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Symplectic diffeomorphisms with inverse shadowing

Abstract

Let f be a symplectic diffeomorphism of a closed C 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 2n-dimensional manifold with Riemannian structure and endowed with a symplectic form ω, and let Diff ω (M) be the set of symplectomorphisms, that is, of diffeomorphisms f defined on M and such that

ω x ( v 1 , v 2 )= ω f ( x ) ( D x f ( v 1 ) , D x f ( v 2 ) ) ,

for xM and v 1 , v 2 T x M. Consider this space endowed with the C 1 Whitney topology. It is well known that Diff ω (M) is a subset of all C 1 volume-preserving diffeomorphisms. Denote by d the distance on M induced from a Riemannian metric on the tangent bundle TM. By the theorem of Darboux [[1], Theorem 1.8], there is an atlas { φ i j : U i R 2 n }, where U i is an open set of M satisfying φ i ω 0 =ω with ω 0 = i = 0 n d y i 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 [24]).

Now, we introduce the inverse shadowing property and some results for the inverse shadowing. Let M Z be the space of all two-sided sequences ξ={ x n :nZ} with elements x n M, endowed with the product topology. Let f:MM be a symplectic diffeomorphism. For a fixed δ>0, let Φ f (δ) denote the set of all δ-pseudo orbits of f. A mapping φ:M Φ f (δ) M Z is said to be a δ-method for f if φ ( x ) 0 =x, and each φ(x) is a δ-pseudo orbit of f through x, where φ ( x ) 0 denotes the 0th component of φ(x). For convenience, write φ(x) for { φ ( x ) k } k Z . The set of all δ-methods for f will be denoted by T 0 (f,δ). 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 T c (f,δ). If g:MM is a homeomorphism with d 0 (f,g)<δ then g induces a continuous δ-method φ g for f by defining

φ g (x)= { g n ( x ) : n Z } .

Let T h (f,δ) denote the set of all continuous δ-methods φ g for f which are induced by a homeomorphism g:MM with d 0 (f,g)<δ, where d 0 is the usual C 0 -metric. Let T d (f,δ) denote by the set of all continuous δ-methods φ g for f, which are induced by g Diff ω (M) with d 1 (f,g)<δ. Then clearly we know that

T d (f) T h (f) T c (f) T 0 (f),

T α (f)= δ > 0 T α (f,δ), α=0,c,h,d. We say that f has the inverse shadowing property with respect to the class T α (f), α=0,c,h,d, if for any ϵ>0 there exists δ>0 such that for any δ-method φ T α (f,δ), and for a point xM there is a point yM such that

d ( f k ( x ) , φ g ( y ) k ) <ϵ,kZ.

We say that f has the orbital inverse shadowing property with respect to the class T α (f), α=0,c,h,d, if for any ϵ>0 there exists δ>0 such that for any δ-method φ T α (f,δ), and for a point xM there is a point yM such that

d H ( O f ( x ) ¯ , O φ g ( y ) ¯ ) <ϵ,

where d H is the Hausdorff metric, and A ¯ is the closure of A. We denote by IS ω , α (M) the set of symplectic diffeomorphisms on M with the inverse shadowing property with respect to the class T α and denote by OIS ω , α (M) the set of symplectic diffeomorphisms on M with the orbital inverse shadowing property respect to the class T α , where α=a,c,h,d. Let int IS ω , α (M) be the C 1 -interior of the set of symplectic diffeomorphisms on M with the inverse shadowing property respect to the class T α , and let int OIS ω , α (M) be the C 1 -interior of the set of symplectic diffeomorphisms on M with the orbital inverse shadowing property respect to the class T α , where α=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 nZ. 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 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 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 δ>0 such that for any gDiff(M), δ- C 0 -closed to f, there is a continuous map h:MM satisfying hg=fh and d(f(x),x)<ϵ for all xM. 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 T d orbital inverse shadowing property with respect to the continuous method 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 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 T d .

We say that Λ is hyperbolic if the tangent bundle T Λ M has a Df-invariant splitting E s E u and there exist constants C>0 and 0<λ<1 such that

D x f n | E x s C λ n and D x f n | E x u C λ n

for all xΛ and n0. If Λ=M then f is Anosov. We define the set F ω (M) as the set of diffeomorphisms f Diff ω (M) which have a C 1 -neighborhood U(f) Diff ω (M) such that if for any gU(f), every periodic point of g is hyperbolic. Then we can see the following.

Lemma 1.2 [6]

If f F ω (M), then f is Anosov.

Note that F ω (M)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 T d (f), then it is Anosov. Let int IS ω (M) be denoted the set of symplectic diffeomorphisms in Diff ω (M) satisfying the inverse shadowing property with respect to the class T d , and let int OIS ω (M) be denoted the set of symplectic diffeomorphisms in Diff ω (M) satisfying the orbital inverse shadowing property with respect to the class 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 T d ’. Now we are in position to state the theorem of our paper.

Theorem 1.3 Let f Diff ω (M). Then

int IS ω (M)=int OIS ω (M)= AN ω (M),

where AN ω (M) is the set of Anosov symplectic diffeomorphisms in Diff ω (M).

2 Proof of Theorem 1.3

Let M be as before, and let f Diff ω (M). Then the following is symplectic version of Franks’ lemma.

Lemma 2.1 [[11], Lemma 5.1]

Let f Diff ω (M) and U(f) be given. Then there are δ 0 >0 and U 0 (f)U(f) such that for any g U 0 (f), a finite set { x 1 , x 2 ,, x n }, a neighborhood U of { x 1 , x 2 ,, x n } and symplectic maps L i : T x i M T g ( x i ) M satisfying L i Dg( x i )< δ 0 for all 1in, there are ϵ 0 >0 and g ˜ U(f) such that

  1. (a)

    g ˜ (x)=g(x) if xMU,

  2. (b)

    g ˜ (x)= φ g ( x i ) L i φ x i 1 (x) if x 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 pM such that f π ( p ) (p)=p, where π(p) is the minimum period of p. We say that a periodic point p is elliptic if D p f π ( 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 π(p) the tangent map D p f π ( 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 π ( 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 Diff ω (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<δ<ϵ be the number of the orbital inverse shadowing property of f. Since f is the identity map, we defined g Diff ω (M) by g(x)=( x 1 +δ/2, x 2 ,, x 2 n ), for x=( x 1 , x 2 ,, x 2 n )M. Then g T d (f). Since f has the orbital inverse shadowing property, there is yM such that for any xM,

d(x,y)<ϵand d H ( O f ( y ) ¯ , O g ( x ) ¯ ) <ϵ.

Since f is the identity map and g is an increasing map, there is kN such that d(y, g k (x))>ϵ. Thus, by the definition of the Hausdorff metric,

d H ( O f ( y ) ¯ , O g ( x ) ¯ ) >ϵ.

This is a contradiction. □

Lemma 2.3 If fint IS ω (M), then every periodic point of f is hyperbolic.

Proof Let fint IS ω (M), and let U 0 (f) be a C 1 -neighborhood of f. Suppose that there is a g 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 |=| a ¯ i |=1, i=1,,n with T p M= E p c i E p c n and dim E p c i =2, i=1,,n. By Lemma 2.1, there are α>0 and g 1 U(f) such that

g 1 (x)={ φ g ( p ) D p g φ p 1 ( x ) if  x B α ( p ) , g ( x ) if  x B 4 α ( p ) .

Then g(p)= g 1 (p)=p.

First, we consider the case E p c 1 (α) 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 E p c 1 (α) φ p 1 ( B α (p)). Take v E p c 1 (α) such that v=α/4. Then we can find a small arc I p = φ p ({tv:α/4tα/4}) B α (p) such that

  1. (i)

    g 1 i ( I p ) g 1 j ( I p )= if 0ijl1, and

  2. (ii)

    g 1 l ( I p )= I p , that is, g 1 l | I p is the identity map.

Then we can choose 0<η<α/4 sufficiently small such that B η ( g 1 i ( I p )) B η ( g 1 j ( I p ))= for all 1ijl1. Take ϵ=η/4, and let 0<δ<ϵ be the number of the definition of the inverse shadowing property of g 1 for ϵ. For the δ>0, we can define T d ( g 1 )-method as follows: Let ψ Diff ω (M) be such that p is a hyperbolic periodic point for ψ with ψ(p)=p and d( g 1 ,ψ)<δ. Then ψ T d ( g 1 ). To simplicity, we may assume that g 1 l = g 1 . Take y B 4 ϵ (p) I p such that d(y,p)=2ϵ, and g 1 n (y)=y for all nZ. Since g 1 has the inverse shadowing property, we can see that for any zM

d ( g 1 n ( y ) , φ ψ ( z ) n ) =d ( g 1 n ( y ) , ψ n ( z ) ) <ϵ

for all nZ. For any z B ϵ (p), if z=p, then since g 1 | I p is the identity map, it is clear that g 1 does not have the inverse shadowing property. If zp, then since p is a hyperbolic periodic point for ψ, there is kZ such that

d ( g 1 k ( y ) , ψ k ( z ) ) =d ( y , ψ 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 E p c i (α) φ p 1 ( B α (p)), i=2,,n. Let K=lcm{ m i :i=2,,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 E p c i (α) φ p 1 ( B α (p)), i=2,,n, by the above argument, there exists a small arc I p B α (p) such that g 2 l | I p is the identity map, for some l>0. Then we can find ψ T d ( g 2 ) such that p is a hyperbolic periodic point for ψ with ψ(p)=p, d( g 2 ,ψ)<δ. By the inverse shadowing property for g 2 , there exists y B 4 ϵ (p) I p such that

d(y,p)=2ϵand g 2 l n (y)=y

for all nZ. Then there exists jZ such that

d ( g 2 j ( y ) , ψ j ( z ) ) =d ( y , ψ j ( z ) ) >ϵ.

This is a contradiction. Thus, every periodic point of fint IS ω (M) is hyperbolic. □

Lemma 2.4 If fint OIS ω (M), then every periodic point of f is hyperbolic.

Proof Let fint OIS ω (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 fint IS ω (M), and let fint OIS ω (M). Suppose that f F ω (M). Then there is g U 0 (f)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 fint IS ω (M) or fint OIS ω (M) then f F ω (M). By Lemma 1.2, f is Anosov. □

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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).

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Lee, M. Symplectic diffeomorphisms with inverse shadowing. J Inequal Appl 2013, 174 (2013). https://doi.org/10.1186/1029-242X-2013-174

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Keywords

  • topological stability
  • inverse shadowing
  • orbital inverse shadowing
  • Anosov
  • symplectic diffeomorphism