcal{O}}_f(y)),$$

where $B_{ϵ}(A)$ denotes the ϵ-neighborhood of a set $A\subset M$. f is said to have the weak shadowing property on Λ (or Λ is weakly shadowable) if for any $ϵ>0$, there exists $\delta >0$ such that for any δ-pseudo-orbit $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda }$, there is a point $y\in M$ such that $\xi \subset B_{ϵ}({\mathcal{O}}_f(y))$. Note that if f has the shadowing property, then f has the orbital shadowing property, but the converse is not true (see [7]). It is easy to see that f has the orbital shadowing property on Λ if and only if $f^n$ has the orbital shadowing property on Λ for $n\in \mathbb{Z}\setminus \{0\}$.

We say that Λ is hyperbolic if the tangent bundle $T_{\mathrm{\Lambda }}M$ has a Df-invariant splitting $E^s\oplus E^u$ and there exist constants $C>0$ and $0<\lambda <1$ such that

$$\parallel D_x{f}^n{|}_{E_x^s}\parallel \le C{\lambda }^n\text{and}\parallel D_x{f}^{-n}{|}_{E_x^u}\parallel \le C{\lambda }^{-n}$$

for all $x\in \mathrm{\Lambda }$ and $n\ge 0$.

We denote by ${\mathcal{F}}_{\mu }(M)$ the set of diffeomorphisms $f\in {Diff}_{\mu }(M)$ which have a $C^1$-neighborhood $\mathcal{U}(f)\subset {Diff}_{\mu }(M)$ such that for any $g\in \mathcal{U}(f)$, every periodic point of g is hyperbolic.

Very recently, Arbieto and Catalan [3] proved that every volume preserving diffeomorphism in ${\mathcal{F}}_{\mu }(M)$ is Anosov. To prove this, they used Mañé’s results in [[1], Proposition II.1] and showed that $\overline{P(f)}$ is hyperbolic if $f\in {\mathcal{F}}_{\mu }(M)$. Thus, we have the following theorem.

Theorem 1.1 [[3], Theorem 1.1]

Every diffeomorphism in ${\mathcal{F}}_{\mu }(M)$ is Anosov.

Let $int{\mathcal{OS}}_{\mu }(M)$ denote the $C^1$-interior of the set of volume preserving diffeomorphisms in ${Diff}_{\mu }(M)$ satisfying the orbital shadowing property. In [7], the authors proved that the $C^1$-interior of the set of diffeomorphisms having the orbital shadowing property coincides with the set of structurally stable diffeomorphisms. Note that if a diffeomorphism satisfies structurally stable then it is not Anosov in general. But the converse is true. Finally, we prove the following theorem.

Theorem 1.2 The set ${\mathcal{AN}}_{\mu }(M)$ of Anosov diffeomorphisms in ${Diff}_{\mu }(M)$ coincides with the $C^1$-interior of the set of diffeomorphisms in ${Diff}_{\mu }(M)$ with orbital shadowing; that is, ${\mathcal{AN}}_{\mu }(M)=int{\mathcal{OS}}_{\mu }(M)$.

2 Proof of Theorem 1.2

Remark 2.1 Let $f\in {Diff}_{\mu }^1(M)$. From Moser’s theorem (see [8]), we can find a smooth conservative change of coordinates ${\phi }_x:U(x)\to T_{x}M$ such that ${\phi }_x(x)=0$, where $U(x)$ is a small neighborhood of $x\in M$.

Recall that f has the orbital shadowing property on Λ (Λ is orbitally shadowable) if for any $ϵ>0$, there is $\delta >0$ such that for any δ-pseudo orbit $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda }$ of f, there is ∈M such that

$${\mathcal{O}}_f(y)\subset B_{ϵ}(\xi )\text{and}\xi \subset B_{ϵ}({\mathcal{O}}_f(y)).$$

Notice that in this definition, only δ-pseudo orbits of f are contained in Λ, but the shadowing point $y\in M$ is not necessarily contained in Λ. To prove our result, we use Franks’ lemma which is proved in [[9], Proposition 7.4].

Lemma 2.2 Let $f\in {Diff}_{\mu }^1(M)$, and $\mathcal{U}$ be a $C^1$-neighborhood of f in ${Diff}_{\mu }^1(M)$. Then there exist a $C^1$-neighborhood ${\mathcal{U}}_0\subset \mathcal{U}$ of f and $ϵ>0$ such that if $g\in {\mathcal{U}}_0$, any finite f-invariant set $E=\{x_1,\dots ,x_m\}$, any neighborhood U of E, and any volume-preserving linear maps $L_j:T_{x_j}M\to T_{g(x_j)}M$ with $\parallel L_j-D_{x_j}g\parallel \le ϵ$ for all $j=1,\dots ,m$, there is a conservative diffeomorphism $g_1\in \mathcal{U}$ coinciding with f on E and out of U, and $D_{x_j}{g}_1=L_j$ for all $j=1,\dots ,m$.

We introduce the notion of normally hyperbolic which was founded in [10]. Let $V\subset M$ be an invariant submanifold of $f\in {Diff}_{\mu }(M)$. We say that V is normally hyperbolic if there is a splitting $T_{V}M=TV\oplus N^s\oplus N^u$ such that

• the splitting depends continuously on $x\in V$,

• $D_{x}f(N_x^{\sigma })=N_{f(x)}^{\sigma }$ ($\sigma =s,u$) for all $x\in V$,

• there are constants $C>0$ and $0<\lambda <1$ such that for every unit vector $x\in T_{x}V$, $v^s\in N_x^s$ and $v^u\in N_x^u$ ($x\in V$), we have

  $$\parallel D_x{f}^n\left(v^s\right)\parallel \le C{\lambda }^n\parallel D_x{f}^n(v)\parallel \text{and}\parallel D_x{f}^n\left(v^u\right)\parallel \ge C^{-1}{\lambda }^{-1}\parallel D_x{f}^n(v)\parallel$$

for all $n\ge 0$.

Proposition 2.3 If $f\in int{\mathcal{OS}}_{\mu }(M)$, then every periodic point of f is hyperbolic.

Proof Take $f\in int{\mathcal{OS}}_{\mu }(M)$, and $\mathcal{U}(f)$ is a $C^1$-neighborhood of $f\in {Diff}_{\mu }(M)$. Let $ϵ>0$ and $\mathcal{V}(f)\subset {\mathcal{U}}_0(f)$ be the number and $C^1$-neighborhood of f corresponding to $\mathcal{U}(f)$ given by Lemma 2.2. To derive a contradiction, we assume that there exists a nonhyperbolic periodic point $p\in P(g)$ for some $g\in \mathcal{V}(f)$. To simplify the notation in the proof, we may assume that $g(p)=p$. Then there is at least one eigenvalue λ of $D_{p}g$ such that $|\lambda |=1$.

By making use of Lemma 2.2, we linearize g at p using Moser’s theorem; that is, by choosing $\alpha >0$ sufficiently small, we construct $g_1$ $C^1$-nearby g such that

$$g_1(x)=\{\begin{array}{cc}{\phi }_p^{-1}\circ D_{p}g\circ {\phi }_p(x)\hfill & \text{if }x\in B_{\alpha }(p),\hfill \\ g(x)\hfill & \text{if }x\notin B_{4\alpha }(p).\hfill \end{array}$$

Then $g_1(p)=g(p)=p$.

First, we may assume that $\lambda \in \mathbb{R}$ with $\lambda =1$. Let v be the associated non-zero eigenvector such that $\parallel v\parallel =\alpha /4$. Then we can get a small arc

$${\mathcal{I}}_v=\{tv:-1\le t\le 1\}\subset {\phi }_p(B_{\alpha }(p)).$$

Take $ϵ=\alpha /8$. Let $0<\delta <ϵ$ be a number of the orbital shadowing property of $g_1$ corresponding to ϵ. Then by our construction of $g_1$,

$${\phi }_p^{-1}({\mathcal{I}}_v)\subset B_{\alpha }(p).$$

Then it is clear that ${\phi }_p^{-1}({\mathcal{I}}_v)$ is normally hyperbolic for $g_1$. Put ${\mathcal{J}}_p={\phi }_p^{-1}({\mathcal{I}}_v)$. Given a constant $\delta >0$, we construct a δ-pseudo orbit $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset {\mathcal{J}}_p$ as follows. For fixed $k\in \mathbb{Z}$, choose distinct points $x_0=p,x_1,x_2,\dots ,x_k$ in ${\mathcal{J}}_p$ such that

• (a) $d(x_i,x_{i+1})<\delta$ for $i=0,1,\dots ,k-1$,

• (b) $d(x_{-i-1},x_{-i})<\delta$ for $i=0,\dots ,k-1$,

• (c) $x_0=x$ and $d(x_{-k},x_k)>2ϵ$.

Now, we define $\xi ={\{x_i\}}_{i\in \mathbb{Z}}$ by $x_{ki+j}=x_j$ for $i\in \mathbb{Z}$ and $j=-k-1,-k-2,\dots ,-1,0,1,\dots ,k-1$. Since $g_1$ has the orbital shadowing property, $g_1{|}_{{\mathcal{J}}_p}$ must have the orbital shadowing property. Thus, we can find a point $y\in M$ such that $\xi \subset B_{ϵ}({\mathcal{O}}_{g_1}(y))$, and ${\mathcal{O}}_{g_1}(y)\subset B_{ϵ}(\xi )$. For any $v\in {\mathcal{I}}_v$, ${\phi }_p^{-1}(v)\in {\mathcal{J}}_p\subset B_{\alpha }(p)$ and

$$g_1({\phi }_p^{-1}(v))={\phi }_p^{-1}\circ D_{p}g\circ {\phi }_p({\phi }_p^{-1}(v)).$$

Then $g_1({\phi }_p^{-1}(v))={\phi }_p^{-1}(v)$. Thus, $g_1^l({\mathcal{J}}_p)={\mathcal{J}}_p$ for some $l>0$. Now, we show that if ${\mathcal{J}}_p$ is normally hyperbolic for $g_1$, then the shadowing points belong to ${\mathcal{J}}_p$. Assume that there is a shadowing point $y\in M\setminus {\mathcal{J}}_p$. Then by the hyperbolicity, there are $l,k\in \mathbb{Z}$ such that $d(g_1^l(y),x_k)>ϵ$, where $x_k\in \xi ={\{x_i\}}_{i\in \mathbb{Z}}$. This is a contradiction since $g_1{|}_{{\mathcal{J}}_p}$ has the orbital shadowing property. Thus, if ${\mathcal{J}}_p$ is normally hyperbolic for $g_1$, then the shadowing point belongs to ${\mathcal{J}}_p$. Since $g_1{|}_{{\mathcal{J}}_p}$ has the orbital shadowing property, from the above facts, we have $y\in {\mathcal{J}}_p$. But $g_1^l({\mathcal{J}}_p)={\mathcal{J}}_p$ and so $g_1^l{|}_{{\mathcal{J}}_p}$ is the identity map. Then $g_1^l{|}_{{\mathcal{J}}_p}$ does not have the orbital shadowing property. Thus, $g_1{|}_{{\mathcal{J}}_p}$ also does not have the orbital shadowing property.

Finally, if $\lambda \in \mathbb{C}$, then to avoid the notational complexity, we may assume that $g(p)=p$. As in the first case, by Lemma 2.2, there are $\alpha >0$ and $g_1\in \mathcal{V}(f)$ such that $g_1(p)=g(p)=p$ and

$$g_1(x)=\{\begin{array}{cc}{\phi }_p^{-1}\circ D_{p}g\circ {\phi }_p(x)\hfill & \text{if }x\in B_{\alpha }(p),\hfill \\ g(x)\hfill & \text{if }x\notin B_{4\alpha }(p).\hfill \end{array}$$

With a $C^1$-small modification of the map $D_{p}g$, we may suppose that there is $l>0$ (the minimum number) such that $D_p{g}^l(v)=v$ for any $v\in {\phi }_p(B_{\alpha }(p))\subset T_{p}M$. Then we can go on with the previous argument in order to reach the same contradiction. Thus, every periodic point of $f\in int{\mathcal{OS}}_{\mu }(M)$ is hyperbolic. □

End of the proof of Theorem 1.2 Let $f\in int{\mathcal{OS}}_{\mu }(M)$. By Proposition 2.3, we see that $f\in {\mathcal{F}}_{\mu }(M)$. Thus, by Theorem 1.1, f is Anosov. □