The ergodic shadowing property and homoclinic classes
© Lee; licensee Springer. 2014
Received: 8 November 2013
Accepted: 6 February 2014
Published: 20 February 2014
In this paper, we show that if a diffeomorphism satisfies a local star condition and it has the ergodic shadowing property then it is hyperbolic.
The notion of structural stability was introduced be Andronov and Pontrjagin . This means that under small perturbations the dynamics are topologically equivalent. The system is Ω-stable; then it is called Axiom A, that is, the non-wandering set is the closure of the set of periodic points and it is hyperbolic. It turned out to be one of the most problems in the differentiable dynamical systems to find if a structurally stable system satisfies the Axiom A property. Let M be a closed manifold. Mañé defined a set of diffeomorphisms having a -neighborhood such that every diffeomorphism inside of has all periodic orbits of hyperbolic. In , Mañé proved that every surface diffeomorphism of satisfies Axiom A. Hayashi has shown in  that every diffeomorphism of satisfies Axiom A. Robinson has proven in  that a dynamical system is structurally stable when the system has the shadowing property. Also, in  Sakai showed that if a dynamical system belongs to the -interior of the set of all systems having the shadowing property then it is a structurally stable diffeomorphism. Lee has shown in  that if a dynamical system belongs to the -interior of the set of all systems having the ergodic shadowing property then it is a structurally stable diffeomorphism. Carvalho proved in  that the -interior of the set of all systems having the two-side limit shadowing property is equal to the set of transitive Anosov diffeomorphisms, Pilyugin has shown in  that the -interior of the set of all systems having the limit shadowing property is equal to the set Ω-stable diffeomorphisms. Recently, in  Sakai proved that for -generically if a diffeomorphism has the s-limit shadowing property on the chain recurrent set then it is a Ω-stable diffeomorphism. From that, we know that the shadowing property is very close to the stability theory (see ). In , Mañé introduced the family of periodic sequences of linear isomorphisms of , and from that we can define the local star condition (see [, Proposition II.1]).
In this paper, we introduce the notion of the local star condition, and study under the local star condition the some shadowing property.
Let Λ be a closed f-invariant set. We say that f has the ergodic shadowing property in Λ if for any ϵ there is such that for any δ-ergodic pseudo orbit of f is ϵ-shadowed in ergodic sense for some point .
By the result of , if a diffeomorphism has the ergodic shadowing property then it is chain transitive, moreover, it is topologically mixing. Thus the diffeomorphism does not contain a sink and sources. We know that a Morse-Smale diffeomorphism has the shadowing property. But the diffeomorphism contains sinks and sources. Thus it does not have the ergodic shadowing property. We say that f is topologically mixing if for any nonempty open sets U and V, there is such that for . In [, Theorem A], Fakhari and Ghane proved that f has the ergodic shadowing property if and only if f has the shadowing property and it is topologically mixing. Let be the set of periodic points of f. Denote by the periodic f-orbit of . Let be a hyperbolic saddle with period , then there are a local stable manifold and a local unstable manifold for some . Then we see that if , then , for and if then for . The stable manifold and the unstable manifold of p are defined as usual. The dimension of the stable manifold is called the index of p, and we denote it by .
A point is called a transversal homoclinic point of f if the above intersection is transversal at x; i.e., . The closure of the set of transversal homoclinic points of f associated to p is called the transversal homoclinic class of f associated to p, and it is denoted by . It is clear that is compact, invariant, and transitive.
for all and .
In , Lee showed that if and f has the average shadowing property in then is hyperbolic. For that, we show the following.
f satisfies the local star condition, and
f has the ergodic shadowing property in .
Then is hyperbolic.
The average shadowing property is not the ergodic shadowing property. Indeed, the map is defined by if , and if . Then the map has two fixed points. In [, Example], the map has the ergodic shadowing property. However, in [, Theorem 3.1], Park and Zhang proved that if the number of the fixed points is greater than two, then the map f does not have the average shadowing property.
2 Proof of Theorem 1.1
for every , where denotes the mininorm of a linear map A. It always extends to a neighborhood which is called an admissible neighborhood of Λ. By Mañé (see ), the family of periodic sequences of linear isomorphisms of generated by Dg (g close to f) along the hyperbolic periodic point is uniformly hyperbolic. This means that there is such that for any g -nearby and any sequence of linear maps with (), , is hyperbolic. Thus if then we have the following.
admits a dominated splitting with .
- (b)For any if has minimum period then
By Proposition 2.1, we get the following, which was found by [, Theorem 3.2].
- (a)admits a dominated splitting with such that for every ,
- (b)For any if then , and if then
Theorem 2.3 [, Proposition 2.3]
for all . Moreover, q can be chosen such that is arbitrarily large.
Let Λ be a closed f-invariant set.
Lemma 2.5 [, Lemma 2.2]
- (a)admits a dominated splitting with such that for every ,
- (b)For any if q is hyperbolic and , then , and
f has the ergodic shadowing property in .
Then is hyperbolic for f.
Let be a subbundle. We say that is contracting if there exist and such that for every and every . We will say that E is expanding if E is contracting respecting .
Since is locally maximal, . Since , (1) is a contradiction by Proposition 2.6(b). This is the proof of Theorem 1.1. □
3 Stably ergodic shadowing property in
Let M be as before, and let . We introduce the notion of the -stably ergodic shadowing property.
Definition 3.1 We say that f has the -stably ergodic shadowing property in Λ if there are a compact neighborhood U of f and a -neighborhood of f such that (locally maximal), and for any , g has the ergodic shadowing property in , where is the continuation of Λ.
For given , we write if for any , there is a δ-pseudo orbit () of f such that and . We write if and . The set of points is called the chain recurrent set of f and is denoted by . It is well known that is a closed and f-invariant set. If we denote the set of periodic points of f by , then . Here is the non-wandering set of f. We write if and . The relation ↭ induces an equivalence relation on , whose classes are called chain components of f. Denote by . Then we know that (see ). In , Lee et al. proved that if f has the -stably shadowing property on then is a hyperbolic homoclinic class. They used Mañé’s ergodic closing lemma. In this section we use Theorem 1.1. We say that Λ is topologically transitive if for any neighborhoods U, V in Λ there is such that . Note that it can be rewritten as follows: there is such that , where is the omega limit set. Note that if Λ is topologically mixing then Λ is topologically transitive. In , Lee showed that if f has the -stably ergodic shadowing property on a transitive set Λ then it admits a dominated splitting. In this section, we will show that if f has the -stably ergodic shadowing property in then it is hyperbolic. The following is the main theorem in this section.
Theorem 3.2 Let Λ be a closed f-invariant set. Suppose that f has the -stably ergodic shadowing property in Λ. Then .
To prove Theorem 3.2, we need the following lemmas.
Lemma 3.3 [, Theorem A]
f has the ergodic shadowing property if and only if f has the shadowing property and it is topologically mixing.
Lemma 3.4 Let be hyperbolic periodic points. If f has the ergodic shadowing property in Λ then and .
Proof Since are hyperbolic periodic points, there are and such that and are defined, where . Suppose that f has the ergodic shadowing property in Λ. By Lemma 3.3, f has the shadowing property in Λ and Λ is topologically mixing. Since f has the shadowing property in Λ, we can take . For that , take be as in the definition of the shadowing property. Since Λ is topologically mixing, Λ is topologically transitive. Then there is a point such that . For simplicity, we may assume that and . Then there exist and such that and . To construct a δ-pseudo orbit of f, we assume that for some . Put (i) for (ii) for and (iii) for all . Then as in the proof of [, Lemma 2.3], we get . The other case is similar. □
We say that f is a Kupka Smale diffeomorphism if every periodic points are hyperbolic and if , then is transversal to . It is well known that if f is a Kupka Smale then f is residual in . Denote by the set of all Kupka Smale diffeomorphisms.
Proof of Theorem 3.2 Since f has the -stably ergodic shadowing property in Λ, there exist a -neighborhood of f and a neighborhood U of Λ such that for any , g has the ergodic shadowing property in . To derive a contradiction, we may assume that . Then there are and such that q is not hyperbolic. Then there is close to g such that has two hyperbolic periodic points with different indices. Then we know or . Without loss of generality, we assume that . Take . Since h is Kupka Smale, , where and are continuations of and , respectively. Since h has the ergodic shadowing property in , h has the shadowing property in and is topologically mixing. Since , by Lemma 3.4 we get a contradiction. □
We say that Λ is a basic set if Λ is transitive, and locally maximal. If the basic set 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 Λ. We say that Λ is an elementary set if Λ is mixing, and locally maximal. Note that every elementary set is a basic set.
Theorem 3.5 Let be the homoclinic class. If f has the -stably ergodic shadowing property in then is a hyperbolic elementary set.
Proof of Theorem 3.5 Suppose that f has the -stably ergodic shadowing property in . By Theorem 3.2, . Since f has the ergodic shadowing property in , by Lemma 3.3 is topologically mixing. Thus by Theorem 1.1, is a hyperbolic elementary set. □
We wish to thank the referee for carefully reading the manuscript and providing us with 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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