# Symplectic diffeomorphisms with inverse shadowing

- Manseob Lee
^{1}Email author

**2013**:174

https://doi.org/10.1186/1029-242X-2013-174

© Lee; licensee Springer 2013

**Received: **5 January 2013

**Accepted: **2 April 2013

**Published: **16 April 2013

## Abstract

Let *f* be a symplectic diffeomorphism of a closed ${C}^{\mathrm{\infty}}$ 2*n*-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.

## Keywords

## 1 Introduction

*M*be a closed ${C}^{\mathrm{\infty}}$ 2

*n*-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

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]).

*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

*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

*f*has the

*inverse shadowing property*with respect to the class ${\mathcal{T}}_{\alpha}(f)$, $\alpha =0,c,h,d$, if for any $\u03f5>0$ 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

*f*has the

*orbital inverse shadowing property*with respect to the class ${\mathcal{T}}_{\alpha}(f)$, $\alpha =0,c,h,d$, if for any $\u03f5>0$ 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

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 $\u03f5>0$, 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 $d(f(x),x)<\u03f5$ 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 ${\mathcal{T}}_{d}$.

*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

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*

*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*${\u03f5}_{0}>0$

*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*$x\in {B}_{{\u03f5}_{0}}({x}_{i})$,

*where* ${B}_{{\u03f5}_{0}}({x}_{i})$ *is the* ${\u03f5}_{0}$-*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 2*k* 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 $\u03f5=1/4$, and let $0<\delta <\u03f5$ 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 $x\in M$,

*f*is the identity map and

*g*is an increasing map, there is $k\in \mathbb{N}$ such that $d(y,{g}^{k}(x))>\u03f5$. Thus, by the definition of the Hausdorff metric,

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

Then $g(p)={g}_{1}(p)=p$.

*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.

*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 $y\in {B}_{4\u03f5}(p)\cap {\mathcal{I}}_{p}$ such that $d(y,p)=2\u03f5$, 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 $z\in M$

*p*is a hyperbolic periodic point for

*ψ*, there is $k\in \mathbb{Z}$ such that

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.

*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 $y\in {B}_{4\u03f5}(p)\cap {\mathcal{I}}_{p}$ such that

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. □

## Declarations

### Acknowledgements

We wish to thank the referee for carefully reading the manuscript and providing us many good suggestions. This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology, Korea (No. 2011-0007649).

## Authors’ Affiliations

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