Symplectic diffeomorphisms with inverse shadowing

Abstract

Let f be a symplectic diffeomorphism of a closed $C^{\mathrm{\infty }}$ 2n-dimensional Riemannian manifold M. In this paper, we prove the equivalence between the following conditions:

(a) f belongs to the $C^1$-interior of the set of symplectic diffeomorphisms satisfying the inverse shadowing property with respect to the continuous methods,

(b) f belongs to the $C^1$-interior of the set of symplectic diffeomorphisms satisfying the orbital inverse shadowing property with respect to the continuous methods,

(c) f is Anosov.

This result extends Bessa and Rocha’s result (Appl. Math. Lett. 25:163-165, 2012).

MSC:37C15, 37C50.

1 Introduction

Let M be a closed $C^{\mathrm{\infty }}$ 2n-dimensional manifold with Riemannian structure and endowed with a symplectic form ω, and let ${Diff}_{\omega }(M)$ be the set of symplectomorphisms, that is, of diffeomorphisms f defined on M and such that

$${\omega }_x(v_1,v_2)={\omega }_{f(x)}(D_{x}f(v_1),D_{x}f(v_2)),$$

for $x\in M$ and $v_1,v_2\in T_{x}M$. Consider this space endowed with the $C^1$ Whitney topology. It is well known that ${Diff}_{\omega }(M)$ is a subset of all $C^1$ volume-preserving diffeomorphisms. Denote by d the distance on M induced from a Riemannian metric $\parallel \cdot \parallel$ on the tangent bundle TM. By the theorem of Darboux [[1], Theorem 1.8], there is an atlas $\{{\phi }_i^j:U_i\to {\mathbb{R}}^{2n}\}$, where $U_i$ is an open set of M satisfying ${\phi }_i^{\ast }{\omega }_0=\omega$ with ${\omega }_0={\sum }_{i=0}^{n}d{y}_i\wedge d{y}_{n+i}$.

The notion of the inverse shadowing property which is a ‘dual’ notion of the shadowing property. Inverse shadowing property was introduced by Corless and Pilyugin in [2], and the qualitative theory of dynamical systems with the property was developed by various authors (see [2–4]).

Now, we introduce the inverse shadowing property and some results for the inverse shadowing. Let $M^{\mathbb{Z}}$ be the space of all two-sided sequences $\xi =\{x_n:n\in \mathbb{Z}\}$ with elements $x_n\in M$, endowed with the product topology. Let $f:M\to M$ be a symplectic diffeomorphism. For a fixed $\delta >0$, let ${\mathrm{\Phi }}_f(\delta )$ denote the set of all δ-pseudo orbits of f. A mapping $\phi :M\to {\mathrm{\Phi }}_f(\delta )\subset M^{\mathbb{Z}}$ is said to be a δ-method for f if $\phi {(x)}_0=x$, and each $\phi (x)$ is a δ-pseudo orbit of f through x, where $\phi {(x)}_0$ denotes the 0th component of $\phi (x)$. For convenience, write $\phi (x)$ for ${\{\phi {(x)}_k\}}_{k\in \mathbb{Z}}$. The set of all δ-methods for f will be denoted by ${\mathcal{T}}_0(f,\delta )$. Say that φ is continuous δ-method for f if φ is continuous. The set of all continuous δ-methods for f will be denoted by ${\mathcal{T}}_c(f,\delta )$. If $g:M\to M$ is a homeomorphism with $d_0(f,g)<\delta$ then g induces a continuous δ-method ${\phi }_g$ for f by defining

$${\phi }_g(x)=\{g^n(x):n\in \mathbb{Z}\}.$$

Let ${\mathcal{T}}_h(f,\delta )$ denote the set of all continuous δ-methods ${\phi }_g$ for f which are induced by a homeomorphism $g:M\to M$ with $d_0(f,g)<\delta$, where $d_0$ is the usual $C^0$-metric. Let ${\mathcal{T}}_d(f,\delta )$ denote by the set of all continuous δ-methods ${\phi }_g$ for f, which are induced by $g\in {Diff}_{\omega }(M)$ with $d_1(f,g)<\delta$. Then clearly we know that

$${\mathcal{T}}_d(f)\subset {\mathcal{T}}_h(f)\subset {\mathcal{T}}_c(f)\subset {\mathcal{T}}_0(f),$$

${\mathcal{T}}_{\alpha }(f)={\bigcup }_{\delta >0}{\mathcal{T}}_{\alpha }(f,\delta )$, $\alpha =0,c,h,d$. We say that f has the inverse shadowing property with respect to the class ${\mathcal{T}}_{\alpha }(f)$, $\alpha =0,c,h,d$, if for any $ϵ>0$ there exists $\delta >0$ such that for any δ-method $\phi \in {\mathcal{T}}_{\alpha }(f,\delta )$, and for a point $x\in M$ there is a point $y\in M$ such that

$$d(f^k(x),{\phi }_g{(y)}_k)<ϵ,k\in \mathbb{Z}.$$

We say that f has the orbital inverse shadowing property with respect to the class ${\mathcal{T}}_{\alpha }(f)$, $\alpha =0,c,h,d$, if for any $ϵ>0$ there exists $\delta >0$ such that for any δ-method $\phi \in {\mathcal{T}}_{\alpha }(f,\delta )$, and for a point $x\in M$ there is a point $y\in M$ such that

$$d_H(\overline{{\mathcal{O}}_f(x)},\overline{{\mathcal{O}}_{{\phi }_g}(y)})<ϵ,$$

where $d_H$ is the Hausdorff metric, and $\overline{A}$ is the closure of A. We denote by ${\mathcal{IS}}_{\omega ,\alpha }(M)$ the set of symplectic diffeomorphisms on M with the inverse shadowing property with respect to the class ${\mathcal{T}}_{\alpha }$ and denote by ${\mathcal{OIS}}_{\omega ,\alpha }(M)$ the set of symplectic diffeomorphisms on M with the orbital inverse shadowing property respect to the class ${\mathcal{T}}_{\alpha }$, where $\alpha =a,c,h,d$. Let $int{\mathcal{IS}}_{\omega ,\alpha }(M)$ be the $C^1$-interior of the set of symplectic diffeomorphisms on M with the inverse shadowing property respect to the class ${\mathcal{T}}_{\alpha }$, and let $int{\mathcal{OIS}}_{\omega ,\alpha }(M)$ be the $C^1$-interior of the set of symplectic diffeomorphisms on M with the orbital inverse shadowing property respect to the class ${\mathcal{T}}_{\alpha }$, where $\alpha =a,c,h,d$. Note that f has the inverse shadowing property if and only if $f^n$ has the inverse shadowing property, for all $n\in \mathbb{Z}$. Lee [3] showed that a diffeomorphism belongs to the $C^1$-interior of the set of diffeomorphisms having the inverse shadowing property with respect to the ${\mathcal{T}}_d(f)$ if and only if it is structurally stable. Pilyugin [4] proved that a diffeomorphism belongs to the $C^1$-interior of the set of f diffeomorphisms having the inverse shadowing property with respect to the class ${\mathcal{T}}_c(f)$ if and only if it is structurally stable.

The notion of topological stability was introduced by Walters [5], and he showed that every Anosov diffeomorphism is topological stable. In [6], Nitecki proved that if f satisfies both Axiom A and the strong transversality condition, then it is topological stable. We say that f is topological stable if for any $ϵ>0$, there is $\delta >0$ such that for any $g\in Diff(M)$, δ-$C^0$-closed to f, there is a continuous map $h:M\to M$ satisfying $h\circ g=f\circ h$ and $d(f(x),x)<ϵ$ for all $x\in M$. Moriyasu [7] showed that the $C^1$-interior of the set of all topologically stable diffeomorphisms is characterized as the set of $C^1$-structurally stable. Very recently, Bessa and Rocha [8] proved that if a symplectic diffeomorphism belongs to the $C^1$-interior of the set of topologically stable, then the diffeomorphism is Anosov.

Remark 1.1 By the definition of the inverse shadowing, we have the following implication: topological stability ⇒ inverse shadowing property with respect to the continuous method ${\mathcal{T}}_d\Rightarrow$ orbital inverse shadowing property with respect to the continuous method ${\mathcal{T}}_d$.

From the above remark, we know that our result is a slight generalization of the main theorem in [8]. In this paper, we omit the phrase ‘with respect to the class ${\mathcal{T}}_d$’ for simplicity. So, we say that f has the inverse shadowing property means that f has the inverse shadowing property with respect to the class ${\mathcal{T}}_d$.

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 $n\ge 0$. If $\mathrm{\Lambda }=M$ then f is Anosov. We define the set ${\mathcal{F}}_{\omega }(M)$ as the set of diffeomorphisms $f\in {Diff}_{\omega }(M)$ which have a $C^1$-neighborhood $\mathcal{U}(f)\subset {Diff}_{\omega }(M)$ such that if for any $g\in \mathcal{U}(f)$, every periodic point of g is hyperbolic. Then we can see the following.

Lemma 1.2 [6]

If $f\in {\mathcal{F}}_{\omega }(M)$, then f is Anosov.

Note that ${\mathcal{F}}_{\omega }(M)\subset \mathcal{F}(M)$ (see [[9], Corollary 1.2]). By a result of Newhouse [10], if the symplectic diffeomorphisms is not Anosov then 1-elliptic points can be created by an arbitrary small $C^1$-perturbations of the symplectic diffeomorphism.

In this paper, we investigate the cases when a symplectic diffeomorphism f is in $C^1$-interior inverse shadowing property with respect to the class ${\mathcal{T}}_d(f)$, then it is Anosov. Let $int{\mathcal{IS}}_{\omega }(M)$ be denoted the set of symplectic diffeomorphisms in ${Diff}_{\omega }(M)$ satisfying the inverse shadowing property with respect to the class ${\mathcal{T}}_d$, and let $int{\mathcal{OIS}}_{\omega }(M)$ be denoted the set of symplectic diffeomorphisms in ${Diff}_{\omega }(M)$ satisfying the orbital inverse shadowing property with respect to the class ${\mathcal{T}}_d$ when we mention the inverse shadowing property (resp. orbital inverse shadowing property); that is, the ‘inverse shadowing property (resp. orbital inverse shadowing property)’ implies the ‘inverse shadowing property (resp. orbital inverse shadowing property) with respect to the class ${\mathcal{T}}_d$’. Now we are in position to state the theorem of our paper.

Theorem 1.3 Let $f\in {Diff}_{\omega }(M)$. Then

$$int{\mathcal{IS}}_{\omega }(M)=int{\mathcal{OIS}}_{\omega }(M)={\mathcal{AN}}_{\omega }(M),$$

where ${\mathcal{AN}}_{\omega }(M)$ is the set of Anosov symplectic diffeomorphisms in ${Diff}_{\omega }(M)$.

2 Proof of Theorem 1.3

Let M be as before, and let $f\in {Diff}_{\omega }(M)$. Then the following is symplectic version of Franks’ lemma.

Lemma 2.1 [[11], Lemma 5.1]

Let $f\in {Diff}_{\omega }(M)$ and $\mathcal{U}(f)$ be given. Then there are ${\delta }_0>0$ and ${\mathcal{U}}_0(f)\subset \mathcal{U}(f)$ such that for any $g\in {\mathcal{U}}_0(f)$, a finite set $\{x_1,x_2,\dots ,x_n\}$, a neighborhood U of $\{x_1,x_2,\dots ,x_n\}$ and symplectic maps $L_i:T_{x_i}M\to T_{g(x_i)}M$ satisfying $\parallel L_i-Dg(x_i)\parallel <{\delta }_0$ for all $1\le i\le n$, there are ${ϵ}_0>0$ and $\tilde{g}\in \mathcal{U}(f)$ such that

• (a) $\tilde{g}(x)=g(x)$ if $x\in M\setminus U$,

• (b) $\tilde{g}(x)={\phi }_{g(x_i)}\circ L_i\circ {\phi }_{x_i}^{-1}(x)$ if $x\in B_{{ϵ}_0}(x_i)$,

where $B_{{ϵ}_0}(x_i)$ is the ${ϵ}_0$-neighborhood of $x_i$.

A periodic point for f is a point $p\in M$ such that $f^{\pi (p)}(p)=p$, where $\pi (p)$ is the minimum period of p. We say that a periodic point p is elliptic if $D_p{f}^{\pi (p)}$ has one nonreal eigenvalue of norm one. We say that a periodic point p is a k-elliptic periodic point if for a periodic point p of period $\pi (p)$ the tangent map $D_p{f}^{\pi (p)}$ has exactly 2k simple nonreal eigenvalues of norm 1 and the other ones have norm different from 1. In dimension 2, then 1-elliptic periodic points are actually elliptic. We say that p is hyperbolic if $D{f}^{\pi (p)}$ has no norm one eigenvalue. The following facts are enough to prove Theorem 1.3 by Lemma 1.2.

Lemma 2.2 If $f\in {Diff}_{\omega }(M)$ has the orbital inverse shadowing property, then f is not the identity map.

Proof Suppose, by contradiction, that f is the identity map. Take $ϵ=1/4$, and let $0<\delta <ϵ$ be the number of the orbital inverse shadowing property of f. Since f is the identity map, we defined $g\in {Diff}_{\omega }(M)$ by $g(x)=(x_1+\delta /2,x_2,\dots ,x_{2n})$, for $x=(x_1,x_2,\dots ,x_{2n})\in M$. Then $g\in {\mathcal{T}}_d(f)$. Since f has the orbital inverse shadowing property, there is $y\in M$ such that for any $x\in M$,

$$d(x,y)<ϵ\text{and}{d}_H(\overline{{\mathcal{O}}_f(y)},\overline{{\mathcal{O}}_g(x)})<ϵ.$$

Since f is the identity map and g is an increasing map, there is $k\in \mathbb{N}$ such that $d(y,g^k(x))>ϵ$. Thus, by the definition of the Hausdorff metric,

$$d_H(\overline{{\mathcal{O}}_f(y)},\overline{{\mathcal{O}}_g(x)})>ϵ.$$

This is a contradiction. □

Lemma 2.3 If $f\in int{\mathcal{IS}}_{\omega }(M)$, then every periodic point of f is hyperbolic.

Proof Let $f\in int{\mathcal{IS}}_{\omega }(M)$, and let ${\mathcal{U}}_0(f)$ be a $C^1$-neighborhood of f. Suppose that there is a $g\in {\mathcal{U}}_0(f)$ such that g have a periodic elliptic point p. To simplify, we may assume that $g(p)=p$. Then $D_{p}g$ has n pairs of nonreal eigenvalues, that is, $|a_i|=|{\overline{a}}_i|=1$, $i=1,\dots ,n$ with $T_{p}M=E_p^{c_i}\oplus \cdots \oplus E_p^{c_n}$ and $dim{E}_p^{c_i}=2$, $i=1,\dots ,n$. By Lemma 2.1, there are $\alpha >0$ and $g_1\in \mathcal{U}(f)$ such that

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

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

First, we consider the case $E_p^{c_1}(\alpha )$ other case is similar. Since p is nonhyperbolic for $g_1$, by our construction, we may assume that there is $l>0$ such that $D_p{g}_1^l(v)=v$ for any $v\in E_p^{c_1}(\alpha )\cap {\phi }_p^{-1}(B_{\alpha }(p))$. Take $v\in E_p^{c_1}(\alpha )$ such that $\parallel v\parallel =\alpha /4$. Then we can find a small arc ${\mathcal{I}}_p={\phi }_p(\{tv:-\alpha /4\le t\le \alpha /4\})\subset B_{\alpha }(p)$ such that

• (i) $g_1^i({\mathcal{I}}_p)\cap g_1^j({\mathcal{I}}_p)=\mathrm{\varnothing }$ if $0\le i\ne j\le l-1$, and

• (ii) $g_1^l({\mathcal{I}}_p)={\mathcal{I}}_p$, that is, $g_1^l{|}_{{\mathcal{I}}_p}$ is the identity map.

Then we can choose $0<\eta <\alpha /4$ sufficiently small such that $B_{\eta }(g_1^i({\mathcal{I}}_p))\cap B_{\eta }(g_1^j({\mathcal{I}}_p))=\mathrm{\varnothing }$ for all $1\le i\ne j\le l-1$. Take $ϵ=\eta /4$, and let $0<\delta <ϵ$ be the number of the definition of the inverse shadowing property of $g_1$ for ϵ. For the $\delta >0$, we can define ${\mathcal{T}}_d(g_1)$-method as follows: Let $\psi \in {Diff}_{\omega }(M)$ be such that p is a hyperbolic periodic point for ψ with $\psi (p)=p$ and $d(g_1,\psi )<\delta$. Then $\psi \in {\mathcal{T}}_d(g_1)$. To simplicity, we may assume that $g_1^l=g_1$. Take $y\in B_{4ϵ}(p)\cap {\mathcal{I}}_p$ such that $d(y,p)=2ϵ$, and $g_1^n(y)=y$ for all $n\in \mathbb{Z}$. Since $g_1$ has the inverse shadowing property, we can see that for any $z\in M$

$$d(g_1^n(y),{\phi }_{\psi }{(z)}_n)=d(g_1^n(y),{\psi }^n(z))<ϵ$$

for all $n\in \mathbb{Z}$. For any $z\in B_{ϵ}(p)$, if $z=p$, then since $g_1{|}_{{\mathcal{I}}_p}$ is the identity map, it is clear that $g_1$ does not have the inverse shadowing property. If $z\ne p$, then since p is a hyperbolic periodic point for ψ, there is $k\in \mathbb{Z}$ such that

$$d(g_1^k(y),{\psi }^k(z))=d(y,{\psi }^k(z))>ϵ.$$

This is a contradiction.

Finally, we may assume that there are $m_i$ (the minimum numbers) such that $D_p{g}_1^{m_i}(v)=v$ for any $v\in E_p^{c_i}(\alpha )\cap {\phi }_p^{-1}(B_{\alpha }(p))$, $i=2,\dots ,n$. Let $K=lcm\{m_i:i=2,\dots ,n\}$. Here, lcm is the lowest common multiple.

To simplify, we assume that $g_2=g_1^K$. Since $D_p{g}_1^{m_i}(v)=v$ for any $v\in E_p^{c_i}(\alpha )\cap {\phi }_p^{-1}(B_{\alpha }(p))$, $i=2,\dots ,n$, by the above argument, there exists a small arc ${\mathcal{I}}_p\subset B_{\alpha }(p)$ such that $g_2^l{|}_{{\mathcal{I}}_p}$ is the identity map, for some $l>0$. Then we can find $\psi \in {\mathcal{T}}_d(g_2)$ such that p is a hyperbolic periodic point for ψ with $\psi (p)=p$, $d(g_2,\psi )<\delta$. By the inverse shadowing property for $g_2$, there exists $y\in B_{4ϵ}(p)\cap {\mathcal{I}}_p$ such that

$$d(y,p)=2ϵ\text{and}{g}_2^{ln}(y)=y$$

for all $n\in \mathbb{Z}$. Then there exists $j\in \mathbb{Z}$ such that

$$d(g_2^j(y),{\psi }^j(z))=d(y,{\psi }^j(z))>ϵ.$$

This is a contradiction. Thus, every periodic point of $f\in int{\mathcal{IS}}_{\omega }(M)$ is hyperbolic. □

Lemma 2.4 If $f\in int{\mathcal{OIS}}_{\omega }(M)$, then every periodic point of f is hyperbolic.

Proof Let $f\in int{\mathcal{OIS}}_{\omega }(M)$. Then as in the proof of Lemma 2.3 and Lemma 2.2, we can obtain a contradiction. □

Proof of Theorem 1.3 Let $f\in int{\mathcal{IS}}_{\omega }(M)$, and let $f\in int{\mathcal{OIS}}_{\omega }(M)$. Suppose that $f\notin {\mathcal{F}}_{\omega }(M)$. Then there is $g\in {\mathcal{U}}_0(f)\subset \mathcal{U}(f)$ such that g have a periodic elliptic point p. By Lemma 2.3 and Lemma 2.4, g does not have a periodic elliptic point. This is a contradiction. Thus, if $f\in int{\mathcal{IS}}_{\omega }(M)$ or $f\in int{\mathcal{OIS}}_{\omega }(M)$ then $f\in {\mathcal{F}}_{\omega }(M)$. By Lemma 1.2, f is Anosov. □