Homoclinic classes with shadowing

Abstract

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.

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 diffeo-morphisms of M endowed with the C¹-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 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 ${\left\{x_i\right\}}_{i=a_{\delta }}^{b\delta }\subset \Lambda \left(a_{\delta }\right)<b_{\delta }$ 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 δ-pseudo 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 non-wandering 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 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

$$\begin{array}{c}{W}^s\left(p\right)=\left\{x\in M:f^{kn}\left(x\right)\to p\text{as}n\to \infty \right\}\text{and}\\ W^u\left(p\right)=\left\{x\in M:f^{-kn}\left(x\right)\to p\text{as}n\to \infty \right\}\end{array}$$

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 $W^s\left(p\right)⋔W^u\left(q\right)\ne \mathrm{\varnothing }$ and $W^u\left(p\right)⋔W^s\left(q\right)\ne \mathrm{\varnothing }$. 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_ΛM has a Df-invariant splitting E^s⊕ E^uand there exist constants C > 0 and 0 < λ < 1 such that

$$∥D_x{f}^n{|}_{E_x^s}∥\le C{\lambda }^n\text{and}∥D_x{f}^{-n}{|}_{E_x^u}∥\le C{\lambda }^n$$

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∈ℤfⁿ(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 (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 $\mathcal{G}\subset \text{Diff}\left(M\right)$ such that for any $f\in \mathcal{G}$ ...".

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

Conjecture. For 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)$\mathcal{R}\left(f\right)$is hyperbolic,

(b)$\mathcal{R}\left(f\right)$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.

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 C¹-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 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$q\in H_f\left(p\right),W^s\left(p\right)\cap W^u\left(q\right)\ne \mathrm{\varnothing }$, 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 $W_{ϵ\left(p\right)}^s\left(p\right)$ and $W_{ϵ\left(q\right)}^u\left(q\right)$ are C¹-embedded disks,

• if d(fⁿ(x), fⁿ(p)) < ϵ(p) for n ≥ 0, then $x\in W_{ϵ\left(p1\right)}^s\left(p\right)$,

• if d(fⁿ(x), fⁿ(q)) < ϵ(q) for n ≤ 0 then $x\in W_{ϵ\left(q\right)}^u\left(q\right)$.

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 n₁ > 0 and n₂ > 0 such that

$$d\left(f^{n_1}\left(x\right),p\right)<\delta /4\text{and}d\left(f^{-n_2}\left(x\right),\gamma \right)<\delta /4.$$

Thus

$$d\left(f^{-n_2}\left(x\right),q\right)<d\left(f^{-n_2}\left(x\right),\gamma \right)+d\left(\gamma ,q\right)<\delta /2.$$

Therefore, we can construct a δ-pseudo orbit;

$$\xi =\left\{...,q,f^{-n_2+1}\left(x\right),...,f^{-1}\left(x\right),x,f\left(x\right),...,f^{n_1-1}\left(x\right),p,...,\right\}.$$

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

$$f^{n_1+l}\left(y\right)\in W_{ϵ}^s\left(p\right)\text{and}{f}^{-n_2-l}\left(y\right)\in W_{ϵ}^u\left(q\right),\text{for}l>0.$$

Therefore, $y\in f^{-n_1-l}\left(W_{ϵ}^s\left(p\right)\right)\cap f^{n_2+l}\left(W_{ϵ}^u\left(q\right)\right)$. 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 (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${\mathcal{G}}_2\subset \text{Diff}\left(M\right)$such that if$f\in {\mathcal{G}}_2$, and f has the shadowing property on H _f (p), then for any hyperbolic periodic point q ∈ H _f (p),

$$W^s\left(p\right)⋔W^u\left(q\right)\ne \mathrm{\varnothing },and{W}^u\left(p\right)⋔W^s\left(q\right)\ne \mathrm{\varnothing }.$$

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

$$W^s\left(p\right)⋔W^u\left(q\right)\ne \mathrm{\varnothing },\text{and}{W}^u\left(p\right)⋔W^s\left(q\right)\ne \mathrm{\varnothing }.$$

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),

$$\prod _{i=0}^{\pi \left(q\right)-1}∥D{f}^m{|}_{{}_{E^s\left(f^{im}\left(q\right)\right)}}∥<{\lambda }^{\pi \left(q\right)},\prod _{i=0}^{\pi \left(q\right)-1}∥D{f}^m{|}_{{}_{E^s\left(f^{im}\left(q\right)\right)}}∥<{\lambda }^{\pi \left(q\right)}$$

and

$$∥D{f}^m{|}_{E^s\left(q\right)}∥\cdot ∥D{f}^{-m}{|}_{E^u\left(f^m\left(q\right)\right)}∥<{\lambda }^2,$$

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${\mathcal{G}}_3\subset \text{Diff}\left(M\right)$such that for any$f\in {\mathcal{G}}_3$, any hyper-bolic periodic point p of f, and

(a) for any η > 0, if for any C¹neighborhood$\mathcal{U}\left(f\right)$of f, some$g\in \mathcal{U}\left(f\right)$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 $\mathcal{R}$ in Diff(M) such that for any f in $\mathcal{R}$, the map f ↦ H _f (p)(= C _f (p)) is semi-continuous.

Remark 2.7 There is a residual set${\mathcal{G}}_4\subset \text{Diff}\left(M\right)$such that for any$f\in {\mathcal{G}}_4$, 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 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 set${\mathcal{G}}_5\subset \text{Diff}\left(M\right)$such that for any$f\in {\mathcal{G}}_5$and any δ > 0, if every periodic point q ~ p has no 2δ-weak eigenvalue, then there is a C¹neighborhood$\mathcal{U}\left(f\right)$of f such that for any$g\in \mathcal{U}\left(f\right)$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${\mathcal{G}}_6\subset \text{Diff}\left(M\right)$such that if$f\in {\mathcal{G}}_6$, 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 δ-weak 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 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 $\mathcal{U}\left(f\right)$ of f, there is $g\in \mathcal{U}\left(f\right)$ such that g has an η/2-simply periodic curve ${\mathcal{I}}_q$, whose endpoints are homoclinically related to p _g , and η/2-simply 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 η-simply periodic curve ${\mathcal{J}}_q$ such that the endpoints of ${\mathcal{J}}_q$ are homoclinically related to p. Then we see that

$${\mathcal{J}}_q\subset B_{2\eta }\left(C_f\left(p\right)\right)\subset U.$$

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

$${\mathcal{J}}_q\subset \bigcap _{n\in \mathbb{Z}}{f}^n\left(B_{2\eta }\left(C_f\left(p\right)\right)\right)\subseteq \bigcap _{n\in \mathbb{Z}}{f}^n\left(U\right)=C_f\left(p\right).$$

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 δ-weak 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 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 $f\in {\mathcal{G}}_6$, 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 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.