Abstract and Applied Analysis

Volume 2013 (2013), Article ID 693032, 4 pages

http://dx.doi.org/10.1155/2013/693032

Research Article

Orbital Shadowing for -Generic Volume-Preserving Diffeomorphisms

Manseob Lee

Department of Mathematics, Mokwon University, Dajeon 302-729, Republic of Korea

Received 13 June 2013; Revised 29 August 2013; Accepted 1 September 2013

Academic Editor: Chun-Gang Zhu

Copyright © 2013 Manseob Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We show that -generically, if a volume-preserving diffeomorphism has the orbital shadowing property, then the diffeomorphism is Anosov.

1. Introduction

In the differentiable dynamical systems, the shadowing theory is a very useful notion for the investigation of the stability condition. In fact, Robinson [1] and Sakai [2] proved that a diffeomorphism belongs to the -interior of the set of diffeomorphisms having the shadowing property coincides the structurally stable; that is, the diffeomorphism satisfies both Axiom A and the strong transversality condition. In general, if a diffeomorphism is -stable, that is, a diffeomorphism satisfies both Axiom A and the no-cycle condition, then there is a diffeomorphism which does not have the shadowing property. Indeed, let a diffeomorphism of the two-dimensional torus . The nonwandering set consists of 4 hyperbolic fixed points, , where is a sink, is a source, and and are saddles such that . It is assumed that the eigenvalues of are with , and the eigenvalues of are with . Then does not have the shadowing property. But it has the orbital shadowing property (see [3]).

In this paper, we study the orbital shadowing property in which it is clear that if a diffeomorphism has the shadowing property, then it has the orbital shadowing property, but the converse is not true. In fact, an irrational rotation map does not have the shadowing property, but it has the orbital shadowing property.

The orbital shadowing property was introduced by Pilyugin et al. [3]. They showed that a diffeomorphism belongs to the -interior of the set of all diffeomorphisms having the orbital shadowing property if and only if the diffeomorphism is structurally stable.

For a conservative diffeomorphism, Bessa and Rocha proved in [4] that if a conservative diffeomorphism belongs to the -interior of the set of all topologically stable conservative diffeomorphisms, then it is Anosov. In [5], Bessa proved that a conservative diffeomorphism is in the -interior of the set of all conservative diffeomorphisms having the shadowing property if and only if it is Anosov. K. Lee and M. Lee [6] proved that a conservative diffeomorphism is in the -interior of the set of all conservative diffeomorphisms having the orbital shadowing property if and only if it is Anosov. Our result is a generalization of the result in [7].

Let be a closed Riemannian manifold endowed with a volume form . Let denote the Lebesgue measure associated to , and let denote the metric induced on by the Riemannian structure. Denote by the set of diffeomorphisms which preserves the Lebesgue measure endowed with the Whitney topology. We know that every volume preserving diffeomorphism satisfying Axiom is Anosov (for more details, see [8]).

For , a sequence of points in is called a -pseudo-orbit of if for all . We say that has the shadowing property if, for every , there is , such that, for any -pseudo-orbit of , there is a point , such that, for all . It is easy to see that has the shadowing property if and only if has the shadowing property for . For each , let be the orbit of through ; that is,

We say that has the orbital shadowing property if, for any , there exists , such that for any -pseudo-orbit , we can find a point such that

where denotes the -neighborhood of a set . It is easy to see that has the orbital shadowing property if and only if has the orbital shadowing property for . Let be a closed -invariant set. We say that is hyperbolic if the tangent bundle has a -invariant splitting and there exist constants and , such that

for all and .

In [3], the authors proved that the -interior of the set of dissipative diffeomorphisms having the orbital shadowing property coincides with the set of structurally stable diffeomorphisms. Note that if a diffeomorphism satisfies structurally stable, then it is not Anosov in general, but the converse is true.

We say that a subset is residual if contains the intersection of a countable family of open and dense subsets of ; in this case, is dense in . A property “” is said to be -generic if “” holds for all diffeomorphisms which belong to some residual subset of . We use the terminology “for -generic ” to express that “there is a residual subset such that for any .” The following is the main result in this paper.

Theorem 1. For -generic , if has the orbital shadowing property, then is Anosov.

Let be a periodic point of with period . We say that is an elementary point if eigenvalues are multiplicatively independent over . Elementary points have simple spectrum, and none of eigenvalues are a root of unity or equal to 1. For a periodic point of , if we consider , then we have three cases. Firstly, is a hyperbolic saddle, that is, real eigenvalues with . Secondly, is an elliptic point; that is, nonreal eigenvalues are conjugated and of norm one. Finally, is a parabolic point; that is, eigenvalues equal 1 or −1. Note that the first and second cases are robust under small perturbations. Elementary elliptic points are associated with an irrational rotation number. In [9], Robinson showed that if , there is a residual set in such that any elementary in this residual displays all its elliptic points of elementary type. In [10, Theorem 1.3], Newhouse showed that -generic volume-preserving diffeomorphisms in surfaces are Anosov, or else the elliptical points are dense. Actually, Newhouse’s proof is strongly supported in the symplectic structure. By Newhouse [10] and Robinson [9], we give a problem as follows: For generic , if has the orbital shadowing property, then is it Anosov?

2. Proof of Theorems 1

Let be as before, and let . Let be a closed -invariant set. We say that is a transitive set if there is a point such that , where is the omega-limit set. If , then we called that is transitive. We say that is a hyperbolic if has no eigenvalues of absolute value one. It is well known that if is a hyperbolic periodic point of with period , then

are -injectively immersed submanifolds of . Let be a hyperbolic periodic point of . We say that and are homoclinically related if

For given hyperbolic periodic points and of , we write if and are homoclinically related. It is clear that if , then . The following result is very useful to prove Theorem 1. Let be a hyperbolic periodic point. For , we say that is a homoclinic point if . Denote by .

Theorem 2 (see [11, Theorem 1.3]). There is a residual set such that, for any , is transitive. Moreover, is a unique homoclinic class.

We denote by the set of diffeomorphisms which has a -neighborhood such that for any , every periodic point of is hyperbolic.

Very recently, Arbieto and Catalan [8] proved that every volume preserving diffeomorphism in is Anosov.

Theorem 3 (see [8, Theorem 1]). Every volume preserving diffeomorphism in is Anosov.

To prove Theorem 1, it is enough to show that . Let be a hyperbolic periodic point of ; there exists such that, for any and , we know that

for all . Then is called the local stable manifold of , and is the local unstable manifold of . It is clear that , and .

Lemma 4. Let , and let . If has the orbital shadowing property, then

where is the set of all hyperbolic periodic points of .

Proof. Let , and let be hyperbolic periodic points of . Take and as before with respect to and . For simplicity, we may assume that and . Take . Let be the number of the orbital shadowing property of for . Since is transitive, there exists such that . Then there exist and such that and . We may assume that for some . Then we get a finite -pseudo-orbit . Now we construct a -pseudo orbit as follows: put (i) for , (ii) for , and (iii) for . Then

Since has the orbital shadowing property, there are points and such that (i) and for all ; (ii) and for all . Then

, and . Thus, .

The study of the Kupka-Smale systems within volume preserving maps was developed by Robinson (see [1]). We say that is Kupka-Smale if every periodic point is hyperbolic and each invariant manifold has transverse intersections. Denote by the set of Kupka-Smale volume-preserving diffeomorphisms. It is well-known that the is residual in .

Lemma 5. There is a residual set such that, for any , if has the orbital shadowing property, then for any , .

Proof. Let , and let . Suppose that has the orbital shadowing property. Let be hyperbolic periodic points of . To derive a contradiction, we may assume that . Then we know that or . Assume that . Since is Kupka-Smale, we have . This is a contradiction by Lemma 4.

Remark 6. In , the index is always constant, and so these arguments cannot used in this low-dimensional case.

To prove our result, we use Franks’ lemma which is proved in [12, Proposition 7.4].

Lemma 7. Let , and let be a -neighborhood of in . Then there exist a -neighborhood of and such that if , any finite -invariant set , any neighborhood of , and any volume-preserving linear maps with for all , there is a conservative diffeomorphism coinciding with on and out of , and for all .

Denote by the set of all periodic points of . The following was proved by [7]. Since the paper is still not published yet, we give the proof for completeness.

Lemma 8. Let , and let be a -neighborhood of . If is not hyperbolic, then there is such that has two periodic points with different indices.

Proof. Let be the nonhyperbolic periodic orbit of period and . By Pugh-Robinson’s closing lemma [13] there is , such that is arbitrarily -close to , with close to by closing some recurrent orbit, since Poincaré recurrence almost every point is recurrent. Moreover, since hyperbolicity holds open and is densely even in the volume-preserving setting, can be chosen to be hyperbolic. Let . After this perturbation (away from the orbit of ), we get such that has a periodic orbit close to , with period . We observe that may not be the analytic continuation of and this is precisely the case when is an eigenvalue of the tangent map . If is not hyperbolic take . If is hyperbolic for , then, since is arbitrarily -close to , the distance between the spectrum of and the unitary circle can be taken arbitrarily close to zero. This means that we are in the presence of a very weak hyperbolicity, that is, of a -weak eigenvalue thus in a position to apply [12, Proposition 7.4] to obtain , such that is a non-hyperbolic periodic orbit. Moreover, this local perturbation can be done far from the periodic point . Once again, we use [12, Proposition 7.4] in order to obtain , such that is hyperbolic and .

The following is a volume-preserving diffeomorphism version of [14, Lemma 2.2].

Lemma 9. There is a residual set such that, for any , if for any -neighborhood of there is such that has two hyperbolic periodic points with different indices, then has two hyperbolic periodic points with different indices.

For any , we say that has a -weak eigenvalue if there is an eigenvalue of , such that, , where is the period of .

Lemma 10. There is a residual set such that for any , if has the orbital shadowing property then there is such that for any , does not have a -weak eigenvalue.

Proof. Let , and let have the orbital shadowing property. To derive a contradiction, we may assume that there is such that, for any , has a -weak eigenvalue. Then by Lemma 7, we can find   -close to , such that is not hyperbolic, where is the continuation of . By Lemma 8, again using the Lemma 7, we take  -close to and also -close to such that has two hyperbolic periodic points with . Since , by Lemma 9, has two hyperbolic periodic points, , with . This is a contradiction by Lemma 5.

Lemma 11 (see [15, Lemma 5.1]). There is a residual set such that, for any , for any , if for any -neighborhood , there is , such that for any , has a -weak eigenvalue, then has a -weak eigenvalue.

Proof of Theorem 1. Let , and let have the orbital shadowing property. The proof is by a contradiction; we may assume that . Then there is a non-hyperbolic periodic point for some   -nearby , such that is -weak eigenvalue. Then by Lemma 11, has -weak eigenvalue. This is a contradiction by Lemma 10.

Acknowledgments

The author wishes to express his deepest appreciation to the referee for his careful reading of the paper, critical comments, and valuable 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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