Open Access

The ergodic shadowing property from the robust and generic view point

Advances in Difference Equations20142014:170

https://doi.org/10.1186/1687-1847-2014-170

Received: 6 January 2014

Accepted: 29 May 2014

Published: 23 June 2014

Abstract

In this paper, we discuss that if a diffeomorphisms has the C 1 -stably ergodic shadowing property in a closed set, then it is a hyperbolic elementary set. Moreover, C 1 -generically: if a diffeomorphism has the ergodic shadowing property in a locally maximal closed set, then it is a hyperbolic basic set.

MSC:34D30, 37C20.

Keywords

ergodic shadowingshadowinglocally maximalgenericAnosov

1 Introduction

Let M be a closed C manifold, and let Diff ( M ) be the space of diffeomorphisms of M endowed with the C 1 -topology. Denote by d the distance on M induced from a Riemannian metric on the tangent bundle TM. Let f Diff ( M ) . For δ > 0 , a sequence of points { x i } i = a b ( a < b ) in M is called a δ-pseudo orbit of f if d ( f ( x i ) , x i + 1 ) < δ for all a i b 1 . For given x , y M , we write x y if for any δ > 0 , there is a δ-pseudo orbit { x i } i = a b ( a < b ) of f such that x a = x and x b = y . Let Λ be a closed f-invariant set. We say that f has the shadowing property in Λ if for every ϵ > 0 there is δ > 0 such that, for any δ-pseudo orbit { x i } i = a b Λ of f ( a < b ), there is a point y Λ such that d ( f i ( y ) , x i ) < ϵ for all a i b 1 . If Λ = M , then f has the shadowing property. The shadowing property usually plays an important role in the investigation of stability theory and ergodic theory. For instance, Sakai [1] proved that if f has the C 1 -robustly shadowing property, then f is structurally stable. Now we introduce the notion of the ergodic shadowing property which was introduced and studied by [2]. Lee has shown in [3] that if f belongs to the C 1 -interior of the set of all diffeomorphisms having the ergodic shadowing property, then it is structurally stable diffeomorphisms. In [4], Lee showed that if f is local star condition and has the ergodic shadowing property on the homoclinic class, then it is hyperbolic. For any δ > 0 , a sequence ξ = { x i } i Z is a δ-ergodic pseudo orbit of f if for N p n + ( ξ , f , δ ) = { i : d ( f ( x i ) , x i + 1 ) δ } { 0 , 1 , , n 1 } , and N p n ( ξ , f , δ ) = { i : d ( f 1 ( x i ) , x i 1 ) δ } { n + 1 , , 1 , 0 }
lim n # N p n + ( ξ , f , δ ) n = 0 and lim n # N p n ( ξ , f , δ ) n = 0 .
Here #A is the number of elements of the set A. We say that f has the ergodic shadowing property in Λ (or f | Λ has ergodic shadowing) if for any ϵ > 0 , there is a δ > 0 such that every δ-ergodic pseudo orbit ξ = { x i } i Z Λ of f there is a point z Λ such that, for N s n + ( ξ , f , z , ϵ ) = { i : d ( f i ( z ) , x i ) ϵ } { 0 , 1 , , n 1 } , and N s n ( ξ , f , z , ϵ ) = { i : d ( f i ( z ) , x i ) ϵ } { n + 1 , , 1 , 0 } ,
lim n # N s n + ( ξ , f , z , ϵ ) n = 0 and lim n # N s n ( ξ , f , z , ϵ ) n = 0 .

Note that f has the ergodic shadowing property on Λ and f has the ergodic shadowing property in Λ are different notions. That is, the shadowing point is in M or Λ. In the first notion, the shadowing point is in M. In the second notion, the shadowing point is in Λ. In this paper we consider the latter case.

We say that Λ is locally maximal if there is a compact neighborhood U of Λ such that
n Z f n ( U ) = Λ f ( U ) = Λ .

Now, we introduce the notion of the C 1 -stably ergodic shadowing property in a closed set.

Definition 1.1 Let Λ be a closed f-invariant set. We say that f has the C 1 -stably ergodic shadowing property in Λ if
  1. (i)

    there is a neighborhood U of Λ and a C 1 -neighborhood U ( f ) of f such that Λ f ( U ) = Λ = n Z f n ( U ) (that is, Λ is locally maximal);

     
  2. (ii)

    for any g U ( f ) , g has the ergodic shadowing property on Λ g ( U ) = n Z g n ( U ) , where Λ g ( U ) is the continuation of Λ.

     
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 x Λ and n 0 . If Λ = M , then f is Anosov. We say that Λ is a basic set (resp. elementary set) if f | Λ is transitive (resp. mixing) and locally maximal. Note that if Λ is hyperbolic, then we can easily show that there is a periodic point such that the orbit of the periodic point is dense in the set. Then we get the following.

Theorem 1.2 [[5], Theorem 3.3]

Let Λ be a closed f-invariant set. If f has the C 1 -stably ergodic shadowing property in Λ, then it is a hyperbolic elementary set.

Corollary 1.3 If f belongs to the C 1 -interior of the set of all diffeomorphisms having the ergodic shadowing property, then it is transitive Anosov.

We say that a subset G Diff ( M ) is residual if contains the intersection of a countable family of open and dense subsets of Diff ( M ) ; in this case is dense in Diff ( M ) . A property P is said to be C 1 -generic if P holds for all diffeomorphisms which belong to some residual subset of Diff ( M ) . We use the terminology ‘for C 1 -generic f’ to express ‘there is a residual subset G Diff ( M ) such that, for any f G  .’ In [6], Abdenur and Díaz proved that if tame diffeomorphisms has the shadowing property, then it is hyperbolic. Still open is the question if C 1 -generically: f is shadowable, then is it hyperbolic?

Recently, Ahn et al. [7] have given a partial answer which is C 1 -generically: if a locally maximal homoclinic class is shadowing, then it is hyperbolic. Lee has shown in [8] that C 1 -generically: if f has the limit shadowing property on the homoclinic class, then it is hyperbolic. Inspired by this, we consider that C 1 -generically: f has the ergodic shadowing property in a locally maximal closed set. Then we have the following.

Theorem 1.4 For C 1 -generic f, if f has the ergodic shadowing property in a locally maximal closed set Λ, then it is a hyperbolic elementary set. Moreover, C 1 -generically: if f has the ergodic shadowing property, then it is transitive Anosov.

2 Proof of Theorem 1.4

Let P ( f ) be the set of periodic points of f. If f | Λ is transitive, then every p Λ P ( f ) is saddle, that is, there is no eigenvalues of D p f π ( p ) with modulus equal to 1, at least one of them is greater than 1, at least one of them is smaller than 1, where π ( p ) is the minimum period of p.

Lemma 2.1 [[2], Corollary 3.5]

If f has the ergodic shadowing property in Λ, then f | Λ is mixing.

By Lemma 2.1, f has the ergodic shadowing property in Λ, then f | Λ is mixing, and so f | Λ is transitive. Thus p Λ P ( f ) is neither a sink nor a source.

Lemma 2.2 [[2], Lemma 3.2]

If f has the ergodic shadowing property in Λ, then f has a finite shadowing property in Λ.

We say that f has the finite shadowing property on Λ if for any ϵ > 0 there is δ > 0 such that, for any finite δ-pseudo orbit { x 0 , x 1 , , x n } Λ , there is y M such that d ( f i ( y ) , x i ) < ϵ for all 0 i < n . In [[9], Lemma 1.1.1], Pilyugin showed that f has a finite shadowing shadowing property on Λ, then f has the shadowing property on Λ.

Lemma 2.3 Let f have the ergodic shadowing in Λ and Λ be locally maximal in U. Then the shadowing point taken from Λ.

Proof Let f have the ergodic shadowing property in Λ, and let U be a locally maximal of Λ. For any ϵ > 0 , let δ > 0 be the number of the ergodic shadowing property of f. Take a sequence γ = { x i } i = 0 n ( n 1 ) such that γ is a δ-pseudo orbit of f and γ Λ . As in the proof of [[2], Lemma 3.1], there is a δ-pseudo orbit η = { x i } i = n 0 such that η Λ . Then we set ξ = { , γ , η , γ , η , } is a δ-ergodic pseudo orbit of f. Clear that ξ Λ . Since f has the ergodic shadowing property in Λ, ξ can be ergodic shadowed by some point y Λ . By Lemma 2.2, there is γ ξ such that d ( f i ( y ) , x i ) < ϵ for 0 i n 1 . By [[9], Lemma 1.1.1], f has the shadowing property on Λ. Since Λ is locally maximal in U, the shadowing point y Λ . □

Let p P ( f ) be a hyperbolic saddle with period π ( p ) > 0 . Then there are the local stable manifold W ϵ s ( p ) and the local unstable manifold W ϵ u ( p ) of p for some ϵ = ϵ ( p ) > 0 . It is easily seen that if d ( f n ( x ) , f n ( p ) ) ϵ for all n 0 , then x W ϵ s ( p ) , and if d ( f n ( x ) , f n ( p ) ) ϵ for all n 0 , then x W ϵ u ( p ) . The stable manifold W s ( p ) and the unstable manifold W u ( p ) defined as following. It is well known that if p is a hyperbolic periodic point of f with period k, then the sets
W s ( p ) = { x M : f k n ( x ) p  as  n } and W u ( p ) = { x M : f k n ( x ) p  as  n }

are C 1 -injectively immersed submanifolds of M.

Lemma 2.4 Let p , q P ( f ) be hyperbolic saddles. If f has the ergodic shadowing property in a closed set Λ, then W s ( p ) W u ( q ) , and W u ( p ) W s ( q ) .

Proof Let p , q P ( f ) be hyperbolic saddles, and let U be a locally maximal neighborhood of Λ. Suppose that f has the ergodic shadowing property in a locally maximal Λ. Since p and q are hyperbolic, there are ϵ ( p ) > 0 and ϵ ( q ) > 0 as in the above. Take ϵ = min { ϵ ( p ) , ϵ ( q ) } / 4 and let 0 < δ ϵ be the number of the ergodic shadowing property of f. For simplicity, we may assume that f ( p ) = p and f ( q ) = q . Since f has the ergodic shadowing property in Λ, f | Λ is chain transitive. Then we can construct a finite δ-pseudo orbit form p to q as follows: x 0 = p , x n = q ( n 1 ), and d ( f ( x i ) , x i + 1 ) < δ for all 0 < i < n 1 . Put (i) x i = f i ( p ) , for all i 0 , and (ii) x n + i = f i ( q ) for all i 0 . Then we have the sequence ξ = { x i } i Z = { , p , x 1 , x 2 , , x n , x n + 1 , } . It is clearly a δ-ergodic pseudo orbit of f. Since f has the ergodic shadowing property in Λ and locally maximal, by Lemma 2.2, f has the finite shadowing property on Λ and so, by [[9], Lemma 1.1.1], f has the shadowing property in Λ. By the shadowing property in Λ, we can show that Orb ( y ) W u ( p ) W s ( q ) , and so W u ( p ) W s ( q ) . The other case is similar. □

A diffeomorphism f is Kupka-Smale if their periodic points of f are hyperbolic and if p , q P ( f ) , then W s ( p ) is transversal to W u ( q ) . Then it is C 1 -residual in Diff ( M ) . Denote by KS ( M ) the set of all Kupka-Smale diffeomorphisms. The following was proved by [10].

Lemma 2.5 [[10], Lemma 2.4]

Let Λ be locally maximal in U, and let U ( f ) be given. If for any g U ( f ) , p Λ g ( U ) P ( g ) is not hyperbolic, then there is g 1 U ( f ) such that g 1 has two hyperbolic periodic points p , q Λ g 1 ( U ) with different indices.

Denote by F ( M ) the set of f Diff ( M ) such that there is a C 1 neighborhood U ( f ) of f such that, for any g U ( f ) , every p P ( g ) is hyperbolic. In [11], Hayashi proved that f F ( M ) if and only if f satisfies both Axiom A and the no-cycle condition. We say that f is the local star condition diffeomorphism if there exist a C 1 -neighborhood U ( f ) and a neighborhood U of Λ such that, for any g U ( f ) , every p Λ g ( U ) P ( g ) is hyperbolic (see [12]). Denote by F ( Λ ) the set of all local star diffeomorphisms. Note that there are a C 1 -neighborhood U ( f ) and a neighborhood U of p such that, for all g U ( f ) , there is a unique hyperbolic periodic point p g U of g with the same period as p and index ( p g ) = index ( p ) . Here index ( p ) = dim E p s , and the point p g is called the continuation of p.

Lemma 2.6 [[13], Lemma 2.2]

There is a residual set G 1 Diff ( M ) such that, for any f G 1 , if for any C 1 -neighborhood U ( f ) of f, there exists g U ( f ) such that two hyperbolic periodic points p g , q g P ( g ) with index ( p g ) index ( q g ) , then f has two hyperbolic periodic points p , q P ( f ) with index ( p ) index ( q ) .

Lemma 2.7 There is a residual set G 2 Diff ( M ) such that, for any f G 2 , if f has the ergodic shadowing property in a locally maximal Λ, then for any p , q Λ P ( f )
index ( p ) = index ( q ) .

Proof Let f G 2 = G 1 KS ( M ) , and let p , q Λ P ( f ) be hyperbolic saddles. Suppose that f has the ergodic shadowing property in a locally maximal Λ. Then by Lemma 2.4 W s ( p ) W u ( q ) and W u ( p ) W s ( q ) . Since f G 2 , W s ( p ) W u ( q ) and W u ( p ) W s ( q ) . This means that p q and so index ( p ) = index ( q ) . □

Let p be a periodic point of f. For 0 < δ < 1 , we say that p has a δ-weak eigenvalue if D f π ( p ) ( p ) has an eigenvalue λ such that ( 1 δ ) π ( p ) < | λ | < ( 1 + δ ) π ( p ) . We say that a periodic point has a real spectrum if all of its eigenvalues are real and simple spectrum if all its eigenvalues have multiplicity one. Denote by P h ( f ) the set of all hyperbolic periodic points of f.

Lemma 2.8 [[14], Lemma 5.1]

There is a residual set G 3 Diff ( M ) such that, for any f G 3 :

  • For any δ > 0 , if for any C 1 -neighborhood U ( f ) of f there exist g U ( f ) and p g P h ( g ) with a δ-weak eigenvalue, then there is p P h ( f ) with a 2δ-weak eigenvalue.

  • For any δ > 0 , if q P h ( f ) with a δ-weak eigenvalue and a real spectrum, then there is p P h ( f ) with a δ-weak eigenvalue with a simple real spectrum.

Lemma 2.9 There is a residual set G 4 Diff ( M ) such that, for any f G 4 , if f has the ergodic shadowing property in a locally maximal Λ, then there exists η > 0 such that, for any q Λ P h ( f ) , q has no η-weak eigenvalues.

Proof Let f G 4 = G 2 G 3 have the ergodic shadowing property in a locally maximal Λ. We will derive a contradiction. Suppose that, for any η > 0 , there is q Λ P h ( f ) such that q has an η-weak eigenvalue. By Franks’ lemma, there is g C 1 -close to f such that p is not hyperbolic. By Franks’ lemma and Lemma 2.5, there is h C 1 -nearby g and C 1 -close to f such that h has tow hyperbolic periodic points q h , γ h with different indices. Since f G 1 , and it is locally maximal, by Lemma 2.6 f has two hyperbolic periodic points q, γ in Λ. Since f has the ergodic shadowing property in Λ, this is a contradiction by Lemma 2.7. □

Proposition 2.10 There is a residual set G 4 Diff ( M ) such that, for any f G 4 , if f has the ergodic shadowing property in Λ, then f F ( Λ ) .

Proof Let f G 4 have the ergodic shadowing property in a locally maximal Λ. Suppose by contradiction that f F ( Λ ) . Then there are g C 1 -close to f and q P ( g ) such that q has a η-weak eigenvalue. Then by Lemma 2.9, we get a contradiction. Thus f F ( Λ ) . □

Proposition 2.11 [[15], Proposition A]

Let f G 4 , and let Λ be locally maximal. If f has the ergodic shadowing property in Λ, then there are m > 0 , C 1 and λ ( 0 , 1 ) such that, for any p Λ P ( f ) with π ( q ) > m , we have
i = 0 π ( p ) 1 D f m | E s ( f m i ( x ) ) C λ π ( p ) , i = 0 π ( p ) 1 D f m | E s ( f m i ( x ) ) C λ π ( p ) and D f m | E s ( x ) D f m | E u ( f m ( x ) ) λ ,

where π ( p ) is the period of p.

Remark 2.12 By Pugh’s closing lemma, there is a residual set G 6 Diff ( M ) such that, for any f G 6 , if f | Λ is transitive, then there is a periodic orbit p n such that Orb ( p n ) Λ in Hausdorff metric.

Lemma 2.13 [[16], Theorem 3.8]

There is residual set G 5 Diff ( M ) such that, for any f G 5 , for any ergodic measure μ of f, there is a sequence of the periodic point p n such that μ p n μ in weak topology and Orb ( p n ) Supp ( μ ) in Hausdorff metric.

The following was proved by Mañé [17]. Denote by M ( f | Λ ) the set of invariant probabilities on the Borel σ-algebra of Λ endowed with the weak topology.

Lemma 2.14 Let Λ M be a closed f-invariant set of f and E T Λ M be a continuous invariant subbundle. If there is m > 0 such that
log D f m | E d μ < 0

for every ergodic μ M ( f m | Λ ) , then E is contracting.

Proof of Theorem 1.4 Let f G 4 G 5 G 6 have the ergodic shadowing property in a locally maximal Λ. Then by Proposition 2.11, we know that Λ admits a dominated splitting T Λ M = E F . Since f has the ergodic shadowing property in Λ, by Lemma 2.1, Remark 2.12, and Lemma 2.13, there is a sequence of periodic points such that Orb ( p n ) Supp ( μ ) = Λ in the Hausdorff metric. By Proposition 2.11, we have
D f m | E d μ = lim n D f m | E d μ p n < 0 .

By Lemma 2.14, E is contracting. Similarly, we can show that F is expanding. □

Corollary 2.15 For C 1 -generic f, if f has the ergodic shadowing property, then f is transitive Anosov.

Declarations

Acknowledgements

This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2014R1A1A1A05002124).

Authors’ Affiliations

(1)
Department of Mathematics, Mokwon University

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