Abstract

Let be a diffeomorphism on a closed surface. In this paper, we show that if has the -stably limit shadowing property, then we have the following: (i) satisfies the Kupka-Smale condition; (ii) if is dense in the nonwandering set () and if there is a dominated splitting on (), then satisfies both Axiom and the strong transversality condition.

1. Introduction

The theory of shadowing was developed intensively in recent years and became a significant part of the qualitative theory of dynamical systems containing a lot of interesting and deep results (see [1]). Let be a closed manifold and let be the space of diffeomorphisms of endowed with the -topology. Denote by the distance on induced from a Riemannian metric on the tangent bundle . Let . A sequence of points in is called a -pseudo orbit of if for all . Let be a closed -invariant subset of . We say that satisfies the shadowing property on if, for every , there is such that, for every -pseudo orbit of , there exists such that for all . If , we say that has the shadowing property.

The limit shadowing property was originally introduced by Eirola et al. [2], and it was slightly modified in [3]. In this paper we will adapt the definition of the limit shadowing property in [3] as follows. We say that has the limit shadowing property on if there is such that, for any -pseudo orbit with as , which is called a -limit pseudo orbit, there is a point such that as . If , we say that has the limit shadowing property. From the numerical point of view this property of a dynamical system means the following: if we apply a numerical method that approximates with “improving accuracy” so that one-step errors tend to zero as time goes to infinity, then the numerically obtained orbits tend to real ones. Such situations arise, for example, when we are not so interested in the initial behaviour of orbits but want to get areas where “interesting things” happen (e.g., attractors) and then improve accuracy.

Note that the limit shadowing property is different from the shadowing property. In fact, the limit shadowing property needs not have the shadowing property as we can see in [3, Example 4].

The following example shows that every irrational rotation map of the unit circle does not have the limit shadowing property which will be used in the proof of our main theorem.

Example 1. Let be the unit circle and let be defined by , where and is irrational.
Assume, by contradiction, that has the limit shadowing property. It is clear that, for any , there exists rational such that . Let . Then for any .
Now, we can construct a -limit pseudo orbit of by Then, since has the limit shadowing property, there is a point such that as . Then we consider the following cases.
Firstly, if then for all . And . For any , Finally, if with then it is clear that, for ,
This is a contradiction. Therefore, the irrational rotation map does not have the limit shadowing property.

It is a well-known fact that maps are dense in , and so we can consider the following. We say that satisfies the -stably shadowing property if there exists a -neighborhood of such that, for every , satisfies the shadowing property. When is a closed surface, Sakai [4] proved that if has the -stably shadowing property then is Kupka-Smale; that is, every periodic point of is hyperbolic and all their invariant manifolds are transverse. If, in addition, the periodic points of are dense in the nonwandering set and there is a dominated splitting on the closure of periodic points of saddle type, then satisfies both Axiom and the strong transversality condition; that is, is structurally stable.

Definition 2. One says that has the -stably limit shadowing property if is in the -interior of the set of all diffeomorphisms having the limit shadowing property.

Let be an invariant set for . We say that a compact -invariant set admits a dominated splitting if the tangent bundle has a continuous -invariant splitting and there exist , such that for all , .

We say that is hyperbolic for if there is a tangent bundle which has a -invariant continuous splitting and constants and such that for all and .

A set is a basic set if it is compact and locally maximal, and is transitive on . A basic set is called of saddle type if for . As usual, we denote by the set of periodic points of and let be the set of periodic points of saddle type. In this paper, we prove the following theorem.

Theorem 3. Let be a closed surface. If has the -stably limit shadowing property, then one has the following:(i) satisfies the Kupka-Smale condition,(ii)if is dense in the nonwandering set and if there is a dominated splitting on , then satisfies both Axiom and the strong transversality condition.

2. Proof of Theorem 3

First, we show that if has the -stably limit shadowing property, then every periodic point of is hyperbolic. For the proof, we need to perturb some maps but, unfortunately, we cannot use a perturbation lemma, so-called “Franks’ Lemma” which only works for the -topology. Therefore, by a technical reason in the proof, we restrict the manifold to a surface. The proof is motivated by [4].

Proposition 4. If has the -stably limit shadowing property, then every is hyperbolic.

Proof. Let have the -stably limit shadowing property and fix with period . Assume that is not hyperbolic. To simplify, suppose . With a -small perturbation, we can find -nearby such that and where is a constant or matrix and is a hyperbolic matrix (with respect to some coordinates), satisfying one of the following three possible cases:(a),(b),(c)the eigenvalues of are of the form , for some real and .Since the dimension of is 2, we can put
In cases (a) and (b), we approximate by -diffeomorphism (with respect to the -topology) such that(i),(ii)there is a --invariant curve , so-called the center manifold of , which is tangent to the eigenspace associated with (case (a)) or (case (b)),(iii)if we consider and is the origin 0 (with respect to corresponding coordinates), then the restriction has the following expressions (see [5, page 38]): for if is small enough. We may assume that (since the condition is satisfied generically).
In case (a), denote the two disjoint components of by and . Take and and consider the limit pseudo orbit in . We can make as small as we want by letting and . If there exists the limit shadowing point , then its forward orbit is contained in . If then its backward orbit is contained in . If then there is such that , where is a small fixed neighborhood of . Therefore, there does not exist the limit shadowing point of . This is a contradiction since has the limit shadowing property.
In case (b), since with respect to the corresponding coordinates, we see that Thus perturbing in a neighborhood of with respect to the -topology, there exists    (-nearby ) which has the limit shadowing property and such that(i)there exists the center manifold of such that ,(ii) for if is small enough.Clearly, is the identity map.
On the other hand, since has the limit shadowing property, has to have the limit shadowing property. However, we can see that the identity map does not satisfy the limit shadowing property (see [3, Example 3]). This is a contradiction.
In case (c), by the proof of [6, page 23, Theorem 5.2 and Remark 5.3], we will derive a contradiction.
First, we may suppose that there exists a smooth arc of diffeomorphisms on (the corresponding map defined by for is ) such that and is a Hopf point unfolding generically (see [6, page 22]). Then, we approximate the arc by an arc (with respect to the -topology) such that the eigenvalues of have the form with irrational and such that the center manifold is (see [6, page 15]).
Finally, apply the arguments in [6, page 23, Theorem 5.2 and Remark 5.3]. Then slightly perturbing the arc if necessary (with respect to the -topology), we may have the following assertions:(i)there is a -invariant attracting (or repelling) circle (in the manifold) near for small enough,(ii)the restriction is conjugated to a rotation map.
Recall that has the limit shadowing property and is hyperbolic, where is the matrix (for corresponding to . Thus we can see that satisfies limit shadowing, but this is a contradiction because any rotation map does not have the limit shadowing property (cf. Example 1) and so complete the proof.

The notion of transversality between the stable and unstable manifolds of basic sets and was introduced in [7] as follows. If there exists , then for we denote by the connected component of in . Let and be the connected components of . Here . We say that and meet transversely at if , , and , for every . Let be an invariant submanifold of . We say that is normally hyperbolic if there is a splitting , , such that(a)the splitting depends continuously on ,(b) for all ,(c)there are constants , and such that, for every triple of unit vectors , , and , we have  for all .

Let be a closed -invariant set. We say that has the limit shadowing property in if there is such that, for any -limit pseudo orbit, there is a point such that as . Note that the definition is different from that we defined before. In fact, if has the limit shadowing property on , the limit shadowing point needs not be in .

Lemma 5. Let be a normally hyperbolic set for . If has the limit shadowing property on , then the limit shadowing point is in .

Proof. Since is compact and normally hyperbolic, for any and for every , there is such that . Let be as in the limit shadowing property of . Since has the limit shadowing property on , for any -limit pseudo orbit of , there exists such that as . Hence if , then as . Thus the limit shadowing point is in .

Proposition 6. Let and be two basic sets of and let . If has the -stably limit shadowing property, then and meet transversely at .

To prove Proposition 6, we need the following two lemmas.

Lemma 7 (see [8, Lemma 3.2]). Let be basic sets of and suppose that . If has the limit shadowing property, then and meet transversely at .

Lemma 8 (see [9, Lemma 2]). Let be basic sets of and suppose that . Then there are and a -diffeomorphism such that and -axis. Here is defined as before.

Proof of Proposition 6. Let be basic sets of and suppose . We will prove that if there is a -neighborhood of such that every has the limit shadowing property, then . By Lemma 8, there are and a -diffeomorphism such that If , then we have ; that is, and meet transversely at . Thus we assume that . It is easy to see that there are and a -function such that If , then, since , and do not meet transversely at . This is inconsistent with Lemma 7 and so . If we denote a -metric by , then for every , there exists such that since and . Thus, by using a standard procedure, for every and every -neighborhood of such that every has the limit shadowing property, we can construct a -diffeomorphism such that (1) , (2) , (3) , and (4) , where is sufficiently small.
From this we have Here are the stable and unstable manifolds of at . By Lemma 7, this is a contradiction since has the limit shadowing property and so the proof is completed.

Let be a diffeomorphism. For given , we write if, for any , there is a finite -pseudo orbit of such that and . For a closed -invariant set , we say that is chain transitive in (or is chain transitive) if for any , .

To prove Theorem 3, we need some results as follows from [10–12]. Since [10] is preprint, we give the proof here.

Lemma 9 (see [10, Lemma 2.1]). If is chain transitive, then has neither sinks nor sources.

Proof. Let be a sink. Then there exist and such that if then for all . Take such that . For any , let be a -pseudo orbit of such that . For simplicity, we may assume that . Then we have and . Thus we obtain Put . Then if is sufficiently small, we can make . This is a contradiction since .

Theorem 10 (see [11, Lemma 2.3]). Let be a locally maximal set. If has the limit shadowing property on then is chain transitive.

Proposition 11 (see [12]). Let be a -diffeomorphism on a closed surface and let be a compact -invariant set having a dominated splitting . Assume that all the periodic points in are hyperbolic of saddle type. Then, , where is hyperbolic and consist of a finite union of normally hyperbolic periodic simple closed curves such that is conjugated to an irrational rotation. Here denotes the minimal number such that .

Proof of Theorem 3. Let be a closed manifold and let have the -stably limit shadowing property. Then (i) follows from Propositions 4 and 6 directly.
To prove (ii), we suppose and there is a dominated splitting on . By Proposition 4, every is hyperbolic. Hence by Proposition 11, we have , where is hyperbolic and consists of a finite union of normally hyperbolic periodic simple closed curves such that is conjugated to an irrational rotation. Here denotes the smallest number such that . Since is normally hyperbolic, by Lemma 5, the limit shadowing point is in . That is satisfies the limit shadowing property on such that the limit shadowing point is in .
On the other hand, by Example 1, the irrational rotation map does not have the limit shadowing property. Since the limit shadowing property is invariant under a topological conjugacy, is concluded. Thus is hyperbolic. Since has the limit shadowing property, by Theorem 10, it is chain transitive and, by Lemma 9, it has neither sinks nor sources. Therefore satisfies Axiom . The strong transversality condition follows from Proposition 6, and so the proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Manseob Lee 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).