Chain components with $C^1$-stably orbital shadowing

Abstract

Let $f:M\to M$ be a diffeomorphism on a $C^{\mathrm{\infty }}$ n-dimensional manifold. Let $C_f(p)$ be the chain component of f associated to a hyperbolic periodic point p. In this paper, we show that (i) if f has the $C^1$-stably orbitally shadowing property on the chain recurrent set $\mathcal{R}(f)$, then f satisfies both Axiom A and no-cycle condition, and (ii) if f has the $C^1$-stably orbitally shadowing property on $C_f(p)$, then $C_f(p)$ is hyperbolic.

MSC:37C50, 34D10, 37C20, 37C29.

1 Introduction

Let M be a closed $C^{\mathrm{\infty }}$ n-dimensional manifold, and let $Diff(M)$ be the space of diffeomorphisms of M endowed with the $C^1$-topology. Denote by d the distance on M induced from a Riemannian metric $\parallel \cdot \parallel$ on the tangent bundle TM. Let $f\in Diff(M)$. For $\delta >0$, a sequence of points ${\{x_i\}}_{i=a}^b$ ($-\mathrm{\infty }\le a<b\le \mathrm{\infty }$) in M is called a δ-pseudo-orbit of f if $d(f(x_i),x_{i+1})<\delta$ for all $a\le i\le b-1$. Let $\mathrm{\Lambda }\subset M$ be a closed f-invariant set. We say that f has the shadowing property on Λ (or Λ is orbitally shadowable) if for every $ϵ>0$, there is $\delta >0$ such that for any δ-pseudo-orbit ${\{x_i\}}_{i=a}^b\subset \mathrm{\Lambda }$ of f ($-\mathrm{\infty }\le a<b\le \mathrm{\infty }$), there is a point $y\in M$ such that $d(f^i(y),x_i)<ϵ$ for all $a\le i\le b-1$. It is easy to see that f has the shadowing property on Λ if and only if $f^n$ has the shadowing property on Λ for $n\in \mathbb{Z}\setminus \{0\}$. The notion of pseudo-orbits often appears in several methods of the modern theory of a dynamical system. Moreover, the shadowing property plays an important role in the investigation of stability theory and ergodic theory. Actually, in [1, 2], the authors showed that every f satisfying both Axiom A and the strong transversality condition has the shadowing property. Since such a system is structurally stable, there is a $C^1$-neighborhood $\mathcal{U}(f)$ of f such that every $g\in \mathcal{U}(f)$ has the shadowing property because f satisfies both Axiom A and the strong transversality condition. And in [3], Sakai proved that if there is a $C^1$-neighborhood $\mathcal{U}(f)$ of f, for any $g\in \mathcal{U}(f)$, g has the shadowing property, then f satisfies both Axiom A and the strong transversality condition.

For each $x\in M$, let ${\mathcal{O}}_f(x)$ be the orbit of f through x; that is, ${\mathcal{O}}_f(x)=\{f^n(x):n\in \mathbb{Z}\}$. We say that f has the orbital shadowing property on Λ (or Λ is orbitally shadowable) if for any $ϵ>0$, there exists $\delta >0$ such that for any δ-pseudo-orbit $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda }$, we can find a point $y\in M$ such that

$${\mathcal{O}}_f(y)\subset B_{ϵ}(\xi )\text{and}\xi \subset B_{ϵ}({\mathcal{O}}_f(y)),$$

where $B_{ϵ}(A)$ denotes the ϵ-neighborhood of a set $A\subset M$. f is said to have the weak shadowing property on Λ (or Λ is weakly shadowable) if for any $ϵ>0$, there exists $\delta >0$ such that for any δ-pseudo-orbit $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda }$, there is a point $y\in M$ such that $\xi \subset B_{ϵ}({\mathcal{O}}_f(y))$.

The orbital shadowing property is a weak version of the shadowing property: the difference is that we do not require a point $x_i$ of a pseudo-orbit ξ and the point $f^i(y)$ of an exact orbit ${\mathcal{O}}_f(y)$ to be close ‘at any time moment’; instead, the sets of the points of ξ and ${\mathcal{O}}_f(y)$ are required to be close. The weak showing property is a slightly weak version of the orbital shadowing property. The difference is that a set of points of a ‘sufficiently precise’ pseudo-orbit ξ is required to be contained in a small neighborhood of some exact orbit ${\mathcal{O}}_f(y)$. We say that Λ is locally maximal if there is a compact neighborhood U of Λ such that

$$\bigcap _{n\in \mathbb{Z}}{f}^n(U)=\mathrm{\Lambda }.$$

It is easy to see that f has the orbital shadowing property on Λ if and only if $f^n$ has the orbital shadowing property on Λ for $n\in \mathbb{Z}\setminus \{0\}$.

Now we introduce the notion of the $C^1$-stably orbitally shadowing property on Λ. We say that f has the $C^1$-stably orbitally shadowing property on Λ if there are a $C^1$-neighborhood $\mathcal{U}(f)$ of f and a compact neighborhood U of Λ such that (i) $\mathrm{\Lambda }={\bigcap }_{n\in \mathbb{Z}}{f}^n(U)$ (locally maximal), (ii) for any $g\in \mathcal{U}(f)$, g has the orbital shadowing property on ${\mathrm{\Lambda }}_g(U)={\bigcap }_{n\in \mathbb{Z}}{g}^n(U)$, where ${\mathrm{\Lambda }}_g(U)$ is the continuation of Λ. We say that f has the $C^1$-stably orbitally shadowing property if $\mathrm{\Lambda }=M$ in the above definition. It is known that if any structurally stable diffeomorphism f has a $C^1$-neighborhood $\mathcal{U}(f)$ of f, then for any $g\in \mathcal{U}(f)$, g has the shadowing property, hence g has the orbital shadowing property. In [4], the authors showed that if f has the $C^1$-stably orbitally shadowing property, then f satisfies both Axiom A and the strong transversality condition. Thus we can restate the above facts as follows.

Theorem 1.1 [4]

Let $f:M\to M$ be a diffeomorphism. f has the $C^1$-stably orbitally shadowing property if and only if f is structurally stable.

For any $x,y\in M$, we write $x⇝y$ if for any $\delta >0$, there is a δ-pseudo orbit ${\{x_i\}}_{i=a_{\delta }}^{b_{\delta }}$ ($a_{\delta }<b_{\delta }$) of f such that $x_{a_{\delta }}=x$ and $x_{b_{\delta }}=y$. The set of points $\{x\in M:x⇝x\}$ is called the chain recurrent set of f and is denoted by $\mathcal{R}(f)$. It is well known that $\mathcal{R}(f)$ is a closed and f-invariant set. If we denote the set of periodic points of f by $P(f)$, then $P(f)\subset \mathrm{\Omega }(f)\subset \mathcal{R}(f)$. Here $\mathrm{\Omega }(f)$ is the non-wandering set of f. We write $x\leftrightsquigarrow y$ if $x⇝y$ and $y⇝x$. The relation ↭ induces on $\mathcal{R}(f)$ an equivalence relation, whose classes are called chain components of f. Note that by [5], the map $f\mapsto \mathcal{R}(f)$ is upper semi-continuous. From the fact, we will show the following result.

Theorem 1.2 Let $\mathcal{R}(f)$ be the chain recurrent set of f. f has the $C^1$-stably orbitally shadowing property on $\mathcal{R}(f)$ if and only if f satisfies Axiom A and the no-cycle condition.

Let f be an Axiom A diffeomorphism. Then, it is well known that $\mathrm{\Omega }(f)=\mathcal{R}(f)$ if and only if f satisfies the no-cycle condition. Hence, f has the $C^1$-stably orbitally shadowing property on $\mathcal{R}(f)$ and is characterized as the Ω-stability of the system by Theorem 1.2.

We say that Λ is hyperbolic for f if the tangent bundle $T_{\mathrm{\Lambda }}M$ has a Df-invariant splitting $E^s\oplus E^u$ and there exist constants $C>0$ and $0<\lambda <1$ such that

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

for all $x\in \mathrm{\Lambda }$ and $n\ge 0$. If $\mathrm{\Lambda }=M$, then f is Anosov.

If $p\in P(f)$ is a hyperbolic saddle with the period $\pi (p)>0$, then there is no eigenvalues of $D_p{f}^{\pi (p)}$ with modulus equal to 1, at least one of them is greater than 1, at least one of them is smaller than 1. Note that there is a $C^1$-neighborhood $\mathcal{U}(f)$ and a neighborhood U of p such that for all $g\in \mathcal{U}(f)$, there is a unique hyperbolic periodic point $p_g\in U$ of g with the same period as p and $index(p_g)=index(p)$. Here $index(p)=dim{E}_p^s$, and the point $p_g$ is called the continuation of p. The stable manifold $W^s(p)$ and the unstable manifold $W^u(p)$ are defined as follows. It is well known that if p is a hyperbolic periodic point of f with period k, then the sets

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

are $C^1$-injectively immersed submanifolds of M.

Denote by $C_f(p)$ the chain component of f containing p. If p is a sink or source periodic point, then $C_f(p)$ is a periodic orbit itself. Therefore, in this paper, we may assume that all periodic points are the saddle type. Let $q\in P(f)$. We say that p and q are homoclinic related, and write $p\sim q$ if $W^s(p)⋔W^u(q)\ne \mathrm{\varnothing }$, and $W^u(p)⋔W^s(q)\ne \mathrm{\varnothing }$. It is clear that if $p\sim q$ then $index(p)=index(q)$.

Denote by $H_f(p)$ the homoclinic points associated with p, that is, $H_f(p)=\{x\in M:x\in W^s(p)\cap W^u(q)\}$, and let $H_f^T(p)$ be the transverse homoclinic points associated with p, that is, $H_f^T(p)=\{x\in M:x\in W^s(p)⋔W^u(p)\}$. Obviously, $H_f^T(p)$ and $H_f(p)$ are closed f-invariant sets, and it is clear that $H_f^T(p)\subset H_f(p)$. Note that by Smale’s transverse homoclinic point theorem, $H_f^T(p)$ coincides with the closure of the set of all $q\in P(f)$ such that $p\sim q$. It is known that $H_f^T(p)$ is a transitive set, and if $H_f^T(p)$ is not hyperbolic, then it may contain periodic points having different indices. Note that $H_f^T(p)\subset H_f(p)\subset C_f(p)$ (see [[6], examples and counter examples]). For the chain component $C_f(p)$, Lee, Moriyasu and Sakai [7] showed that f has the $C^1$-stably shadowing property on $C_f(p)$ if and only if the chain component $C_f(p)$ is a hyperbolic homoclinic class of p. From the fact, we consider the orbital shadowing property and the chain component. The following result is the main theorem in this paper.

Theorem 1.3 Let p be a hyperbolic periodic point of f, and let $C_f(p)$ be the chain component of f containing p. f has the $C^1$-stably orbitally shadowing property on $C_f(p)$ if and only if $C_f(p)$ is the hyperbolic homoclinic class of p.

If $C_f(p)$ is hyperbolic, then it is locally maximal, that is, there is a compact neighborhood U of $C_f(p)$ such that ${\bigcap }_{n\in \mathbb{Z}}{f}^n(U)=C_f(p)$. By the local stability of a hyperbolic set, there is a $C^1$-neighborhood $\mathcal{U}(f)$ of f such that for any $g\in \mathcal{U}(f)$, ${\mathrm{\Lambda }}_g(U)={\bigcap }_{n\in \mathbb{Z}}{g}^n(U)$ is hyperbolic. Then we know that g has the shadowing property on ${\mathrm{\Lambda }}_g$, and so g has the orbital shadowing property on ${\mathrm{\Lambda }}_g(U)$. Thus we get the ‘if’ part. Thus, we need to show that f has the $C^1$-stably orbitally shadowing property on $C_f(p)$, then $C_f(p)=H_f^T(p)$ is hyperbolic.

2 Proof of Theorem 1.2

Let M be as before, and for $ϵ>0$, we denote by $B_{ϵ}(A)$ the closed ϵ-ball $\{x\in M:d(x,A)\le ϵ\}$ of a subset A of M.

Proposition 2.1 Let $f\in Diff(M)$. Suppose that f has the $C^1$-stably orbitally shadowing property on $\mathcal{R}(f)$. Then f satisfies Axiom A and the no-cycle condition.

Denote by $\mathcal{H}(M)$ the set of homeomorphisms of M. For the proof of the following lemma, see [5].

Lemma 2.2 Let $f\in \mathcal{H}(M)$, and let $\mathcal{R}(f)$ be the chain recurrent set of f. For any $ϵ>0$, there is $\delta >0$ such that if ${\rho }_0(f,g)<\delta$ ($g\in \mathcal{H}(M)$), then $\mathcal{R}(g)\subset B_{ϵ}(\mathcal{R}(f))$.

The following Franks’ lemma will play essential roles in our proofs.

Lemma 2.3 [8]

Let $\mathcal{U}(f)$ be any given $C^1$-neighborhood of f. Then there exist $ϵ>0$ and a $C^1$-neighborhood ${\mathcal{U}}_0(f)\subset \mathcal{U}(f)$ of f such that for given $g\in {\mathcal{U}}_0(f)$, a finite set $\{x_1,x_2,\dots ,x_N\}$, a neighborhood U of $\{x_1,x_2,\dots ,x_N\}$ and linear maps $L_i:T_{x_i}M\to T_{g(x_i)}M$ satisfying $\parallel L_i-D_{x_i}g\parallel \le ϵ$ for all $1\le i\le N$, there exists $\hat{g}\in \mathcal{U}(f)$ such that $\hat{g}(x)=g(x)$ if $x\in \{x_1,x_2,\dots ,x_N\}\cup (M\setminus U)$ and $D_{x_i}\hat{g}=L_i$ for all $1\le i\le N$.

Denote by $\mathfrak{F}(M)$ the set of $f\in Diff(M)$ such that there is a $C^1$-neighborhood $\mathcal{U}(f)$ of f with the property that every $p\in P(g)$ ($g\in \mathcal{U}(f)$) is hyperbolic. It is proved by Hayashi [9] that $f\in \mathfrak{F}(M)$ if and only if f satisfies both Axiom A and the no-cycle condition. Let $\mathrm{\Lambda }\subset M$ be an invariant submanifold of f. We say that Λ is normally hyperbolic if there is a splitting $T_{\mathrm{\Lambda }}M=T\mathrm{\Lambda }\oplus N^s\oplus N^u$ such that

• (a) the splitting depends continuously on $x\in \mathrm{\Lambda }$,

• (b) $D_{x}f(N_x^{\sigma })=N_{f(x)}^{\sigma }$ ($\sigma =s,u$) for all $x\in \mathrm{\Lambda }$,

• (c) there are constants $C>0$ and $\lambda \in (0,1)$ such that for every triple of unit vectors $v\in T_x\mathrm{\Lambda }$, $v^s\in N_x^s$ and $v^u\in N_x^u$ ($x\in \mathrm{\Lambda }$), we have

  $$\frac{\parallel D_x{f}^n(v^s)\parallel }{\parallel D_x{f}^n(v)\parallel }\le C{\lambda }^n\text{and}\frac{\parallel D_x{f}^n(v^u)\parallel }{\parallel D_x{f}^n(v)\parallel }\ge C^{-1}{\lambda }^{-n}$$

for all $n\ge 0$.

Proof of Proposition 2.1 First we suppose that f satisfies both Axiom A and the no-cycle condition. Then $\mathcal{R}(f)=\mathrm{\Omega }(f)=\overline{P(f)}$ is hyperbolic, and so $\mathcal{R}(f)$ is locally maximal. By the stability of locally maximal hyperbolic sets, we can choose a compact neighborhood U of $\mathcal{R}(f)$ and a $C^1$-neighborhood $\mathcal{U}(f)$ of f such that $\mathcal{R}(f)={\bigcap }_{n\in \mathbb{Z}}{f}^n(U)$, and for any $g\in \mathcal{U}(f)$, ${\mathrm{\Lambda }}_g(U)={\bigcap }_{n\in \mathbb{Z}}{g}^n(U)$ is hyperbolic for g. Since $\mathcal{R}(f)$ is hyperbolic for f, f has the shadowing property on $\mathcal{R}(f)$, and so f has the orbital shadowing property on $\mathcal{R}(f)$. Thus, we know that f has the $C^1$-stably orbitally shadowing property on $\mathcal{R}(f)$. Finally, we show that if f has the $C^1$-stably orbitally shadowing property on $\mathcal{R}(f)$, then f satisfies both Axiom A and the no-cycle condition. From the above facts, to complete the proof of the theorem, it is enough to show that if f has the $C^1$-stably orbitally shadowing property on $\mathcal{R}(f)$, then $f\in \mathfrak{F}(M)$.

Suppose that f has the $C^1$-stably orbitally shadowing property on $\mathcal{R}(f)$. Then there are a compact neighborhood U of $\mathcal{R}(f)$ and a $C^1$-neighborhood $\mathcal{U}(f)$ of f such that for any $g\in \mathcal{U}(f)$, g has the orbital shadowing property on ${\mathrm{\Lambda }}_g(U)={\bigcap }_{n\in \mathbb{Z}}{g}^n(U)$, where ${\mathrm{\Lambda }}_g(U)={\bigcap }_{n\in \mathbb{Z}}{g}^n(U)$ is the continuation of Λ. Choose $ϵ>0$ satisfying $B_{ϵ}(\mathcal{R}(f))\subset U$. By Lemma 2.2, there is $\delta >0$ such that if ${\rho }_1(f,g)<\delta$ for $g\in \mathcal{U}(f)$, then

$$\mathcal{R}(g)\subset B_{ϵ}(\mathcal{R}(f))\subset U,$$

(1)

where ${\rho }_1$ is the usual $C^1$-metric on $Diff(M)$. Put ${\mathcal{U}}_0(f)=\{g\in \mathcal{U}(f):{\rho }_1(f,g)<\delta \}$. Then for each $g\in {\mathcal{U}}_0(f)$, $\mathcal{R}(g)\subset U$ and so $\mathcal{R}(g)\subset g^n(U)$ for all $n\in \mathbb{Z}$. This means that $\mathcal{R}(g)\subset {\bigcap }_{n\in \mathbb{Z}}{g}^n(U)={\mathrm{\Lambda }}_g(U)$ for $g\in {\mathcal{U}}_0(f)$. Since g has the orbital shadowing property on ${\mathrm{\Lambda }}_g(U)$, g has the orbital shadowing property on $\mathcal{R}(g)$.

Let $ϵ>0$, and let $\tilde{U}(f)\subset {\mathcal{U}}_0(f)$ be a $C^1$-neighborhood of f which is given by Lemma 2.3 with respect to ${\mathcal{U}}_0(f)$. Let $p\in P(f)$, and let k be the period of p. Let

$$T_{p}M=E^c(p)\oplus E^s(p)\oplus E^u(p),$$

where $E^{\sigma }(p)$, $\sigma =c,s,u$, are $D_p{f}^k$-invariant subspaces corresponding to eigenvalues λ of $D_p{f}^k$ for $|\lambda |=1$, $|\lambda |<1$ and $|\lambda |>1$, respectively. It is sufficient to show that each $p\in P(f)$ is hyperbolic. Suppose this is not true. Then there exist eigenvalues ${\lambda }_1,\dots ,{\lambda }_c$ of $D_p{f}^k$ with $|{\lambda }_j|=1$ or ${\lambda }_j=e^{i{\theta }_j}$, $j=1,2,\dots ,c$. Let ${\mathcal{U}}_{{ϵ}_0}(f)\subset \tilde{U}(f)$ be the $C^1$ ${ϵ}_0$-ball of f. Set $C={sup}_{x\in M}\{\parallel D_{x}f\parallel \}$. For $0<{ϵ}_1<{ϵ}_0$, we can obtain a linear automorphism $\mathcal{O}:T_{p}M\to T_{p}M$ such that

(a.i) $\parallel \mathcal{O}-\mathit{id}\parallel <\frac{{ϵ}_1}{C}$,

(a.ii) keeps $E^{\sigma }$ invariant, where $\sigma =c,s,u$,

(a.iii) all eigenvalues of $\mathcal{O}\circ D_p{f}^k$, say ${\tilde{\lambda }}_j$, $j=1,2,\dots ,c$, are roots of unity.

Let F be a finite set $\{p,f(p),\dots ,f^{k-1}(p)\}$. Define

$$G_j=\{\begin{array}{cc}{D}_{f^j(p)}f,\hfill & j=0,1,\dots ,k-2,\hfill \\ \mathcal{O}\circ D_{f^{k-1}(p)}f,\hfill & j=k-1.\hfill \end{array}$$

Observe that $\parallel G_{k-1}-D_{f^{k-1}(p)}f\parallel \le \parallel \mathcal{O}-\mathit{id}\parallel \parallel D_{f^{k-1}(p)}f\parallel <{ϵ}_0$. Thus $\parallel G_j-D_{f^j(p)}f\parallel <{ϵ}_0$ for all $j=0,1,\dots ,k-1$. By Lemma 2.3, we can find a diffeomorphism $g\in {\mathcal{U}}_{{ϵ}_0}(f)$ and ${\delta }_0>0$ such that

(b.i) $B_{4{\delta }_0}(f^j(p))\subset U$,

(b.ii) $B_{4{\delta }_0}(f^i(p))\cap B_{4{\delta }_0}(p)=\mathrm{\varnothing }$, $0\le i,j\le k-1$, $i\ne j$,

(b.iii) $g=f$ on $F\cup (M-{\bigcup }_{j=0}^{k-1}{B}_{4{\delta }_0}(f^j(p)))$, and

(b.iv) $g={exp}_{f^{j+1}(p)}\circ G_j\circ {exp}_{f^j(p)}^{-1}$ on $B_{{\delta }_0}(f^j(p))$, $0\le j\le k-1$.

Define

$$G=\mathcal{O}\circ D_p{f}^k=\prod _{j=0}^{k-1}{G}_j.$$

Then by (a.iii) we can find $m>0$ such that $G^m{|}_{E^c(p)}=\mathit{id}{|}_{E^c(p)}$. Choose a small ${\delta }_1$ satisfying $0<4{\delta }_1<{\delta }_0$ such that

$$G^{mk}(T_{p}M(4{\delta }_1))\subset T_{p}M({\delta }_0),$$

where $T_{p}M({\delta }_1)=\{v\in T_{p}M|\parallel v\parallel \le {\delta }_1\}$. Then by (b.iv) we have

$${\left(g^k\right)}^m=g^{mk}={exp}_p\circ G^m\circ {exp}_p^{-1}$$

on ${exp}_p(T_{p}M(4{\delta }_1))$. Since ${exp}_p(T_{p}M(4{\delta }_1))\subset B_{{\delta }_0}(f^j(p))$, we get

$${exp}_p(T_{p}M(4{\delta }_1))\subset \bigcap _{n\in \mathbb{Z}}{g}^n\left(B_{{\delta }_0}(f^j(p))\right)\subset {\mathrm{\Lambda }}_g(U)=\bigcap _{n\in \mathbb{Z}}{g}^n(U).$$

We write

$$T_{p}M({\delta }_1)=E^c(p,{\delta }_1)\oplus E^s(p,{\delta }_1)\oplus E^u(p,{\delta }_1),$$

where $E^{\sigma }(p,{\delta }_1)=E^{\sigma }(p)\cap T_{p}M({\delta }_1)$, $\sigma =c,s,u$. Then ${exp}_p(E^c(p,4{\delta }_1))$ is ${(g^k)}^m$-invariant. If the eigenvalue is real, then $dim({exp}_p(E^c(p,4{\delta }_1)))=1$ and ${exp}_p(E^c(p,4{\delta }_1))$ is an arc ${\mathfrak{I}}_{\mathfrak{p}}$ centered at p; and if the eigenvalue is complex, then $dim({exp}_p(E^c(p,4{\delta }_1)))=2$ and ${exp}_p(E^c(p,4{\delta }_1))$ is a disk ${\mathfrak{D}}_{\mathfrak{p}}$ centered at p. We know that ${\mathfrak{I}}_{\mathfrak{p}}\subset \mathcal{R}(g)$ and ${\mathfrak{D}}_{\mathfrak{p}}\subset \mathcal{R}(g)$. By (1), we get

$${\mathfrak{I}}_{\mathfrak{p}}\subset \mathcal{R}(g)\subset {\mathrm{\Lambda }}_g(U)=\bigcap _{n\in \mathbb{Z}}{g}^n(U)$$

and

$${\mathfrak{D}}_{\mathfrak{p}}\subset \mathcal{R}(g)\subset {\mathrm{\Lambda }}_g(U)=\bigcap _{n\in \mathbb{Z}}{g}^n(U).$$

Since g has the orbital shadowing property on $\mathcal{R}(g)$, g must have the orbital shadowing property on ${\mathfrak{I}}_{\mathfrak{p}}$ and ${\mathfrak{D}}_{\mathfrak{p}}$. Since ${\mathfrak{I}}_{\mathfrak{p}}$ and ${\mathfrak{D}}_{\mathfrak{p}}$ are normally hyperbolic, we can see that a shadowing point is in ${\mathfrak{I}}_{\mathfrak{p}}$ and ${\mathfrak{D}}_{\mathfrak{p}}$. Observe that ${(g^k)}^m=\mathit{id}$ on ${exp}_p(E^c(p,4{\delta }_1))$. By our construction, ${(g^k)}^m$ is the identity on the arc ${\mathfrak{I}}_{\mathfrak{p}}$ as well as on the disk ${\mathfrak{D}}_{\mathfrak{p}}$. Since the identity map does not have the orbital shadowing property. Indeed, let $\xi ={\{x_i\}}_{i\in \mathbb{Z}}\subset {\mathfrak{I}}_{\mathfrak{p}}$ be a δ-pseudo orbit. Then, by the orbital shadowing property and normally hyperbolicity, there is a point $y\in {\mathfrak{I}}_{\mathfrak{p}}$ such that ${\mathcal{O}}_g(y)\subset B_{ϵ}({\mathcal{O}}_g(y))$ and $\xi \subset B_{ϵ}({\mathcal{O}}_g(y))$. But since g is the identity map on ${\mathfrak{I}}_{\mathfrak{p}}$ for all $n\in \mathbb{Z}$, $g^n(y)=y$. Thus we have $\xi \not\subset B_{ϵ}({\mathcal{O}}_g(y))$. This is a contradiction. This completes the proof of Proposition 2.1. □

3 Proof of Theorem 1.3

Let $f\in Diff(M)$, and let $p\in P(f)$ be a hyperbolic saddle with period $\pi (p)>0$. Then there are the local stable manifold $W_{ϵ}^s(p)$ and the unstable manifold $W_{ϵ}^u(p)$ of p for some $ϵ=ϵ(p)>0$. It is easily seen that if $d(f^n(x),f^n(p))\le ϵ$, for all $n\ge 0$, then $x\in W_{ϵ}^s(p)$, and if $d(f^n(x),f^n(p))\le ϵ$ for all $n\le 0$, then $x\in W_{ϵ}^u(p)$. Note that the local stable manifold $W_{ϵ}^s(p)\subset W^s(p)$ (resp. the local unstable manifold $W_{ϵ}^u(p)\subset W^u(p)$).

Lemma 3.1 Let p be a hyperbolic periodic point of f, and let $C_f(p)$ be the chain component of f containing p. If f has the orbital shadowing property on $C_f(p)$, then $C_f(p)=H_f(p)$.

Proof We know that $H_f(p)\subset C_f(p)$. We now show that $C_f(p)\subset H_f(p)$. Let $x\in C_f(p)$. Then $x⇝p$ and $p⇝x$. Thus, for any $\eta >0$, there is a periodic η-pseudo orbit ${\{x_i\}}_{i=-n_1}^{n_2}$ of f such that $x_{-n_1}=p$, $x_0=x$ and $x_{n_2}=p$ for some $n_1=n_1(\eta )$ and $n_2=n_2(\eta )>0$. By [[10], Proposition 1.6], ${\{x_i\}}_{i=-n_1}^{n_2}\subset C_f(p)$. To simplify, we may assume that $f(p)=p$. Then we extend the pseudo orbit as follows:

• (i) $x_i=f^{n_1+i}(p)$ for all $i\le -n_1$ and

• (ii) $x_i=f^{i-n_2}(p)$ for all $i\ge n_2$.

Thus we see that $x_i\to p$ as $i\to \pm \mathrm{\infty }$. Then we get an η-pseudo orbit:

$$\begin{array}{rcl}\xi & =& \{\dots ,f^{-n_1}(p)=p,x_{-n_1+1},\dots ,x_{-1},x,x_1,\dots ,x_{n_2-1},f^{n_2}(p)=p,\dots \}\\ =& \{\dots ,x_{-n_1-1},x_{-n_1},x_{-n_1+1},\dots ,x_{-1},x,x_1,\dots ,x_{n_2-1},x_{n_2},x_{n_2+1},\dots \}.\end{array}$$

Since p is a hyperbolic periodic point of f, we can take an $ϵ(p)>0$ such that $d(f^i(x),f^i(p))=d(f^i(x),p)\le ϵ(p)$ for all $i\ge 0$ implies $x\in W_{ϵ(p)}^s(p)$ and $d(f^{-i}(x),f^{-i}(p))=d(f^{-i}(x),p)\le ϵ(p)$ for all $i\ge 0$ implies $x\in W_{ϵ(p)}^u(p)$. Take $ϵ=min\{{ϵ}_0,d(x,p)/4\}$, and let $0<\delta =\delta (ϵ)<ϵ$ be the number in the definition of the orbital shadowing property for f. From the above, we set $\eta =\delta$. Then we see that $\xi \subset C_f(p)$. Since f has the orbital shadowing property on $C_f(p)$, there is $y\in M$ such that

$${\mathcal{O}}_f(y)\subset B_{ϵ}(\xi )\text{and}\xi \subset B_{ϵ}({\mathcal{O}}_f(y)).$$

There are $l_1>0$ and $l_2>0$ such that $d(f^{-n_1}(p),f^{-l_1}(y))=d(p,f^{-l_1}(y))<ϵ$, and $d(f^{n_2}(p),f^{l_2}(y))=d(p,f^{l_2}(y))<ϵ$.

Then $d(p,f^{-l_1-i}(y))<ϵ$ for all $i\ge 0$, and $d(p,f^{l_2+i}(y))<ϵ$ for all $i\ge 0$. Hence $f^{-l_1}(y)\in W_{ϵ(p)}^u(p)$ and $f^{l_2}(y)\in W_{ϵ(p)}^s(p)$. Therefore, we see that $y\in W^s(p)\cap W^u(p)$. Thus we can find $j\in \mathbb{Z}$ such that $f^j(y)\in W^s(p)\cap W^u(p)\cap B_{ϵ}(x)\subset H_f(p)$. □

Let Λ be a closed f-invariant set. Note that if f has the orbital shadowing property on a locally maximal Λ, then the shadowing point can be taken from Λ.

Lemma 3.2 Suppose that f has the $C^1$-stably orbitally shadowing property on Λ. Then for every $p\in \mathrm{\Lambda }\cap P(f)$ is hyperbolic.

Proof Suppose that f has the $C^1$-stably orbitally shadowing property on Λ. Then there exist a compact neighborhood U of Λ and a $C^1$-neighborhood $\mathcal{U}(f)$ of f such that for any $g\in \mathcal{U}(f)$, g has the orbital shadowing property on ${\mathrm{\Lambda }}_g(U)={\bigcap }_{n\in \mathbb{Z}}{g}^n(U)$. Take a $C^1$-neighborhood ${\mathcal{U}}_0(f)\subset \mathcal{U}(f)$ of f as in Lemma 2.3. Then arguing similarly as in the proof of Theorem 1.2, ${\mathcal{U}}_0(f)$ is seen to be a desired neighborhood of f. In fact, suppose $p\in {\mathrm{\Lambda }}_g$ is a non-hyperbolic periodic point for some $g\in {\mathcal{U}}_0(f)$. Observe that we can choose smaller ${\mathcal{U}}_0(f)$ if necessary so that $p\in intU$. Then, by making use of Lemma 2.3, we can construct a diffeomorphism $h\in {\mathcal{U}}_0(f)$ $C^1$-close to g which has the invariant hyperbolic small arc ${\mathfrak{I}}_{\mathfrak{q}}$ and disk ${\mathfrak{D}}_{\mathfrak{q}}$, centered at q, contained in ${\mathrm{\Lambda }}_h(U)$, where ${\mathrm{\Lambda }}_h(U)={\bigcap }_{n\in \mathbb{Z}}{h}^n(U)$. Note that $h_{|\mathfrak{J}}^k=\mathit{id}$ for some $k>0$, where either $\mathfrak{J}={\mathfrak{I}}_{\mathfrak{q}}$ or $\mathfrak{J}={\mathfrak{D}}_{\mathfrak{q}}$. Since the identity map does not have the orbital shadowing property and $h^k$ has the orbital shadowing property on ${\mathfrak{I}}_{\mathfrak{q}}$ as well as on ${\mathfrak{D}}_{\mathfrak{q}}$, we have a contradiction. This completes the proof. □

For $f\in Diff(M)$, we say that a compact f-invariant set Λ admits a dominated splitting if the tangent bundle $T_{\mathrm{\Lambda }}M$ has a continuous Df-invariant splitting $E\oplus F$ and there exist $C>0$, $0<\lambda <1$ such that for all $x\in \mathrm{\Lambda }$ and $n\ge 0$, we have

$$\parallel D{f}^n{|}_{E(x)}\parallel \cdot \parallel D{f}^{-n}{|}_{F(f^n(x))}\parallel \le C{\lambda }^n\dots .$$

Remark 3.3 If Λ admits a dominated splitting $T_{\mathrm{\Lambda }}M=E\oplus F$ such that for any $x\in \mathrm{\Lambda }$, $dimE(x)$ is constant, then there are a $C^1$-neighborhood $\mathcal{U}(f)$ of f and a compact neighborhood V of Λ such that for any $g\in \mathcal{U}(f)$, ${\bigcap }_{n\in \mathbb{Z}}{g}^n(V)$ admits a dominated splitting $T_{{\bigcap }_{n\in \mathbb{Z}}{g}^n(V)}M=E^{\prime }\oplus F^{\prime }$ with $dimE=dim{E}^{\prime }$.

From Lemma 3.2, for any $g\in \mathcal{U}(f)$, the family of periodic sequences of linear isomorphisms of ${\mathbb{R}}^{dimM}$ generated by Dg along the hyperbolic periodic points $q\in {\mathrm{\Lambda }}_g(U)\cap P(g)$ is uniformly hyperbolic (see [11]). Indeed, there is $ϵ>0$ such that for any $g\in \mathcal{U}(f)$, $q\in {\mathrm{\Lambda }}_g(U)\cap P(g)$, and any sequence of linear maps $L_i:T_{g^i(q)}M\to T_{g^{i+1}(q)}M$ with $\parallel L_i-D_{g^i(q)}g\parallel <ϵ$ for $i=1,2,\dots ,\pi (q)-1$, and ${\prod }_{i=0}^{\pi (q)-1}{L}_i$ is hyperbolic. Here $\mathcal{U}(f)$ is the $C^1$-neighborhood of f. Then we can apply Proposition 2.1 in [11] to obtain the following proposition.

Proposition 3.4 Suppose that f has the $C^1$-stably orbitally shadowing property on $C_f(p)$. Then there exist a $C^1$-neighborhood $\mathcal{U}(f)$ of f, constants $C>0$, $0<\lambda <1$ and $m\in {\mathbb{Z}}^{+}$ such that

• (1) for each $g\in \mathcal{U}(f)$, if q is a periodic point of g in ${\mathrm{\Lambda }}_g(U)={\bigcap }_{n\in \mathbb{Z}}{g}^n(U)$ with period $\pi (q,g)$ ($\pi (q,g)\ge m$), then

  $$\prod _{i=0}^{k-1}\parallel D{g}^m{|}_{E^s(g^{im}(q))}\parallel <C{\lambda }^k\mathit{\text{and}}\prod _{i=0}^{k-1}\parallel D{g}^{-m}{|}_{E^u(g^{-im}(q))}\parallel <C{\lambda }^k,$$

where $k=[\pi (q,g)/m]$.

• (2) $C_f(p)$ admits a dominated splitting $T_{C_f(p)}M=E\oplus F$ with $dimE=index(p)$.

Remark 3.5 By Proposition 3.4(2) and [[12], Theorem A], the homoclinic class $H_f(p)$ is the transverse homoclinic class $H_f^T(p)$. Thus if f has the $C^1$-stably orbital shadowing property on $C_f(p)$, then we see that $H_f^T(p)=H_f(p)=C_f(p)$.

Let j denote the $index(p)$, $0<j<dimM$, and let

$$\begin{array}{rcl}{P}_i(f{|}_{C_f(p)})& =& P_i(f{|}_{H_f^T(p)})=P_i(f{|}_{H_f(p)})\\ =& \{q\in H_f^T(p)\cap P(f):index(q)=i\},\end{array}$$

where $0<i<dim(M)$. We write ${\mathrm{\Lambda }}_i(f)=\overline{P_i(f{|}_{H_f(p)})}$. Then we know that these are basic sets and ${\mathrm{\Lambda }}_i(f)=H_f(p)$. In general, a non-hyperbolic homoclinic class $H_f(p)$ contains saddle periodic points with different indices. Since $H_f(p)\subset C_f(p)$, we know that the chain component $C_f(p)$ may contain saddle periodic points with different indices in general. However, if f has the $C^1$-stably orbitally shadowing property on $C_f(p)$, then such a case cannot happen. Thus we need the following proposition to prove Theorem 1.3.

Proposition 3.6 Suppose that f has the $C^1$-stably orbitally shadowing property on $C_f(p)$. Then for any $q\in C_f(p)\cap P(f)$, $index(p)=index(q)$.

To prove Proposition 3.6, we need the following lemma.

Lemma 3.7 Suppose that f has the orbital shadowing property on $C_f(p)$. Then, for any hyperbolic $q\in C_f(p)\cap P(f)$,

$$W^s(p)\cap W^u(q)\ne \mathrm{\varnothing }\mathit{\text{and}}{W}^u(p)\cap W^s(q)\ne \mathrm{\varnothing }.$$

Proof Suppose that f has the orbital shadowing property on $C_f(p)$. Let $ϵ(p)>0$ and $ϵ(q)>0$ be as in the definition of $W_{ϵ(p)}^{s,u}(p)$ and $W_{ϵ(q)}^{s,u}(q)$ with respect to p and q. To simplify notation in this proof, we may assume that $f(p)=p$ and $f(q)=q$. Take $ϵ=min\{ϵ(p),ϵ(q)\}$, and let $0<\delta =\delta (ϵ)\le ϵ$ be the number of the definition of the orbital shadowing property for f.

For $x\in C_f(p)$, there is a finite δ-pseudo orbit ${\{x_i\}}_{i=-n}^n$ ($n\ge 1$) such that

• (i) ${\{x_i\}}_{i=-n}^n\subset C_f(p)$, and

• (ii) $x_{-n}=q$, $x_0=x$ and $x_n=p$.

We extend the finite δ-pseudo orbit as follows: Put

• (i) $x_{-n-i}=f^i(q)$ for all $i\ge 0$, and

• (ii) $x_{n+i}=f^i(p)$ for all $i\ge 0$.

Then we get a δ-pseudo orbit

$$\xi =\{\dots ,q,q,x_{-n+1},\dots ,x_0(=x),\dots ,x_{n-1},p,p,\dots \}.$$

Since f has the orbital shadowing property on $C_f(p)$, there is a point $y\in M$ such that

$${\mathcal{O}}_f(y)\subset B_{ϵ}(\xi )\text{and}\xi \subset B_{ϵ}({\mathcal{O}}_f(y)).$$

Then, we see that ${\mathcal{O}}_f(y)\cap W_{ϵ}^u(q)\cap W_{ϵ}^s(p)\ne \mathrm{\varnothing }$. Thus $W^u(q)\cap W^s(p)\ne \mathrm{\varnothing }$. The other case is similar. □

A diffeomorphism f is said to be Kupka-Smale if the periodic points of f are hyperbolic, and if $p,q\in P(f)$, then $W^s(p)$ is transversal to $W^u(q)$. It is well known that the set of Kupka-Smale diffeomorphisms is a $C^1$-residual set in $Diff(M)$.

Proof of Proposition 3.6 Suppose that f has the $C^1$-stably orbitally shadowing property on $C_f(p)$. Let $\mathcal{U}(f)$ be a $C^1$-neighborhood of f and U be a compact neighborhood of $C_f(p)$ as in the definition. Note that $C_f(p)$ is upper semi continuous and $H_f(p)$ is lower semi continuous. By Lemma 3.1, $C_f(p)=H_f(p)$, $C_f(p)$ is semi continuous. By the definition, $C_g(p_g)\subset {\mathrm{\Lambda }}_g(U)={\bigcap }_{n\in \mathbb{Z}}{g}^n(U)$. To derive a contradiction, we may assume that there is a point $q\in C_f(p)\cap P(f)$ such that for any $g\in {\mathcal{U}}_0(f)\subset \mathcal{U}(f)$, $q\in C_g(p_g)\cap P(g)$ and $index(p)<index(q)$, where $p_g$ is the continuation. By Lemma 3.2, for every $q\in C_f(p)\cap P(f)$ is hyperbolic. Then we know that $dim{W}^s(p)+dim{W}^u(q)<dimM$. Take a Kupka-Smale diffeomorphism $g\in {\mathcal{U}}_0(f)$. Then there are the $p_g$ and $q_g$ that are the continuations of p and q respectively, and $q_g\in C_g(p_g)\cap P(g)\subset {\mathrm{\Lambda }}_g(U)\cap P(g)$. Since $dim{W}^s(p_g)=dim{W}^s(p)$ and $dim{W}^u(q_g)=dim{W}^u(q)$, we know that $W^s(p_g)\cap W^u(q_g)=\mathrm{\varnothing }$, where $W^s(p_g)$ and $W^u(q_g)$ are the stable and the unstable manifolds of $p_g$ and $q_g$ with respect to g. On the other hand, since $g\in {\mathcal{U}}_0(f)$, g has the orbital shadowing property on ${\mathrm{\Lambda }}_g(U)$. Thus g has the shadowing property on $C_g(p_g)$. By Lemma 3.7, $W^s(p_g)\cap W^u(q_g)\ne \mathrm{\varnothing }$. This is a contradiction. □

Let us recall Mañé’s ergodic closing lemma obtained in [11]. Denote by $B_{ϵ}(f,x)$ an ϵ-tubular neighborhood of ${\mathcal{O}}_f(x)$; that is,

$$B_{ϵ}(f,x)\{y\in M:d(f^n(x),y)<ϵ\text{ for some }n\in \mathbb{Z}\}.$$

Let ${\mathrm{\Sigma }}_f$ be the set of points $x\in M$ such that for any $C^1$-neighborhood $\mathcal{U}(f)$ and $ϵ>0$, there are $g\in \mathcal{U}(f)$ and $\gamma \in P(g)$ such that $g=f$ on $M\setminus B_{ϵ}(f,x)$ and $d(f^i(x),g^i(\gamma ))\le ϵ$, for $0\le i\le \pi (\gamma )$. The following lemma is in [11].

Lemma 3.8 [11]

For any f-invariant probability measure μ, we have $\mu ({\mathrm{\Sigma }}_f)=1$.

End of the Proof of Theorem 1.3 Suppose that f has the $C^1$-stably orbitally shadowing property on $C_f(p)$. Let ${\mathcal{U}}_0(f)$ be the $C^1$-neighborhood of f given by Proposition 3.4. To get the conclusion, it is sufficient to show that ${\mathrm{\Lambda }}_i(f)$ is hyperbolic, where ${\mathrm{\Lambda }}_i(f)=\overline{P_i(f{|}_{C_f(p)})}$, and i is the index of p. Fix any neighborhood $U_i\subset U$ of ${\mathrm{\Lambda }}_i(f)$. Note that by Proposition 3.6, ${\mathrm{\Lambda }}_j(f)=\overline{P_j(f{|}_{C_f(p)})}=\mathrm{\varnothing }$ if $i\ne j$.

Thus we show the following: Let $\mathcal{V}(f)\subset {\mathcal{U}}_0(f)$ be a small connected $C^1$-neighborhood of f. If any $g\in \mathcal{V}(f)$ satisfies $q=f$ on $M\setminus U_i$, then $index(p)=index(q)$ for any $p,q\in {\mathrm{\Lambda }}_g(U)\cap P(g)$. Indeed, suppose not, then there are $g_1\in \mathcal{V}(f)$ and $q\in {\mathrm{\Lambda }}_g(U)\cap P(g_1)$ such that $g_1=f$ on $M\setminus U_i$ and $index(p)\ne index(q)$. Suppose that $g_1^n(q)=q$, $k=index(q)$, and define $\gamma :\mathcal{V}(f)\to \mathbb{Z}$ by

$$\gamma (g)=\mathrm{♯}\{y\in {\mathrm{\Lambda }}_g(U)\cap P(g):g^n(y)=y\text{ and }index(y)=k\}.$$

By Lemma 3.2, the function γ is continuous, and since $\mathcal{V}(f)$ is connected, it is constant. But the property of $g_1$ implies $\gamma (g_1)>\gamma (f)$. This is a contradiction.

Finally, to prove Theorem 1.3, we use the proof of Theorem B in [11]. Thus we show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel D_x{f}^n{|}_{E_x}\parallel =0\text{and}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel D_x{f}^{-n}{|}_{F_x}\parallel =0$$

for all $x\in C_f(p)$, and thus, the splitting is hyperbolic.

More precisely, we will prove the case of ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}\parallel D_x{f}_{|E}^n\parallel =0$ (the other case is similar). It is enough to show that for any $x\in C_f(p)$, there exists $n=n(x)>0$ such that

$$\prod _{j=0}^{n-1}\parallel D{f}^m{|}_{E_{f^{mj}(x)}}\parallel <1.$$

We will derive a contraction. If it is not true, then there is $x\in C_f(p)$ such that

$$\prod _{j=0}^{n-1}\parallel D{f}^m{|}_{E_{f^{mj}(x)}}\parallel \ge 1$$

for all $n\ge 0$. Thus

$$\frac{1}{n}\sum _{j=0}^{n-1}log\parallel D{f}^m{|}_{E_{f^{mj}(x)}}\parallel \ge 0$$

for all $n\ge 0$. Define a probability measure

$${\mu }_n:=\frac{1}{n}\sum _{j=0}^{n-1}{\delta }_{f^j(x)}.$$

Then there exists ${\mu }_{n_k}$ ($k\ge 0$) such that ${\mu }_{n_k}\to {\mu }_0\in {\mathcal{M}}_f(M)$, as $k\to \mathrm{\infty }$, where M is a compact metric space. Thus

$$\begin{array}{rcl}\int log\parallel Df{|}_{E_x}\parallel d{\mu }_0& =& \underset{k\to \mathrm{\infty }}{lim}\int log\parallel Df{|}_{E_x}\parallel d{\mu }_{n_k}\\ =& \underset{k\to \mathrm{\infty }}{lim}\int \frac{1}{n}\sum _{j=0}^{n-1}log\parallel Df{|}_{E_{f^j(x)}}\parallel \ge 0.\end{array}$$

By Mañè [[11], p.521],

$${\int }_{C_f(p)}log\parallel Df{|}_{E_x}\parallel d{\mu }_0={\int }_{C_f(p)}\frac{1}{n}\sum _{j=0}^{n-1}log\parallel D_{f^j(r)}f{|}_{E_{f^j(x)}}\parallel d{\mu }_0\ge 0,$$

where ${\mu }_0$ is an f-invariant measure. Let

$$B_{ϵ}(f,x)=\{y\in M:d(f^n(x),y)<ϵ\text{ for some }n\in \mathbb{Z}\},$$

and ${\mathrm{\Sigma }}_f$ = {$x\in M:d(f^n(x),y)<ϵ$, there exist $g\in \mathcal{U}(f)$ and $y\in P(g)$ such that $g=f$ on $M\setminus B_{ϵ}(f,x)$ and $d(f^i(x),f^i(y))\le ϵ$ for $0\le i\le \pi (y)$}.

Note that if $x\notin P(f)$, $0\le \pi (y)=N$ such that $d(f^N(x),f^N(y))=d(f^N(x),y)\to 0$ as $N\to \mathrm{\infty }$, then $d(x,y)\to 0$. So, this is a contradiction.

For any $\mu \in {\mathcal{M}}_f(M)$, $\mu ({\mathrm{\Sigma }}_f)=1$. Then, for any $\mu \in {\mathcal{M}}_f(C_f(p))$,

$$\mu (C_f(p)\cap {\mathrm{\Sigma }}_f)=1,$$

since $\mu (C_f(p))=1$ and $\mu ({\mathrm{\Sigma }}_f)=1$. Thus, $C_f(p)=C_f(p)\cap \mathrm{\Sigma }(f)$ almost everywhere. Therefore,

$${\int }_{C_f(p)\cap \mathrm{\Sigma }(f)}\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{j=0}^{n-1}log\parallel Df{|}_{E_{f^j(x)}}\parallel d\mu \ge 0.$$

By Birkhoff’s theorem and the ergodic closing lemma, we can take $z_0\in C_f(p)\cap \mathrm{\Sigma }(f)$ such that

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{j=0}^{n-1}log\parallel Df{|}_{E_{f^j(z_0)}}\parallel \ge 0.$$

By Proposition 3.4, this is a contradiction.

Thus, by Proposition 3.4, $z_0\notin P(f)$.

Let $K>0$, $m>0$ and $\lambda \in (0,1)$ be given by Proposition 3.4 and take $\lambda <{\lambda }_0<1$ and $n_0>0$ such that

$$\frac{1}{n}\sum _{j=0}^{n-1}log\parallel D{f}_{{|}_{E_{f^{mj}(z_0)}}}^m\parallel \ge log{\lambda }_0\text{if }n\ge n_0.$$

Then, by Mañé’s ergodic closing lemma, we can find $g\in {\mathcal{V}}_0(f)$ $g=f$ on $M\setminus U_j$ and $r_g\in {\mathrm{\Lambda }}_g\cap P(g)$ near by r.

Moreover, we know that $index(r_g)=index(p)$ since $g=f$ on $M\setminus U_j$. By applying Lemma 2.3, we can construct $h\in {\mathcal{V}}_0(f)$ ($\subset \mathcal{V}(f)$) $C^1$-nearby g such that

$${\lambda }_0^k\le \prod _{i=0}^{k-1}\parallel g_1^{im}(r_{g_1})h_{|E_{g_1^{im}(r_{g_1})}}^m\parallel$$

(see [[11], pp.523-524]). On the other hand, by Proposition 3.4, we see that

$$\prod _{i=0}^{k-1}\parallel D_{g_1^{im}(r_{g_1})}{h}_{|E_{g_1^{im}(r_{g_1})}}^m\parallel <K{\lambda }^k.$$

We can choose the period $\pi (r_{g_1})$ ($>n_0$) of $r_{g_1}$ as large as ${\lambda }_0^k\ge K{\lambda }^k$. Here $k=[\pi (r_{g_1})/m]$. This is a contradiction. Thus,

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel D_x{f}_{|E_x}^n\parallel =0$$

for all $x\in C_f(p)$. Therefore, $C_f(p)$ is hyperbolic. This completes the proof of the ‘only if’ part of Theorem 1.3. □