Topologically Stable Chain Recurrence Classes for Diffeomorphisms

Abstract

Let $f:M\to M$ be a diffeomorphism of a finite dimension, smooth compact Riemannian manifold M. In this paper, we demonstrate that if a diffeomorphism f lies within the $C^1$ interior of the set of all chain recurrence class-topologically stable diffeomorphisms, then the chain recurrence class is hyperbolic.

1. Introduction

Through the paper we assume that M is a finite dimensional, smooth, compact, and boundaryless Riemannian manifold and $f:M\to M$ is a $C^1$ diffeomorphism. Let d be the distance in M induced from a Riemannian metric $\parallel ·\parallel$ in the tangent bundle $TM$. A closed subset $\Lambda$ of M is hyperbolic if $\Lambda$ is f-invariant and there is an invariant splitting $T_{x}M=E_x^s\oplus E_x^u$ for each $x\in \Lambda$, a constant $\lambda >1$ such that:

(a)

$\parallel D{f}_x\left(u\right)\parallel \le {\lambda }^{-1}\parallel u\parallel$ for $x\in \Lambda$ and $u\in E_x^s,$ and

(b)

$\parallel D{f}_x\left(v\right)\parallel \ge \lambda \parallel v\parallel$ for $x\in \Lambda$ and $v\in E_x^u.$

Notice that a diffeomorphism f of M is Anosov if M is hyperbolic for $f.$ Let $Hom\left(M\right)$ be the set of all homeomorphisms of M. A diffeomorphism f is topologically stable if for any positive $ϵ$, there is a $\delta >0$ such that any $g\in Hom\left(M\right)$ with $d_0(f,g)<\delta$, there is a continuous map $h:M\to M$ for which $h\circ g=f\circ h$ and $d_0(h,id)<ϵ$, where $d_0(f,g)=sup\{d(f\left(x\right),g\left(x\right)),d(f^{-1}\left(x\right),g^{-1}\left(x\right)):x\in M\},$, and $id$ is the identity map. Note that if $f,g:M\to M$ are $C^r(r\ge 1)$ diffeomorphisms then we define the $C^r$ distance between f and g to be:

$$d_r(f,g)=sup\left\{\right|f\left(x\right)-g\left(x\right)|,|Df\left(x\right)-Dg\left(x\right)|,\dots ,|D^{r}f\left(x\right)-D^{r}g\left(x\right)|:x\in M\},$$

where $|·|$ is the operator norm.

Walters [1] proved that if a diffeomorphism f is Anosov, then f is topologically stable. A periodic point p with a period $\pi \left(p\right)$ is hyperbolic if $D_p{f}^{\pi \left(p\right)}$ has no eigenvalues with a norm of $1.$ We define $C^1$ immersed manifolds $W^s\left(p\right)$, which are called the stable manifolds of p, and $W^u\left(p\right)$, which are called the unstable manifolds of p, as follows: $W^s\left(p\right)=\{x\in M:f^{n\pi \left(p\right)}\left(x\right)\to p$ as $n\to \infty \}$ and $W^u\left(p\right)=\{x\in M:f^{n\pi \left(p\right)}\left(x\right)\to p$ as $n\to \infty \}.$$P\left(f\right)$ denotes the set of all periodic points of $f.$ A diffeomorphism f satisfies Axiom A if the non-wandering set $\Omega \left(f\right)$ is hyperbolic and comprises the closure of $P\left(f\right).$ A diffeomorphism f satisfies the strong transversality condition if for any hyperbolic $p,q\in P\left(f\right)$, the stable and unstable manifolds $W^s\left(p\right)$ and $W^u\left(q\right)$ are transverse. A diffeomorphism f is structurally stable if there is a $C^1$ neighborhood $\mathcal{U}\left(f\right)$ such that there is a given $g\in \mathcal{U}\left(f\right)$, and there is a homeomorphism $h:M\to M$ such that $h\circ f=g\circ h.$ For the sake of simplicity, we write that any g$C^1$ in the neighborhood of f, g is equivalent to f. Robinson proved in [2] that a diffeomorphism f is structurally stable if and only if it satisfies Axiom A and the strong transversality condition. Nitecki proved in [3] that if a diffeomorphism f is structurally stable, then f is topologically stable. Moriyasu proved in [4] that if a diffeomorphism f lies within the $C^1$ interior of the set of all topologically stable diffeomorphisms, then it is structurally stable. If a diffeomorphism f satisfies Axiom A, then $\Omega \left(f\right)={\Lambda }_1\cup {\Lambda }_2\cup \cdots \cup {\Lambda }_m,$ where ${\Lambda }_i$ are closed, disjoint, and invariant sets, and each ${\Lambda }_i$ contains dense periodic orbits. The sets ${\Lambda }_1,\dots ,{\Lambda }_m$ are referred to as basic sets.

We say that a diffeomorphism f is Ω-stable if for every diffeomorphism g in the neighborhood of f, ${g|}_{\Omega \left(g\right)}$ is equivalent to ${f|}_{\Omega \left(f\right)},$ where ${f|}_{\Omega \left(f\right)}:\Omega \left(f\right)\to \Omega \left(f\right).$ Smale [5] proved that if a diffeomorphism f is Axiom A and has no-cycles, then it is $\Omega$-stable. Conversely, Palis [6] proved that if a diffeomorphism f is $\Omega$-stable, then f is Axiom A and has no-cycles.

Moriyasu [4] introduced the concept of $\Omega$-topological stability. We say that a diffeomorphism f is Ω-topologically stable if for any positive $ϵ$, there is a positive $\delta$ such that given $g\in Hom\left(M\right)$ with $d_0(f,g)<\delta$, one can choose a continuous map $h:\Omega \left(g\right)\to \Omega \left(f\right)\left(h\right(\Omega \left(g\right))\subset \Omega (f\left)\right)$ such that $h\circ g=f\circ h$ on $\Omega \left(g\right)$ and $d_0(h,id)<ϵ.$

Nitecki proved in [3] that if a diffeomorphism f is Axiom A and has no-cycles, then it is $\Omega$-topologically stable. Conversely, Moryasu proved in [4] that if a diffeomorphism f lies within the $C^1$ interior of the set of all $\Omega$-topologically stable diffeomorphisms, then it is Axiom A and has no-cycles.

For a given $\delta >0$, a bi-sequence of points ${\left\{x_i\right\}}_{i\in \mathbb{Z}}$ of M is said to be a δ-pseudo orbit of f if $d(f\left(x_i\right),x_{i+1})<\delta$$\forall i\in \mathbb{Z}.$ For a given $x,y\in M$, we denote $x⇝y$ if for any $\delta >0$, there is a finite $\delta$-pseudo orbit (or $\delta$-chain from x to y) ${\left\{x_i\right\}}_{i=0}^n(n\ge 1)$ of f such that $x_0=x$ and $x_n=y.$ We denote $x\leftrightsquigarrow y$ if $x⇝y$ and $y⇝x.$ The set $\{x\in M:x\leftrightsquigarrow x\}$ is referred to as the chain recurrent set of f and is denoted as $\mathcal{R}\left(f\right).$ It is seen that $\overline{P\left(f\right)}\subset \Omega \left(f\right)\subset \mathcal{R}\left(f\right).$ The relationship ↭ induces an equivalence relationship on $\mathcal{R}\left(f\right)$, whose classes are called chain recurrence classes of f and are denoted as $C_f$. In general, a chain recurrent class $C_f$ is a closed and f-invariant set. It is known that if $\mathcal{R}\left(f\right)$ is hyperbolic, then $\mathcal{R}\left(f\right)=\overline{P\left(f\right)}$. Therefore, if the chain recurrent set $\mathcal{R}\left(f\right)$ is hyperbolic, then it satisfies Axiom A.

Let $C_f\left(p\right)=\{x\in M:x\leftrightsquigarrow p\}.$ Note that if a hyperbolic $p\in P\left(f\right)$, then there exist a $C^1$ neighborhood $\mathcal{U}\left(f\right)$ of f and a neighborhood U of p such that there is a given $g\in \mathcal{U}\left(f\right)$, the maximal invariant set ${\bigcap }_{n\in \mathbb{Z}}{g}^n\left(U\right)$ of f in U consists of a single hyperbolic $p_g\in P\left(g\right)$, which it has the same period of p and $\mathrm{index}\left(p\right)=\mathrm{index}\left(p_g\right),$ where $P\left(g\right)$ is the set of periodic points of $g.$

Wen and Wen [7] introduced a local version of structural stability. We say that a chain recurrence class $C_f\left(p\right)$ is $C^1$-structurally stable if there is a $C^1$ neighborhood $\mathcal{U}\left(f\right)$ of f such that any $g\in \mathcal{U}\left(f\right)$, one can take a homeomorphism $h:C_f\left(p\right)\to C_g\left(p_g\right)$ such that $h\circ f=g\circ h$ on $C_f\left(p\right)$, where $p_g$ is a continuation of p and $C_g\left(p_g\right)$ is the chain recurrence class of g associated with $p_g.$ Furthermore, they proved that if the codimension one-chain recurrence class $C_f\left(p\right)$ is $C^1$-structurally stable, then it is hyperbolic. Wang [8] proved that if a chain recurrence class $C_f\left(p\right)$ is $C^1$-structurally stable, then it is also hyperbolic, which is a generalization of the result presented by Wen and Wen [7]. Based on the definition below, we consider a local version of the topological stability of the chain recurrence class $C_f\left(p\right).$

For a hyperbolic $p\in P\left(f\right)$, and a closed f-invariant set $\Lambda \subset M$, we say that a diffeomorphism f is chain recurrence class $C_f\left(p\right)$-topologically stable if for a given positive $ϵ$, there is a positive $\delta$ such that there is a given $g\in Hom\left(M\right)$ with $d_0(f,g)<\delta$, there is a continuous surjective map $h:\Lambda \to C_f\left(p\right)$ for which $h\circ g=f\circ h$ on $\Lambda$, where $d\left(h\right(x),x)<ϵ$ for all $x\in \Lambda$.

Remark 1.

In the above notion, Λ is a chain recurrence class for $g.$ For any point $x\in C_f\left(p\right)$, let $\{x_i:x_0=x,x_n=p,x_{2n}=x\}\subset C_f\left(p\right)$ be a δ-chain from x to x. Since $h(\Lambda )=C_f\left(p\right)$, for any $x_i\in C_f\left(p\right)$, there is a $y_i\in \Lambda$ such that $h\left(y_i\right)=x_i$ for all $i=0,1,\dots ,2n,$ where $y_0=y$. Therefore, we know that:

$$d(f\left(x_i\right),x_{i+1})=d(f\left(h\left(y_i\right)\right),h\left(y_{i+1}\right))=d(h\left(g\left(y_i\right)\right),h\left(y_{i+1}\right)),$$

for all $i=0,1,\dots ,2n.$ Since $d\left(h\right(x),x)<ϵ$, one can see that $d(g\left(y_i\right),y_{i+1})<3ϵ$ and ${\left\{y_i\right\}}_{n=0}^{2n}\subset \Lambda ,$ meaning ${\left\{y_i\right\}}_{n=0}^{2n}$ is an $3ϵ$-chain from y to y. Therefore, Λ is a chain recurrence class for g.

Remark 2.

In the above notion, because $h:\Lambda \to C_f\left(p\right)$ is a continuous surjective map, $h(\Lambda )=C_f\left(p\right)$, $h\circ g=f\circ h$ on Λ, and $d\left(h\right(x),x)<ϵ$ for all $x\in \Lambda ,$ meaning we have $h^{-1}\left(p\right)\ne \varnothing$. Therefore, one can take $q\in h^{-1}\left(p\right)$. However, $q\in h^{-1}\left(p\right)$ is not unique. It is clear that the point q is a periodic point of $g.$ Therefore, we set $q=p_g$. Furthermore, we define $C_g\left(p_g\right)$, which is the chain recurrence class of g associated with $p_g.$ Here $p_g$ is not the continuation of p but an element of $h^{-1}\left(p\right).$

For the property of a continuation, we can see in [7,9]. According to Remark 2, the definition can be written as follows.

Definition 1.

Let $p\in P\left(f\right)$ be hyperbolic. We say that a diffeomorphism f is chain recurrence class $C_f\left(p\right)$-topologically stable if for a given positive $ϵ$, there is a positive $\delta$ such that given $g\in Hom\left(M\right)$ with $d_0(f,g)<\delta$, there is a continuous subjective map $h:C_g\left(p_g\right)\to C_f\left(p\right)$ for which $h\circ g=f\circ h$ on $C_g\left(p_g\right)$, where $d\left(h\right(x),x)<ϵ$ for all $x\in C_g\left(p_g\right)$.

$\mathcal{TS}\left(C_f\left(p\right)\right)$ denotes the set of all chain recurrence class $C_f\left(p\right)$-topologically stable diffeomorphism. We say that a diffeomorphism f lies within the $C^1$ interior of the set of all $C_f\left(p\right)$-topologically stable diffeomorphisms if there exists a $C^1$ neighborhood $\mathcal{U}\left(f\right)$ of f such that given $g\in \mathcal{U}\left(f\right)$, g is $C_g\left(p_g\right)$-topologically stable, where $C_g\left(p_g\right)$ is the chain recurrence class of g and $p_g$ is the continuation of p. Here, since g is a diffeomorphism, it guarantees that $p_g$ is the continuation of p. Note that in the definition above, g is a homeomorphism, it does not belong to $int\mathcal{TS}\left(C_g\left(p_g\right)\right)).$

It is known that if $C_f\left(p\right)$ is $C^1$-structurally stable then $C_f\left(p\right)$ ix topologically stable (see [8]). But, the converse is not true. So, we consider the $C^1$ interior elements of $C_f\left(p\right)$-topologically stable diffeomorphisms.

$int\mathcal{TS}\left(C_f\left(p\right)\right)$ denotes the set of $C^1$ interior elements of $\mathcal{TS}\left(C_f\left(p\right)\right).$ The following theorem is the main conclusion of our research.

Theorem A

Let $p\in P\left(f\right)$ be hyperbolic. If a diffeomorphism $f\in int\mathcal{TS}\left(C_f\left(p\right)\right)$, then $C_f\left(p\right)$ is hyperbolic for f.

2. Proof of Theorem A

Let M be defined as shown previously and let $\mathrm{Diff}\left(M\right)$ be the set of all diffeomorphisms of $M.$ For a closed f-invariant set $A\subset M$, A is called normally hyperbolic for f if there is a $Df$-invariant splitting $T_{\Lambda }M=E^s\oplus E^u\oplus TA$ and $\lambda \in (0,1)$ such that for all $x\in A$:

$$\parallel D_x{f|}_{E_x^s}\parallel <\lambda ,\parallel D_x{f}^{-1}{|}_{E_x^u}\parallel <\lambda ,\mathrm{and}$$

$$\parallel D_x{f|}_{E_x^s}\parallel ·\parallel D_{f\left(x\right)}{f}^{-1}{|}_{D_{f\left(x\right)}A}\parallel <\lambda ,\parallel D_x{f}^{-1}{|}_{E_x^u}\parallel ·\parallel D_{f^{-1}\left(x\right)}f{|}_{D_{f^{-1}\left(x\right)}A}\parallel <\lambda .$$

It is known that if $x\in M\backslash A$ then x is hyperbolic point of $f.$

Remark 3.

For a closed f-invariant set $L\subset M$, if the derivative map $D_{x}f$ has an eigenvalue λ ($x\in L$) such that $\left|\lambda \right|=1$, then for some g$C^1$ close to f, we can construct a small closed curve $\mathcal{J}$ such that ${g|}_{\mathcal{J}}:\mathcal{J}\to \mathcal{J}$ is the identity map, meaning $\mathcal{J}$ is a normally hyperbolic set of $g.$

Regarding Remark 3, we have the following.

Lemma 1.

For a diffeomorphism $f:M\to M$, if a closed f-invariant set $\mathcal{I}\subset C_f\left(p\right)$ is normally hyperbolic and ${f|}_{\mathcal{I}}:\mathcal{I}\to \mathcal{I}$ is the identity map, then f is not $C_f\left(p\right)$-topologically stable.

Proof. 

To derive a contradiction, we assume that f is $C_f\left(p\right)$-topologically stable. Let $\mathrm{diam}\mathcal{I}=l$ and take $0<ϵ<l/8$. Since f is $C_f\left(p\right)$-topologically stable, there is a $C^0$ neighborhood ${\mathcal{U}}_0\left(f\right)$ of f such that given $g\in {\mathcal{U}}_0\left(f\right),$ there is a continuous surjective map $h:C_g\left(p_g\right)\to C_f\left(p\right)$ for which $h\circ f=g\circ h$ on $C_g\left(p_g\right)$ and $d\left(h\right(x),x)<ϵ$ for all $x\in C_g\left(p_g\right)$. For any $x\in \mathcal{I}$, there is a $y\in C_g\left(p_g\right)$ such that $h\left(y\right)=x.$ Since ${f|}_{\mathcal{I}}:\mathcal{I}\to \mathcal{I}$ is the identity map, for any $x\in \mathcal{I}$, one can see that $f^i\left(x\right)=x$ for all $i\in \mathbb{Z}.$

We take $c,d\in C_g\left(p_g\right)$ such that (i) $d(c,d)<ϵ/4$, (ii) $h\left(c\right)\in \mathcal{I}$, and $h\left(d\right)\in C_f\left(p\right)\backslash \mathcal{I},$ and (iii) $d(g^k\left(c\right),g^k\left(d\right))<ϵ$ for some $k\in \mathbb{Z}.$

Let $h\left(c\right)=a$ and $h\left(d\right)=b$. Since $\mathcal{I}\subset C_f\left(p\right)$ is normally hyperbolic and $b\in C_f\left(p\right)\backslash \mathcal{I}$, by hyperbolicity of b there is a $j\in \mathbb{Z}$ such that $d(f^j\left(b\right),a)=8ϵ.$ Since $h\circ f=g\circ h$ on $C_g\left(p_g\right)$ and $d\left(h\right(x),x)<ϵ$ for all $x\in C_g\left(p_g\right)$, we have:

$$\begin{array}{cc}\hfill 8ϵ& =d(f^j\left(b\right),a)=d(f^j\left(b\right),f^j\left(a\right))=d(f^j\left(h\left(d\right)\right),f^j\left(h\left(c\right)\right))=d(h\left(g^j\left(d\right)\right),h\left(g^j\left(c\right)\right))\hfill \\ & \le d(h\left(g^j\left(d\right)\right),g^j\left(d\right))+d(g^j\left(d\right),g^j\left(c\right))+d(g^j\left(c\right),h\left(g^j\left(c\right)\right))\hfill \\ & <ϵ+ϵ+ϵ=3ϵ.\hfill \end{array}$$

This creates a contradiction. Therefore, f is not $C_f\left(p\right)$-topologically stable if $\mathcal{I}\subset C_f\left(p\right)$ is normally hyperbolic and ${f|}_{\mathcal{I}}:\mathcal{I}\to \mathcal{I}$ is the identity map.  □

The following lemma is called the Franks’ lemma [10]. It plays an essential role in our proofs.

Lemma 2.

For any $C^1$ neighborhood $\mathcal{U}\left(f\right)$ of f, one can take a positive ϵ and $C^1$ neighborhood ${\mathcal{U}}_0\left(f\right)\subset \mathcal{U}\left(f\right)$ of f such that given $g\in {\mathcal{U}}_0\left(f\right)$, there exists a finite set $S=\{x_1,x_2,\cdots ,x_N\}$, neighborhood U of S, and linear maps $L_i:T_{x_i}M\to T_{g\left(x_i\right)}M$ for which $\parallel L_i-D_{x_i}g\parallel \le ϵ$ ( $1\le i\le N$), one can find a diffeomorphism $g_1\in \mathcal{U}\left(f\right)$ such that:

(a) 

$g_1\left(x\right)=g\left(x\right)$ if $x\in S\cup (M\backslash U)$ and

(b) 

$D_{x_i}{g}_1=L_i$ for all $1\le i\le N$.

Lemma 3.

Suppose that a diffeomorphism $f\in int\mathcal{TS}\left(C_f\left(p\right)\right)$. Then, every periodic point $q\in C_f\left(p\right)$ is hyperbolic.

Proof. 

Let $f\in int\mathcal{TS}\left(C_f\left(p\right)\right)$ and let $\mathcal{U}\left(f\right)$ be a $C^1$ neighborhood of $f.$ Suppose that there are $g\in \mathcal{U}\left(f\right)$ and a periodic point $q\in C_g\left(p_g\right)$ such that q is not hyperbolic. For simplicity, we can assume that $g\left(q\right)=q$ (other cases are similar). Since q is not hyperbolic, there is an eigenvalue $\lambda$ of $D_{q}g$ such that $\left|\lambda \right|=1.$ Note that if all eigenvlaues of $D_{q}g$ are one then we can get a similar result as this proof. Let $E_q^c$ be the eigenspace corresponding to $\lambda .$ Then we have a splitting $T_{d}M=E_q^s\oplus E_q^c\oplus E_q^u$, where $E_q^s$ associated to all eigenvlaues that are less than one and $E_q^u$ associated to all eigenvalues that are greater than one.

First, we consider $\dim E_q^c=1.$ The case means that the eignevlaue $\lambda$ is real. For simplicity, we assume that $\lambda =1$ (other cases are similar). Then, by Lemma 2, there is a $\alpha >0$ and $g_1\in {\mathcal{U}}_0\left(f\right)\subset \mathcal{U}\left(f\right)$ with the following properties:

(i)

$g_1\left(q\right)=g\left(q\right)=q$,

(ii)

$g_1\left(x\right)={\exp }_q\circ D_{q}g\circ {\exp }_q^{-1}\left(x\right)$ for $x\in B_{\alpha }\left(q\right)$, and

(iii)

$g_1\left(x\right)=g\left(x\right)$ for $x\notin B_{4\alpha }\left(q\right),$ where $B_{\alpha }\left(q\right)$ is an $\alpha$ neighborhood of q.

Consider ${\alpha }_1<\alpha$ and define $E_q^c(p,{\alpha }_1)=E_q^c\cap T_{q}M\left({\alpha }_1\right).$ Here, $T_{q}M\left({\alpha }_1\right)=\{v\in T_{q}M:\parallel v\parallel \le {\alpha }_1\}.$ Therefore, it is clear that $g_1{|}_{{\exp }_q\left(E^c(q,{\alpha }_1)\right)}$ is the identity map of ${\exp }_q\left(E^c(q,{\alpha }_1)\right).$

Since $g_1$ is a diffeomorphism and $p_g$ is a hyperbolic periodic point of g, we have $p_{g_1}\in P\left(g_1\right)$ and we can define the chain recurrence class $C_{g_1}\left(p_{g_1}\right)$ associated with $p_{g_1}.$ Since $g_1\in \mathcal{TS}\left(C_{g_1}\left(p_{g_1}\right)\right)$, according to Definition 1.4, for any $ϵ>0$, there is a $\delta >0$ such that for any $g_2\in Hom\left(M\right)$ with $d_0(g_1,g_2)<\delta$, there is a continuous map $h:C_{g_2}\left(p_{g_2}\right)\to C_{g_1}\left(p_{g_1}\right)$ such that $d\left(h\right(x),x)<ϵ$, and $g_2\circ h=h\circ g_1$ for all $x\in C_{g_2}\left(p_{g_2}\right)$, where $p_{g_2}\in h^{-1}\left(p_{g_1}\right)$ and $C_{g_2}\left(p_{g_2}\right)$ is the chain recurrence class associated with $p_{g_2}.$

Let ${\mathcal{J}}_q={\exp }_q\left(E^c(q,{\delta }_1)\right)$. Since $g_1{|}_{{\mathcal{J}}_q}$ is the identity map, every point in ${\mathcal{J}}_q$ is chain transitive, meaning its points are mutually chain equivalent. Therefore, we know that ${\mathcal{J}}_q\subset C_{g_1}\left(p_{g_1}\right).$ Furthermore, by Remark 3, we can see that ${\mathcal{J}}_q$ is a normally hyperbolic set.

Therefore, according to Lemma 1, $g_1$ is not $C_{g_1}\left(p_{g_1}\right)$-topologically stable. This creates a contradiction.

Finally, we consider dim $E_q^c=2.$ The case means that the eigenvalue $\lambda$ is complex. In this case, to avoid notational complexity, we can assume that $g\left(q\right)=q$ for some $g\in \mathcal{U}\left(f\right)$. Then, similar to the proof of the first case, we can take ${ϵ}_0>0$ and $g_1$$C^1$ close to g with the following properties: (i) $g_1\left(q\right)=g\left(q\right)=q$ and (ii) $g_1$ has a small arc ${\mathcal{L}}_q$, where ${\mathcal{L}}_q={\exp }_q\left(\{t·v_0:1\le t\le 1+{ϵ}_0/4\}\right)$ for some ${ϵ}_0>0$. The small arc ${\mathcal{L}}_q$ has the following properties for $g_1$:

(a)

$g_1^i\left({\mathcal{L}}_q\right)\cap g_1^j\left({\mathcal{L}}_q\right)=\varnothing$ for $0\le i\ne j<l-1$,

(b)

$g_1^l\left({\mathcal{L}}_q\right)={\mathcal{L}}_q$ for some $l>0$, and

(c)

$g_1^l{|}_{{\mathcal{L}}_q}$ is the identity map.

Since $g_1^l{|}_{{\mathcal{L}}_q}$ is the identity map, we can easily show that ${\mathcal{L}}_q\subset C_{g_1}\left(p_{g_1}\right).$ Let $g_1^l=g_1$. Then, just as in the argument for the first case, we can derive a contradiction. Therefore, Lemma 3 is proved.  □

Let p and q be hyperbolic periodic points. We say that q is homoclinically related to p if $W^s\left(p\right)⋔W^u\left(q\right)\ne \varnothing$ and $W^u\left(p\right)⋔W^s\left(q\right)\ne \varnothing .$ Then, $W^s\left(p\right)⋔W^u\left(q\right)\ne \varnothing$ and $W^u\left(p\right)⋔W^s\left(q\right)\ne \varnothing$ are denoted by $p\sim q$. Next, we define $H_f\left(p\right)=\overline{\{q\in P_h\left(f\right):q\sim p\}}$, where $P_h\left(f\right)$ is the set of all hyperbolic periodic points of $f.$

Let p be a periodic point with a period $\pi \left(p\right)$ of a diffeomorphism f. If ${\lambda }_1,{\lambda }_2,\dots ,$ and ${\lambda }_d$ are the eigenvalues of $D_p{f}^{\pi \left(p\right)}$, then the numbers ${\chi }_i=\frac{1}{\pi \left(p\right)}log\left|{\lambda }_i\right|(i=1,\dots ,d)$ are called the Lyapunov exponents of p.

Lemma 4.

A diffeomorphism f in a dense ${\mathcal{G}}_{\delta }$ subset $\mathcal{G}$ in $\mathrm{Diff}\left(M\right)$ has the following properties:

(a) 

A chain recurrence class $C_f\left(p\right)$ is a homoclinical class $H_f\left(p\right)$ for some hyperbolic periodic point p (see [11]);

(b) 

If a homoclinical class $H_f\left(p\right)$ is not hyperbolic, then one can find a periodic point q that is homoclinically related to p and has a Lyapunov exponent arbitrarily close to 0 (see [8]).

Lemma 5.

Let p be a hyperbolic periodic point of f. If the point q is homoclinically related to p and has a Lyapunov exponent arbitrarily close to 0, then there is a g$C^1$ close to f such that $D_{q_g}{g}^{\pi \left(q\right)}$ has an eigenvalue λ such that $\left|\lambda \right|=1.$

Proof. 

Suppose that there is a periodic point q that is homoclinically related to p and has a Lyapunov exponent arbitrarily close to 0. Then, we know that there is an eigenvalue ${\lambda }_i$ of $D_q{f}^{\pi \left(q\right)}$ such that ${\lambda }_i$ is close to 1 for some $i=1,\dots ,d.$ From Lemma 2, there is a g$C^1$ close to f such that $D_{q_g}{g}^{\pi \left(q_g\right)}$ has an eigenvalue ${\mu }_i$ such that $|{\mu }_i|=1.$  □

Proof of Theorem A.

By contradiction, suppose that $C_f\left(p\right)$ is not hyperbolic. Let $\mathcal{U}\left(f\right)$ be a $C^1$ neighborhood of $f\in \mathrm{Diff}\left(M\right).$ Since $f\in int\mathcal{TS}\left(C_f\left(p\right)\right)$, there is a $g\in \mathcal{U}\left(f\right)\cap \mathcal{G}$ with the following properties:

(i)

$g\in \mathcal{TS}\left(C_g\left(p_g\right)\right)$,

(ii)

$C_g\left(p_g\right)=H_g\left(p_g\right)$, and

(iii)

There is a hyperbolic periodic point $q\in C_g\left(p_g\right)$ with a Lyapunov exponent arbitrarily close to 0 such that $q\sim p_g$.

Since q has a Lyapunov exponent arbitrarily close to 0, from Lemma 5, there is a $g_1\in \mathcal{U}\left(g\right)\subset \mathcal{U}\left(f\right)\cap \mathcal{G}$ such that $D_{q_{g_1}}{g}_1^{\pi \left(q_{g_1}\right)}$ has an eigenvalue $\lambda$ such that $\left|\lambda \right|=1.$ Just as in the proof of Lemma 3, there is a $g_2\in \mathcal{U}\left(g_1\right)\subset \mathcal{U}\left(f\right)\cap \mathcal{G}$ such that $g_2$ has a small arc ${\mathcal{J}}_{q_{g_2}}$ centered at $q_{g_2}.$ Then, we have that $g_2^{\pi \left(q_{g_2}\right)}$ is $\pm id$ on ${\mathcal{J}}_{q_{g_2}}$ and ${\mathcal{J}}_{q_{g_2}}\subset C_{g_2}\left(p_{g_2}\right),$ where $id$ is the identity map. Additionally, ${\mathcal{J}}_{q_{g_2}}$ is normally hyperbolic. From Lemma 1, this creates a contradiction. Therefore, if $f\in int\mathcal{TS}\left(C_f\left(p\right)\right)$, then $C_f\left(p\right)$ is hyperbolic. □