Homoclinic classes with shadowing

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 x i i = a b - ∞ ≤ a < b ≤ ∞ 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 x i i = a δ b δ ⊂ Λ a δ < b δ of f such that x a δ = x and x b δ = y. For given x, y ∈ M, we write x ⇝ y if for any δ > 0, there is a δ-pseudo orbit x i i = a b ( a < b ) 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 R ( f ). If we denote the set of periodic points f by P(f), then P ( f ) ⊂ Ω ( f ) ⊂ R ( f ), where Ω(f) is the non-wandering set of f. The relation ↭ on R ( f ) 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(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 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 W s ( p ) ⋔ W u ( q ) ≠ ∅ and W u ( p ) ⋔ W s ( q ) ≠ ∅. 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∈ℤf ⁿ (U) = Λ. We say that a subset G ⊂ Diff ( M ) is residual if G contains the intersection of a countable family of open and dense subsets of Diff (M); in this case, G 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 G ⊂ Diff ( M ) such that for any f ∈ G ...".

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 C ¹ generic f, if f is tame then the following two conditions are equivalent:

(a)R ( f )is hyperbolic,

(b)R ( f )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 C¹generic 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 U ( f ) of f such that for any g ∈ U ( f ), 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 pointq ∈ H f ( p ) , W s ( p ) ∩ W u ( q ) ≠ ∅, andW u ( p ) ∩ W s ( q ) ≠ ∅.

Proof. We will only show that W s ( p ) ∩ W u ( q ) ≠ ∅. Since p and q are hyperbolic saddles, there are ϵ(p) > 0 and ϵ(q) > 0 such that

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 ( p ) with respect to ϵ. To simplify, we assume that f(p) = p and f(q) = q. Since H f ( p ) = q ∈ P ( f ) : q ~ p ¯, 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(f ⁱ (y),x _i ) < ϵ, for i ∈ ℤ. Thus

Therefore, y ∈ f - n 1 - l W ϵ s ( p ) ∩ f n 2 + l W ϵ u ( q ). This means y ∈ W ^s (p) ∩ W ^u (q), and so W s ( p ) ∩ W u ( q ) ≠ ∅.

Lemma 2.2 There is a residual set G 1 ⊂ Diff ( M ) such that f ∈ G 1 satisfies the following properties:

(b) C _f (p) = H _f (p), where p is a hyperbolic periodic point ([5]).

Lemma 2.3 There is a residual setG 2 ⊂ Diff ( M )such that iff ∈ G 2, and f has the shadowing property on H _f (p), then for any hyperbolic periodic point q ∈ H _f (p),

Proof. Let G 2 be as in Lemma 2.2(a), and let f ∈ 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 ( p ) ∩ W u ( q ) ≠ ∅ and W u ( p ) ∩ W s ( q ) ≠ ∅ by Lemma 2.1. Since f ∈ G 2, we know that

Proposition 2.4 For C¹generic 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

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 setG 3 ⊂ Diff ( M )such that for anyf ∈ G 3, any hyper-bolic periodic point p of f, and

(a) for any η > 0, if for any C¹neighborhoodU ( f )of f, someg ∈ U ( f )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 d₁denotes 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 R in Diff(M) such that for any f in R, the map f ↦ H _f (p)(= C _f (p)) is semi-continuous.

Remark 2.7 There is a residual setG 4 ⊂ Diff ( M )such that for anyf ∈ 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 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 d₁is the C¹metric on M.

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

Remark 2.8[6]There is a residual setG 5 ⊂ Diff ( M )such that for anyf ∈ G 5and any δ > 0, if every periodic point q ~ p has no 2δ-weak eigenvalue, then there is a C¹neighborhoodU ( f )of f such that for anyg ∈ U ( f )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 setG 6 ⊂ Diff ( M )such that iff ∈ 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 δ-weak eigenvalue.

Proof. Let G 6 = ⋂ i = 1 5 G i and let f ∈ 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 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 U ( f ) of f, there is g ∈ U ( f ) such that g has an η/2-simply periodic curve I q, whose endpoints are homoclinically related to p _g , and η/2-simply periodic curve I q ⊂ C g ( p g ) ⊂ B 2 η ( C f ( p ) ) (for more details, see Theorem B in [8]). Then we know that for some l > 0 , I q is a g ^l -invariant small curve containing p _g (q is the center of I q), where q ~ p _g . By Lemma 2.5(a), for the above η > 0 f has a η-simply periodic curve J q such that the endpoints of J q are homoclinically related to p. Then we see that

Then for some l ′ > 0 , 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 J q is a simple periodic curve of f, f | J q is the identity map. Since f has the shadowing property on C _f (p), f must have the shadowing property on J q. But it is a contradiction. Thus every periodic point q ∈ H _f (p) has no δ-weak eigenvalue.

Proof of Proposition 2.4. Let f ∈ G 6, 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 f ∈ 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 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.