𝐶 2 -Stably Limit Shadowing Diffeomorphisms

Let f be a diffeomorphism on a 𝐶 ∞ closed surface. In this paper, we show that if f has the 𝐶 2 -stably limit shadowing property, then we have the following: (i) f satisfies the Kupka-Smale condition; (ii) if P ( f ) is dense in the nonwandering set Ω ( f ) and if there is a dominated splitting on 𝑃 𝑠 ( f ), then f satisfies both Axiom A and the strong transversality condition.


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 Diff( ) be the space of diffeomorphisms of endowed with the 1 -topology. Denote by the distance on induced from a Riemannian metric ‖ ⋅ ‖ on the tangent bundle . Let ∈ Diff( ). A sequence { } ∈Z of points in is called a -pseudo orbit of if ( ( ), +1 ) < for all ∈ Z. Let Λ be a closed -invariant subset of . We say that satisfies the shadowing property on Λ if, for every > 0, there is > 0 such that, for every -pseudo orbit { } ∈Z ⊂ Λ of , there exists ∈ such that ( ( ), ) < for all ∈ Z. If Λ = , 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 Λ if there is > 0 such that, for any -pseudo orbit { } ∈Z ⊂ Λ with ( ( ), +1 ) → 0 as → ±∞, which is called a -limit pseudo orbit, there is a point ∈ such that ( ( ), +1 ) → 0 as → ±∞. If Λ = , 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 = 1 be the unit circle and let : 1 → 1 be defined by ( ) = + (mod 1), where ∈ [0, 1) and is irrational.
It is a well-known fact that ∞ maps are dense in Diff ( ) ( ≥ 1), and so we can consider the following. We say that satisfies the 2 -stably shadowing property if there exists a 2 -neighborhood U( ) of such that, for every ∈ U( ), satisfies the shadowing property. When is a ∞ closed surface, Sakai [4] proved that if has the 2stably 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 2 -stably limit shadowing property if is in the 2 -interior of the set of all diffeomorphisms having the limit shadowing property.
Let Λ be an invariant set for ∈ Diff( ). We say that a compact -invariant set Λ admits a dominated splitting if the tangent bundle Λ has a continuous -invariant splitting ⊕ and there exist > 0, 0 < < 1 such that for all ∈ Λ, ≥ 0. We say that Λ is hyperbolic for if there is a tangent bundle Λ which has a -invariant continuous splitting ⊕ and constants > 0 and 0 < < 1 such that for all ∈ Λ and ≥ 0.
A set Λ is a basic set if it is compact and locally maximal, and is transitive on Λ. A basic set Λ is called of saddle type if 0 < dim ( ) < dim for ∈ Λ. 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 2stably limit shadowing property, then one has the following: there is a dominated splitting on ( ), then satisfies both Axiom and the strong transversality condition.

Proof of Theorem 3
First, we show that if has the 2 -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 1 -topology. Therefore, by a technical reason in the proof, we restrict the manifold to a surface. The proof is motivated by [4].
Proof. Let have the 2 -stably limit shadowing property and fix ∈ ( ) with period > 0. Assume that is not hyperbolic. To simplify, suppose = 1. With a 2 -small perturbation, we can find 2 -nearby such that ( ) = and where is a constant ∈ R or 2 × 2 matrix and is a hyperbolic matrix (with respect to some coordinates), satisfying one of the following three possible cases: (c) the eigenvalues of are of the form 1 = , 2 = − for some real ̸ = and ∈ Z.
Since the dimension of is 2, we can put = ( ) .
In cases (a) and (b), we approximate by 3 -diffeomorphism 1 (with respect to the 2 -topology) such that , so-called the center manifold of , which is tangent to the eigenspace associated with = 1 (case (a)) or = −1 (case (b)), (iii) if we consider ( , 1 ) ⊂ R and is the origin 0 (with respect to corresponding coordinates), then the restriction 1 | ( , 1 ) has the following expressions (see [5, page 38]): Abstract and Applied Analysis 3 for ∈ ( , 1 ) (⊂ R) if | | is small enough. We may assume that > 0 (since the condition is satisfied generically).
In case (b), since with respect to the corresponding coordinates, we see that Thus perturbing 1 in a neighborhood of with respect to the 2 -topology, there exists 2 ( 3 -nearby 1 ) which has the limit shadowing property and 0 > 0 such that (i) there exists the center manifold ( , 2 ) of such that Clearly, 2 2 | ( , 2 )∩ 0 ( ) is the identity map. On the other hand, since 2 has the limit shadowing property, 2 2 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 { } ∈R of diffeomorphisms on (the corresponding map : × R → × R defined by ( , ) = ( ( ), ) for ( , ) ∈ × R is ∞ ) such that 0 = and ( , 0) ∈ × R is a Hopf point unfolding generically (see [6, page 22]). Then, we approximate the arc by an arc { } ∈R (with respect to the ∞ -topology) such that the eigenvalues of 0 have the form 2 with irrational and such that the center manifold Finally, apply the arguments in [6, page 23, Theorem 5.2 and Remark 5.3]. Then slightly perturbing the arc { } ∈R if necessary (with respect to the ∞ -topology), we may have the following assertions: (i) there is a -invariant attracting (or repelling) circle C (in the manifold) near for > 0 small enough, (ii) the restriction | C 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 | C 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 0 transversality between the stable and unstable manifolds of basic sets Λ and Λ was introduced in be an invariant submanifold of . We say that Λ is normally hyperbolic if there is a splitting Λ = Λ ⊕ , = , , such that (a) the splitting depends continuously on ∈ Λ, (b) ( ) = ( ) ( = , ) for all ∈ Λ, (c) there are constants > 0, and ∈ (0, 1) such that, for every triple of unit vectors V ∈ Λ, V ∈ , and V ∈ ( ∈ Λ), we have for all ≥ 0.
Let Λ be a closed -invariant set. We say that has the limit shadowing property in Λ if there is > 0 such that, for any -limit pseudo orbit, there is a point ∈ Λ such that ( ( ), ) → 0 as → ±∞. Note that the definition is different from that we defined before. In fact, if has the limit shadowing property on Λ, the limit shadowing point needs not be in Λ.

Lemma 5. Let Λ ⊂
be a normally hyperbolic set for . If has the limit shadowing property on Λ, then the limit shadowing point is in Λ.
To prove Proposition 6, we need the following two lemmas.
From this we have Here ( , ℎ) ( = , ) are the stable and unstable manifolds of ℎ at . By Lemma 7, this is a contradiction since ℎ has the limit shadowing property and so the proof is completed.
Theorem 10 (see [11,Lemma 2.3]). Let Λ be a locally maximal set. If has the limit shadowing property on Λ then | Λ is chain transitive.
Proposition 11 (see [12]). Let be a 2 -diffeomorphism on a ∞ closed surface and let Λ be a compact -invariant set having a dominated splitting Λ = ⊕ . Assume that all the periodic points in Λ are hyperbolic of saddle type. Then, Λ = Λ 1 ∪ Λ 2 , where Λ 1 is hyperbolic and Λ 2 consist of a finite union of normally hyperbolic periodic simple closed curves C 1 ∪ ⋅ ⋅ ⋅ ∪ C such that : C → C is conjugated to an irrational rotation. Here denotes the minimal number such that (C ) = C .
Proof of Theorem 3. Let be a ∞ closed manifold and let have the 2 -stably limit shadowing property. Then (i) follows from Propositions 4 and 6 directly.
To prove (ii), we suppose ( ) = Ω( ) and there is a dominated splitting on ( ). By Proposition 4, every ∈ ( ) is hyperbolic. Hence by Proposition 11, we have ( ) = Λ 1 ∪ Λ 2 , where Λ 1 is hyperbolic and Λ 2 consists of a finite union of normally hyperbolic periodic simple closed curves C 1 , . . . , C such that : C → C is conjugated to an irrational rotation. Here denotes the smallest number such that (C ) = C . Since C is normally hyperbolic, by Lemma 5, the limit shadowing point is in C . That is, satisfies the limit shadowing property on C such that the limit shadowing point is in C .

5
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, Λ 2 = 0 is concluded. Thus ( ) = Λ 1 ∪ Λ 2 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.