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Homoclinic classes with shadowing
Journal of Inequalities and Applications volume 2012, Article number: 97 (2012)
Abstract
We show that for C^{1} generic diffeomorphisms, an isolated homoclinic class is shadowable if and only if it is a hyperbolic basic set.
Mathematics Subject Classification 2000: 37C20; 37C05; 37C29; 37D05.
1 Introduction
Let M be a closed C^{∞} manifold, and denote by d the distance on M induced from the Riemannian metric ∥ · ∥ on the tangent bundle TM. Denote by Diff(M) the space of diffeomorphisms of M endowed with the C^{1}topology. Let f ∈ Diff(M). For δ > 0, a sequence of points {\left\{{x}_{i}\right\}}_{i=a}^{b}\left(\infty \le a<b\le \infty \right) in M is called a δpseudo orbit of f if d(f(x_{ i }),x_{i+1}) < δ for all a ≤ i ≤ b  1. A closed finvariant set Λ ⊂ M is said to be chain transitive if for any points x, y ⇝ Λ and δ > 0, there is a δpseudo orbit {\left\{{x}_{i}\right\}}_{i={a}_{\delta}}^{b\delta}\subset \Lambda \left({a}_{\delta}\right)<{b}_{\delta} of f such that {x}_{{a}_{\delta}}=x and {x}_{{b}_{\delta}}=y. For given x, y ∈ M, we write x ⇝ y if for any δ > 0, there is a δpseudo orbit {\left\{{x}_{i}\right\}}_{i=a}^{b}\left(a<b\right) of f such that x_{ a }= x and x_{ b }= y. Write x ↭ y if x ⇝ y and y ⇝ x. The set of points {x ∈ M : x ↭ x} is called the chain recurrent set of f and is denoted by \mathcal{R}\left(f\right). If we denote the set of periodic points f by P(f), then P\left(f\right)\subset \Omega \left(f\right)\subset \mathcal{R}\left(f\right), where Ω(f) is the nonwandering set of f. The relation ↭ on \mathcal{R}\left(f\right) induces an equivalence relation, whose classes are called chain components of f. Every chain component of f is a closed finvariant set.
Denote by f_{Λ} the restriction of f to the set Λ. We say that f_{Λ} has the shadowing property (or, Λ is shadowable for f) if for any ϵ > 0 there is δ > 0 such that for any δpseudo orbit {x_{ i }}_{i∈ℤ}⊂ Λ of f there is y ∈ M such that d(f^{i}(y),x_{ i }) < ϵ, for i ∈ ℤ.
It is well known that if p is a hyperbolic periodic point f with period k then the sets
are C^{1}injectively immersed submanifolds of M. Every point in the transversal intersection (W^{s}(p) ⋔ W^{u}(p)) of W^{s}(p) and W^{u}(p) is called the homoclinic point of f associated to p. The closure of the homoclinic points of f associated to p is called the homoclinic class of f and it is denoted by H_{ f }(p).
Note that the homoclinic class H_{ f }(p) is a subset of the chain component C_{ f }(p) of f containing p. We consider only the hyperbolic periodic orbits of saddle type. We say that two hyperbolic periodic points p and q are homoclinically related, and write p ~ q, if {W}^{s}\left(p\right)\u22d4{W}^{u}\left(q\right)\ne \mathrm{\varnothing} and {W}^{u}\left(p\right)\u22d4{W}^{s}\left(q\right)\ne \mathrm{\varnothing}. We know that if p ~ q then index(p) = index(q). Here index(p) denotes the dimension of the stable manifold W^{s}(p) of p. By Smale's transverse homoclinic theorem, we know that the closure of the set of homoclinically related points with a hyperbolic periodic point p is the homoclinic class H_{ f }(p) of f associated to p.
We say that Λ is hyperbolic if the tangent bundle T_{Λ}M has a Dfinvariant splitting E^{s}⊕ E^{u}and there exist constants C > 0 and 0 < λ < 1 such that
for all x ∈ Λ and n ≥ 0. It is wellknown that if Λ is hyperbolic, then Λ is shadowable.
We say that Λ is isolated (or locally maximal) if there is a compact neighborhood U of Λ such that ∩_{n∈ℤ}f^{n}(U) = Λ. We say that a subset \mathcal{G}\subset \text{Diff}\left(M\right) is residual if \mathcal{G} contains the intersection of a countable family of open and dense subsets of Diff (M); in this case, \mathcal{G} is dense in Diff(M). A property "P" is said to be (C^{1})generic if "P" holds for all diffeomorphisms which belong to a residual subset of Diff(M). We use the terminology "for C^{1} generic f" to express "there is a residual subset \mathcal{G}\subset \text{Diff}\left(M\right) such that for any f\in \mathcal{G} ...".
In [1], Abdenur and Díaz posed the following conjecture:
Conjecture. For C^{1} generic f, f is shadowable if and only if it is hyperbolic.
In this article, we gives a partial answer to the above conjecture. First, we show that C^{1}generically, the chain recurrent set is hyperbolic if and only if it has the shadowing property. Next, we prove that C^{1}generically, the isolated homoclinic class containing a hyperbolic periodic point is shadowable if and only if it is hyperbolic.
It is explain in [2] that every C^{1}generic diffeomorphism come in one of two types: tame diffeomorphisms, which have a finite number of homoclinic classes and whose nonwandering sets admits partitions into a finite number of disjoint transitive sets; and wild diffeomorphisms, which have an infinite number of homoclinic classes and whose nonwandering sets admit no such partitions. It is easy to show that if a diffeomorphism has a finite number of chain components, then every chain component is locally maximal, and therefore, every chain component of a tame diffeomorphism is locally maximal. Hence, we can get the following result.
Theorem 1.1 For C ^{1} generic f, if f is tame then the following two conditions are equivalent:
(a)\mathcal{R}\left(f\right)is hyperbolic,
(b)\mathcal{R}\left(f\right)is shadowable.
We say that a closed finvariant set Λ is basic , if Λ is isolated, f_{Λ} is transitive and the periodic orbits are dense in Λ. The main result of this article is the following.
Theorem 1.2 For C^{1}generic f, the isolated homoclinic class H_{ f }(p) of f containing a hyperbolic periodic point p is shadowable if and only if it is a hyperbolic basic set.
A similar result for locally maximal chain transitive sets was proved in [3]. More precisely, it is proved that C^{1}generically, every locally maximal chain transitive set is hyperbolic if it is shadowable.
2 Proof of Theorem 1.2
Let M and f ∈ Diff(M) be as before. In this section, to prove Theorem 1.2, we use the techniques developed by Mañé [4]. Let Λ_{ j }(f) be the closure of the set of hyperbolic periodic points of f with index j(0 ≤ j ≤ dimM). If there is a C^{1}neighborhood \mathcal{U}\left(f\right) of f such that for any g\in \mathcal{U}\left(f\right), any periodic points of g are hyperbolic, then f satisfies both Axiom A and the nocycle condition. To prove our result, we first note that if p is homoclinically related to q, then H_{ f }(p) = H_{ f }(q).
Lemma 2.1 Suppose that f has the shadowing property on H_{ f }(p). Then for any hyperbolic periodic pointq\in {H}_{f}\left(p\right),{W}^{s}\left(p\right)\cap {W}^{u}\left(q\right)\ne \mathrm{\varnothing}, and{W}^{u}\left(p\right)\cap {W}^{s}\left(q\right)\ne \mathrm{\varnothing}.
Proof. We will only show that {W}^{s}\left(p\right)\cap {W}^{u}\left(q\right)\ne \mathrm{\varnothing}. Since p and q are hyperbolic saddles, there are ϵ(p) > 0 and ϵ(q) > 0 such that

both {W}_{\u03f5\left(p\right)}^{s}\left(p\right) and {W}_{\u03f5\left(q\right)}^{u}\left(q\right) are C^{1}embedded disks,

if d(f^{n}(x), f^{n}(p)) < ϵ(p) for n ≥ 0, then x\in {W}_{\u03f5\left(p1\right)}^{s}\left(p\right),

if d(f^{n}(x), f^{n}(q)) < ϵ(q) for n ≤ 0 then x\in {W}_{\u03f5\left(q\right)}^{u}\left(q\right).
Set ϵ = min{ϵ(p), ϵ(q)}, and let 0 < δ = δ(ϵ) < ϵ be a number which satisfies the shadowing property of f{}_{{H}_{f}\left(p\right)} with respect to ϵ. To simplify, we assume that f(p) = p and f(q) = q. Since {H}_{f}\left(p\right)=\overline{\left\{q\in P\left(f\right):q~p\right\}}, we can choose a hyperbolic periodic point γ such that γ is homoclinically related to p, and d(q, γ) < δ/2. For x ∈ W^{s}(p) ⋔ W^{u}(γ), choose n_{1} > 0 and n_{2} > 0 such that
Thus
Therefore, we can construct a δpseudo orbit;
Then we have ξ ⊂ H_{ f }(p). Since f has the shadowing property on H_{ f }(p), there is a point y ∈ M such that d(f^{i}(y),x_{ i }) < ϵ, for i ∈ ℤ. Thus
Therefore, y\in {f}^{{n}_{1}l}\left({W}_{\u03f5}^{s}\left(p\right)\right)\cap {f}^{{n}_{2}+l}\left({W}_{\u03f5}^{u}\left(q\right)\right). This means y ∈ W^{s}(p) ∩ W^{u}(q), and so {W}^{s}\left(p\right)\cap {W}^{u}\left(q\right)\ne \mathrm{\varnothing}.
Lemma 2.2 There is a residual set {\mathcal{G}}_{1}\subset \text{Diff}\left(M\right) such that f\in {\mathcal{G}}_{1} satisfies the following properties:
(a) Every periodic point of f is hyperbolic and all their invariant manifolds are intersect transversely (KupkaSmale).
(b) C_{ f }(p) = H_{ f }(p), where p is a hyperbolic periodic point ([5]).
Lemma 2.3 There is a residual set{\mathcal{G}}_{2}\subset \text{Diff}\left(M\right)such that iff\in {\mathcal{G}}_{2}, and f has the shadowing property on H_{ f }(p), then for any hyperbolic periodic point q ∈ H_{ f }(p),
Proof. Let {\mathcal{G}}_{2} be as in Lemma 2.2(a), and let f\in {\mathcal{G}}_{2}. Take a hyperbolic saddle point q ∈ H_{ f }(p) ∩P(f). Since f has the shadowing property on H_{ f }(p), we know that {W}^{s}\left(p\right)\cap {W}^{u}\left(q\right)\ne \mathrm{\varnothing} and {W}^{u}\left(p\right)\cap {W}^{s}\left(q\right)\ne \mathrm{\varnothing} by Lemma 2.1. Since f\in {\mathcal{G}}_{2}, we know that
Proposition 2.4 For C^{1}generic f, if the homoclinic class H_{ f }(p) is isolated and shadowable, then there exist constants m > 0 and 0 < λ < 1 such that for any periodic point q ∈ H_{ f }(p), index(q) = dimW^{s}(q),
and
where π(q) denotes the period of q.
We introduce the following notion which was introduced in [6]. For η > 0 and f ∈ Diff(M), a C^{1} curve γ is called an ηsimply periodic curve of f if

γ is diffeomorphic to [0,1] and its two end points are hyperbolic periodic points of f,

γ is periodic with period π(γ), i.e., f^{π(γ)}(γ) = γ, and l(f^{i}(γ)) < η for any 0 ≤ i ≤ π (γ)  1, where l(γ) denotes the length of γ.

γ is normally hyperbolic.
Let p be a periodic point of f. For δ ∈ (0,1), we say p has a δweak eigenvalue if D f^{π(p)}(p) has an eigenvalue μ such that (1  δ)^{π(p)}< μ < (1 + δ)^{π(p)}.
Lemma 2.5[6]There is a residual set{\mathcal{G}}_{3}\subset \text{Diff}\left(M\right)such that for anyf\in {\mathcal{G}}_{3}, any hyperbolic periodic point p of f, and
(a) for any η > 0, if for any C^{1}neighborhood\mathcal{U}\left(f\right)of f, someg\in \mathcal{U}\left(f\right)has an ηsimply periodic curve γ such that two endpoints of γ are homoclinic related with p_{ g }, then f has an 2ηsimply periodic curve α such that two endpoints of α are homoclinically related with p;
(b) for any δ > 0, if f has a periodic point q ~ p with δweak eigenvalue, then f has a periodic point q' ~ p with δweak eigenvalue, whose eigenvalues are all real.
The following lemma shows that the map f ↦ C_{ f }(p) is upper semicontinuous.
Lemma 2.6 For any ϵ > 0, there is δ > 0 such that if d_{1} (f,g) < δ then C_{ g }(p_{ g }) ⊂ B_{ϵ}(C_{ f }(p)), where d_{1}denotes the C^{1}metric on Diff(M).
Proof. See [[7], Lemma].
Let H_{ f }(p) be the homoclinic class of f associated to p. It is known that the map f ↦ H_{ f }(p) is lower semicontinuous. Thus by Lemma 2.3(b), there is a residual set \mathcal{R} in Diff(M) such that for any f in \mathcal{R}, the map f ↦ H_{ f }(p)(= C_{ f }(p)) is semicontinuous.
Remark 2.7 There is a residual set{\mathcal{G}}_{4}\subset \text{Diff}\left(M\right)such that for anyf\in {\mathcal{G}}_{4}, we have the following property. Let C_{ f }(p) be the isolated chain component of f containing p in an open set U in M. If C_{ f }(p) is semicontinuous, then for any ϵ > 0, there is δ > 0 such that if d_{1}(f,g) < δ then C_{ g }(p_{ g }) ⊂ B_{ϵ} (C_{ f }(p)) and C_{ f }(p) ⊂ B_{ϵ} (C_{ g }(p_{ g })), where d_{1}is the C^{1}metric on M.
Let p be a hyperbolic periodic point f ∈ Diff(M).
Remark 2.8[6]There is a residual set{\mathcal{G}}_{5}\subset \text{Diff}\left(M\right)such that for anyf\in {\mathcal{G}}_{5}and any δ > 0, if every periodic point q ~ p has no 2δweak eigenvalue, then there is a C^{1}neighborhood\mathcal{U}\left(f\right)of f such that for anyg\in \mathcal{U}\left(f\right)every periodic point q which is homoclinically related to p_{ g }has no δweak eigenvalue, where p_{ g }is the continuation of p.
Lemma 2.9 There is a residual set{\mathcal{G}}_{6}\subset \text{Diff}\left(M\right)such that iff\in {\mathcal{G}}_{6}, and the isolated homoclinic class H_{ f }(p) is shadowable, then there is a δ > 0 such that every periodic point q ∈ H_{ f }(p) has no δweak eigenvalue.
Proof. Let {\mathcal{G}}_{6}={\bigcap}_{i=1}^{5}{\mathcal{G}}_{i} and let f\in {\mathcal{G}}_{6}. Assume that f has the shadowing property on the homoclinic class H_{ f }(p). Note that H_{ f }(p) = C_{ f }(p). Suppose that for any δ > 0 there is a hyperbolic periodic point q ∈ C_{ f }(p) such that q has a δweak eigenvalue. Since f has the shadowing property on C_{ f }(p), for the point q ∈ C_{ f }(p) ∩ P(f), we see that q ~ p by Lemma 2.3. Let U be an isolated neighborhood of C_{ f }(p), and let η > 0 be a sufficiently small constant such that B_{2η}(C_{ f }(p)) ⊂ U. Then for any C^{1}neighborhood \mathcal{U}\left(f\right) of f, there is g\in \mathcal{U}\left(f\right) such that g has an η/2simply periodic curve {\mathcal{I}}_{q}, whose endpoints are homoclinically related to p_{ g }, and η/2simply periodic curve {\mathcal{I}}_{q}\subset {C}_{g}\left(pg\right)\subset {B}_{2\eta}\left({C}_{f}\left(p\right)\right) (for more details, see Theorem B in [8]). Then we know that for some l>0,{\mathcal{I}}_{q} is a g^{l}invariant small curve containing p_{ g }(q is the center of {\mathcal{I}}_{q}), where q ~ p_{ g }. By Lemma 2.5(a), for the above η > 0 f has a ηsimply periodic curve {\mathcal{J}}_{q} such that the endpoints of {\mathcal{J}}_{q} are homoclinically related to p. Then we see that
Then for some {l}^{\prime}>0,{\mathcal{J}}_{p} is a f^{l'}invariant small curve center at q. To simplify, we denote f^{l'}by f. Since C_{ f }(p) is an isolated in U, we have
Since {\mathcal{J}}_{q} is a simple periodic curve of f, f{}_{{\mathcal{J}}_{q}} is the identity map. Since f has the shadowing property on C_{ f }(p), f must have the shadowing property on {\mathcal{J}}_{q}. But it is a contradiction. Thus every periodic point q ∈ H_{ f }(p) has no δweak eigenvalue.
Proof of Proposition 2.4. Let f\in {\mathcal{G}}_{6}, and suppose that there is a nonhyperbolic periodic point q ∈ H_{ f }(p). Then q has a δweak eigenvalue. This contradicts Lemma 2.9, and completes the proof of Proposition 2.4 by Mañé [4].
Proof of Theorem 1.2. Let f\in {\mathcal{G}}_{6}, and let C_{ f }(p) be isolated in an open set U. Assume that f has the shadowing property on C_{ f }(p). Then C_{ f }(p) satisfy the assumptions of Proposition 2.4. Since f has the shadowing property on C_{ f }(p), C_{ f }(p) is hyperbolic by the main result in [9]. Consequently, we have proved that C^{1}generically, H_{ f }(p) is a hyperbolic basic set.
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Acknowledgements
We wish to thank the referee for carefully reading the manuscript and providing us many good suggestions. J. Ahn was supported by the BK21 Mathematics Vision 2013 Project. K. Lee was supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (No. 20110015193). M. Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 20110007649).
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Ahn, J., Lee, K. & Lee, M. Homoclinic classes with shadowing. J Inequal Appl 2012, 97 (2012). https://doi.org/10.1186/1029242X201297
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DOI: https://doi.org/10.1186/1029242X201297