Skip to main content

Advertisement

The ergodic shadowing property from the robust and generic view point

Article metrics

  • 774 Accesses

  • 2 Citations

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.

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 fDiff(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 aib1. For given x,yM, we write xy 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 yΛ such that d( f i (y), x i )<ϵ for all aib1. 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,,n1}, 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 =0and 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 zΛ such that, for N s n + (ξ,f,z,ϵ)={i:d( f i (z), x i )ϵ}{0,1,,n1}, 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 =0and 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 gU(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 xΛ and n0. 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 GDiff(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 GDiff(M) such that, for any fG .’ 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 pΛP(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 pΛP(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 yM such that d( f i (y), x i )<ϵ for all 0i<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 (n1) such that γ is a δ-pseudo orbit of f and γΛ. As in the proof of [[2], Lemma 3.1], there is a δ-pseudo orbit η= { x i } i = n 0 such that ηΛ. Then we set ξ={,γ,η,γ,η,} is a δ-ergodic pseudo orbit of f. Clear that ξΛ. Since f has the ergodic shadowing property in Λ, ξ can be ergodic shadowed by some point yΛ. By Lemma 2.2, there is γξ such that d( f i (y), x i )<ϵ for 0in1. By [[9], Lemma 1.1.1], f has the shadowing property on Λ. Since Λ is locally maximal in U, the shadowing point yΛ. □

Let pP(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 n0, then x W ϵ s (p), and if d( f n (x), f n (p))ϵ for all n0, 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,qP(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,qP(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<δϵ 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 (n1), and d(f( x i ), x i + 1 )<δ for all 0<i<n1. Put (i) x i = f i (p), for all i0, and (ii) x n + i = f i (q) for all i0. 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,qP(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 gU(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 fDiff(M) such that there is a C 1 neighborhood U(f) of f such that, for any gU(f), every pP(g) is hyperbolic. In [11], Hayashi proved that fF(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 gU(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 gU(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 gU(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,qP(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,qΛP(f)

index(p)=index(q).

Proof Let f G 2 = G 1 KS(M), and let p,qΛP(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 pq 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 anyδ>0, if for any C 1 -neighborhoodU(f)of f there existgU(f)and p g P h (g)with a δ-weak eigenvalue, then there isp P h (f)with a 2δ-weak eigenvalue.

  • For anyδ>0, ifq P h (f)with a δ-weak eigenvalue and a real spectrum, then there isp 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 qΛ 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 qΛ 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 fF(Λ).

Proof Let f G 4 have the ergodic shadowing property in a locally maximal Λ. Suppose by contradiction that fF(Λ). Then there are g C 1 -close to f and qP(g) such that q has a η-weak eigenvalue. Then by Lemma 2.9, we get a contradiction. Thus fF(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, C1 and λ(0,1) such that, for any pΛP(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 ΛM 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 dμ<0

for every ergodic μM( 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=EF. 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 dμ= 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.

References

  1. 1.

    Sakai K: Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds. Osaka J. Math. 1994, 31: 373–386.

  2. 2.

    Fakhari A, Ghane FH: On shadowing: ordinary and ergodic. J. Math. Anal. Appl. 2010, 364: 151–155. 10.1016/j.jmaa.2009.11.004

  3. 3.

    Lee M: Diffeomorphisms with robustly ergodic shadowing. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2013, 20: 747–753.

  4. 4.

    Lee M: The ergodic shadowing property and homoclinic classes. J. Inequal. Appl. 2014., 2014: Article ID 90

  5. 5.

    Barzanouni A, Honary B: C 1 -Stable ergodic shadowable invariant sets and hyperbolicity. Gen. Math. Notes 2012, 9: 1–6.

  6. 6.

    Abdenur F, Díaz LJ:Pseudo-orbit shadowing in the C 1 topology. Discrete Contin. Dyn. Syst. 2007, 17: 223–245.

  7. 7.

    Ahn J, Lee K, Lee M: Homoclinic classes with shadowing. J. Inequal. Appl. 2012., 2012: Article ID 97

  8. 8.

    Lee M: Usual limit shadowable homoclinic classes of generic diffeomorphisms. Adv. Differ. Equ. 2012., 2012: Article ID 91

  9. 9.

    Pilyugin S Lect. Notes in Math. 1706. In Shadowing in Dynamical Systems. Springer, Berlin; 1999.

  10. 10.

    Sakai K, Sumi N, Yamamoto K: Diffeomorphisms satisfying the specification property. Proc. Am. Math. Soc. 2009, 138: 315–321.

  11. 11.

    Hayashi S:Diffeomorphisms in F 1 (M) satisfy Axiom A. Ergod. Theory Dyn. Syst. 1992, 12: 233–253.

  12. 12.

    Dai X:Dominated splitting of differentiable dynamics with C 1 -topological weak-star property. J. Math. Soc. Jpn. 2012, 64: 1249–1295. 10.2969/jmsj/06441249

  13. 13.

    Lee M, Lee S: Robustly transitive sets with generic diffeomorphisms. Commun. Korean Math. Soc. 2013, 28: 581–587. 10.4134/CKMS.2013.28.3.581

  14. 14.

    Arbieto A: Periodic orbits and expansiveness. Math. Z. 2011, 269: 801–807. 10.1007/s00209-010-0767-5

  15. 15.

    Yang D, Gan S: Expansive homoclinic classes. Nonlinearity 2009, 22: 729–733. 10.1088/0951-7715/22/4/002

  16. 16.

    Abdenur F, Bonatti C, Crovisier C:Non-uniform hyperbolicity for C 1 -generic diffeomorphisms. Isr. J. Math. 2011, 183: 1–60. 10.1007/s11856-011-0041-5

  17. 17.

    Mañé R:A proof of the C 1 stability conjecture. Publ. Math. Inst. Hautes Études Sci. 1987, 66: 161–210. 10.1007/BF02698931

Download references

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).

Author information

Correspondence to Manseob Lee.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Keywords

  • ergodic shadowing
  • shadowing
  • locally maximal
  • generic
  • Anosov