Homoclinic classes with shadowing

We show that for C¹ generic diffeomorphisms, an isolated homoclinic class is shadow-able if and only if it is a hyperbolic basic set.

Mathematics Subject Classification 2000: 37C20; 37C05; 37C29; 37D05.

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 diffeo-morphisms of M endowed with the C¹-topology. Let f ∈ Diff(M). For δ > 0, a sequence of points in M is called a δ-pseudo orbit of f if d(f(x_i),x_i+1) < δ for all a ≤ i ≤ b - 1. A closed f-invariant set Λ ⊂ M is said to be chain transitive if for any points x, y ⇝ Λ and δ > 0, there is a δ-pseudo orbit of f such that and . For given x, y ∈ M, we write x ⇝ y if for any δ > 0, there is a δ-pseudo orbit 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 . If we denote the set of periodic points f by P(f), then , where Ω(f) is the non-wandering set of f. The relation ↭ on induces an equivalence relation, whose classes are called chain components of f. Every chain component of f is a closed f-invariant 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 δ-pseudo orbit {x_i}_i∈ℤ ⊂ Λ of f there is y ∈ M such that d(fⁱ(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 C¹-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 and . 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_ΛM has a Df-invariant 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 well-known 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 fⁿ(U) = Λ. We say that a subset is residual if contains the intersection of a countable family of open and dense subsets of Diff (M); in this case, is dense in Diff(M). A property "P" is said to be (C¹)-generic if "P" holds for all diffeo-morphisms which belong to a residual subset of Diff(M). We use the terminology "for C¹ generic f" to express "there is a residual subset such that for any ...".

In [1], Abdenur and Díaz posed the following conjecture:

Conjecture. For C¹ 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 C¹-generically, the chain recurrent set is hyperbolic if and only if it has the shadowing property. Next, we prove that C¹-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 C¹-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 diffeomor-phisms, 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 C¹ generic f, if f is tame then the following two conditions are equivalent:

(a) is hyperbolic,

(b) is shadowable.

We say that a closed f-invariant 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 C¹ generic f, the isolated homoclinic class H_f(p) of f containing a hy-perbolic 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 C¹-generically, every locally maximal chain transitive set is hyperbolic if it is shadowable.

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 C¹-neighborhood of f such that for any , any periodic points of g are hyperbolic, then f satisfies both Axiom A and the no-cycle 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 point , and .

Proof. We will only show that . Since p and q are hyperbolic saddles, there are ϵ(p) > 0 and ϵ(q) > 0 such that

• both and are C¹-embedded disks,

• if d(fⁿ(x), fⁿ(p)) < ϵ(p) for n ≥ 0, then ,

• if d(fⁿ(x), fⁿ(q)) < ϵ(q) for n ≤ 0 then .

Set ϵ = min{ϵ(p), ϵ(q)}, and let 0 < δ = δ(ϵ) < ϵ be a number which satisfies the shadowing property of with respect to ϵ. To simplify, we assume that f(p) = p and f(q) = q. Since , 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 n₁ > 0 and n₂ > 0 such that

Thus

Therefore, we can construct a δ-pseudo 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(fⁱ(y),x_i) < ϵ, for i ∈ ℤ. Thus

Therefore, . This means y ∈ W^s(p) ∩ W^u(q), and so .

Lemma 2.2 There is a residual set such that satisfies the following properties:

(a) Every periodic point of f is hyperbolic and all their invariant manifolds are intersect transversely (Kupka-Smale).

(b) C_f(p) = H_f(p), where p is a hyperbolic periodic point ([5]).

Lemma 2.3 There is a residual set such that if , and f has the shadowing property on H_f(p), then for any hyperbolic periodic point q ∈ H_f(p),

Proof. Let be as in Lemma 2.2(a), and let . Take a hyperbolic saddle point q ∈ H_f(p) ∩P(f). Since f has the shadowing property on H_f(p), we know that and by Lemma 2.1. Since , we know that

Proposition 2.4 For C¹ 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 C¹ curve γ is called an η-simply 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(fⁱ (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 δ-weak 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 such that for any , any hyper-bolic periodic point p of f, and

(a) for any η > 0, if for any C¹ neighborhood of f, some has an η-simply periodic curve γ such that two endpoints of γ are homoclinic related with p_g, then f has an 2η-simply 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 δ-weak eigenvalue, then f has a periodic point q' ~ p with δ-weak eigenvalue, whose eigenvalues are all real.

The following lemma shows that the map f ↦ C_f(p) is upper semi-continuous.

Lemma 2.6 For any ϵ > 0, there is δ > 0 such that if d₁ (f,g) < δ then C_g(p_g) ⊂ B_ϵ(C_f(p)), where d₁ denotes the C¹-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 semi-continuous. Thus by Lemma 2.3(b), there is a residual set in Diff(M) such that for any f in , the map f ↦ H_f(p)(= C_f(p)) is semi-continuous.

Remark 2.7 There is a residual set such that for any , 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 semi-continuous, then for any ϵ > 0, there is δ > 0 such that if d₁(f,g) < δ then C_g(p_g) ⊂ B_ϵ (C_f(p)) and C_f(p) ⊂ B_ϵ (C_g(p_g)), where d₁ is the C¹ metric on M.

Let p be a hyperbolic periodic point f ∈ Diff(M).

Remark 2.8 [6]There is a residual set such that for any and any δ > 0, if every periodic point q ~ p has no 2δ-weak eigenvalue, then there is a C¹ neighborhood of f such that for any every periodic point q which is homoclinically related to p_g has no δ-weak eigenvalue, where p_g is the continuation of p.

Lemma 2.9 There is a residual set such that if , 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 δ-weak eigenvalue.

Proof. Let and let . 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 δ-weak 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 C¹-neighborhood of f, there is such that g has an η/2-simply periodic curve , whose endpoints are homoclinically related to p_g, and η/2-simply periodic curve (for more details, see Theorem B in [8]). Then we know that for some is a g^l -invariant small curve containing p_g (q is the center of ), where q ~ p_g. By Lemma 2.5(a), for the above η > 0 f has a η-simply periodic curve such that the endpoints of are homoclinically related to p. Then we see that

Then for some 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 is a simple periodic curve of f, is the identity map. Since f has the shadowing property on C_f(p), f must have the shadowing property on . But it is a contradiction. Thus every periodic point q ∈ H_f(p) has no δ-weak eigenvalue.

Proof of Proposition 2.4. Let , and suppose that there is a nonhyperbolic periodic point q ∈ H_f(p). Then q has a δ-weak eigenvalue. This contradicts Lemma 2.9, and com-pletes the proof of Proposition 2.4 by Mañé [4].

Proof of Theorem 1.2. Let , 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 Propo-sition 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 C¹-generically, H_f(p) is a hyperbolic basic set.

The authors declare that they have no competing interests.

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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. 2011-0015193). 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. 2011-0007649).

1. Abdenur, F, Díaz, LJ: Pseudo-orbit shadowing in the C¹ topology. Disc Contin Dynam Syst. 17, 223–245 (2007)

2. Abdenur, F: Generic robustness of spectral decompositions. Ann Scient Ec Norm Sup. 36, 213–224 (2003)

3. Lee, K, Wen, X: Shadowable chain transitive sets of C¹-generic diffeomorphisms. Bull Korean Math Soc. 49, 263–270 (2012)

4. Mañé, R: An ergodic closing lemma. Ann Math. 116, 503–540 (1982)

5. Bonatti, C, Crovisier, S: Recurrence and genericity. Invent Math. 158, 33–104 (2004)

6. Yang, D, Gan, S: Expansive homoclinic classes. Nonlinearity. 22, 729–733 (2009)

7. Sakai, K: A note on C¹-stably shadowable chain components. Far East J Dyn Syst. 11, 297–300 (2009)

8. Sakai, K: C¹-stably shadowable diffeomorphisms. Ergod Th & Dynam Syst. 28, 987–1029 (2008)

9. Wen, X, Gan, S, Lan, W: C¹-stably shadowable chain component is hyperbolic. J Diff Equ. 246, 340–357 (2009)