Positively Continuum-Wise Expansiveness for C¹ Differentiable Maps

Abstract

We show that if a differentiable map f of a compact smooth Riemannian manifold M is $C^1$ robustly positive continuum-wise expansive, then f is expanding. Moreover, $C^1$-generically, if a differentiable map f of a compact smooth Riemannian manifold M is positively continuum-wise expansive, then f is expanding.

1. Introduction and Statements

Starting with Utz [1], expansive dynamical systems have been studied by researchers. Regarding this concept, many researchers suggest various expansivenesses (e.g., N-expansive [2], measure expansive [3] and continuum-wise expansive [4]). These concepts were used to show chaotic systems (see References [3,5,6,7]) and hyperbolic structures (see References [8,9,10,11,12,13,14]).

For chaoticity, Morales and Sirvent proved in Reference [3] that every Li-Yorke chaotic map in the interval or the unit circle are measure-expansive. Kato proved in Reference [7] that, if a homeomorphism f of a compactum X with $\dim X>0$ is continuum-wise expansive and Z is a chaotic continuum of f, then either f or $f^{-1}$ is chaotic in the sense of Li and Yorke on almost all Cantor sets $C\subset Z.$ Hertz [5,6] proved that if a homeomorphism f of locally compact metric space X or Polish continua X is expansive or continuum-wise expansive then f is sensitive dependent on the initial conditions.

For hyperbolicity, Mañé proved in Reference [12] that if a diffeomorphism f of a compact smooth Riemannian manifold M is robustly expansive then it is quasi-Anosov. Arbieto proved in Reference [8] that, $C^1$ generically, if a diffeomorphism f of a compact smooth Riemannian manifold M is expansive then it is Axiom A and has no cycles. Sakai proved in Reference [13] that, if a diffeomorphism f of a compact smooth Riemannian manifold M is robustly expansive then it is quasi-Anosov. Lee proved in Reference [9] that, $C^1$ generically, if a diffeomorphism f of a compact smooth Riemannian manifold M is continuum-wise expansive then it is Axiom A and has no cycles.

Through these results, we are interested in general concepts of expansiveness. Actively researching positive expansivities (positively expansive [15], positively measure-expansive [16,17]) is a motivation of this paper. In this paper, we study positively continuum-wise expansiveness, which is the generalized notion of positive expansiveness and positive measure expansiveness.

In this paper, we assume that M is a compact smooth Riemannian manifold. A differentiable map $f:M\to M$ is positively expansive(write $f\in \mathcal{PE}$) if there exists a constant $\delta >0$ such that for any $x,y\in M$, if $d(f^i(x),f^i(y))\le \delta$$\forall i\ge 0$ then $x=y$. From Reference [18], if a differentiable map $f\in \mathcal{PE}$ then f is open and a local homeomorphism. For any $\delta >0$, we define a dynamical $\delta$-ball for $x\in M$ such as $\{y\in M:d(f^i(x),f^i(y))\le \delta$ $\forall i\ge 0\}$. Put ${\Gamma }_{\delta }^{+}(x)=\{y\in M:d(f^i(x),f^i(y))\le \delta$$\forall i\ge 0\}$. Note that if a differentiable map $f\in \mathcal{PE}$, then ${\Gamma }_{\delta }^{+}(x)=\left\{x\right\}$ for any $x\in M$. Here $\delta >0$ is called an expansive constant of $f.$

Let us introduce a generalization of the positively expansive called the positively measure-expansive (see Reference [3]). Let $\mathcal{M}(M)$ be the space of a Borel probability measure of M. A measure $\mu \in \mathcal{M}(M)$ is atomic if $\mu (\{x\})\ne 0,$ for some point $x\in M.$ Let $\mathcal{A}(M)$ be the set of atomic measures of M. Note that $\mathcal{A}(M)$ is dense in $\mathcal{M}(M).$ Let ${\mathcal{M}}^{*}(M)=\mathcal{M}(M)\setminus \mathcal{A}(M).$ A differentiable map $f:M\to M$ is positively measure-expansive (write $f\in \mathcal{PME}$) if there exists a constant $\delta >0$ such that $\mu ({\Gamma }_{\delta }(x))=0$ for any $\mu \in {\mathcal{M}}^{*}(M),$ where $\delta >0$ is called a measure expansive constant. In Reference [17], the authors found that there exists a differentiable map $f:S^1\to S^1$ that is positively $\mu$-expansive for any $\mu \in {\mathcal{M}}_f^{*}(S^1)$ but not positively expansive where ${\mathcal{M}}_f^{*}(M)$ is the set of non-atomic invariant measures of M.

Now, we introduce another generalization of the positive expansiveness, which is called positively continuum-wise expansiveness (see Reference [4]). We say that C is a continuum if it is compact and connected.

Definition 1.

A differentiable map f is positively continuum-wise expansive (write $f\in \mathcal{PCWE}$) if there is a constant $e>0$ such that if $C\subset M$ is a non-trivial continuum, then there is $n\ge 0$ such that $\mathrm{diam}{f}^n(C)>e$, where if C is a trivial, then C is a one point set.

Note that $f\in \mathcal{PCWE}$ if and only if $f^n\in \mathcal{PCWE}$$\forall n\ge 1.$ We say that f is countably expansive (write $f\in \mathcal{CE}$) if there is a constant $\delta >0$ such that for all $x\in M$, ${\Gamma }_{\delta }^{+}(x)=\{y\in M:d(f^i(x),f^i(y))\le \delta$$\forall i\in \mathbb{Z}\}$ is countable. In Reference [19], the authors showed that if a homeomorphism $f:M\to M$ is measure expansive then f is countably expansive. Moreover, the converse is true. Then, as in the proof of Theorem 2.1 in Reference [19], it is easy to show that f is positively countable-expansive if and only if f is positively measure expansive. In this paper, we consider the relationship between the positively measure-expansive and the positively continuum-wise expansive (see Lemma 1). We can know that if f is positively measure-expansive then it is not positively continuum-wise expansive because a continuum is not countable, in general.

Definition 2.

A differentiable map $f:M\to M$ is expanding if there exist constants $C>0$ and $\lambda >1$ such that

$$\parallel D_x{f}^n(v)\parallel \ge C{\lambda }^n\parallel v\parallel ,$$

for any vector $v\in T_{x}M(x\in M)$ and any $n\ge 0$.

Note that a positively measure-expansive differentiable map is not necessarily expanding. However, under the $C^1$ robust or $C^1$ generic condition, it is true.

A differentiable map f is $C^1$ robustly positive $\mathfrak{P}$ if there exists a $C^1$ neighborhood $\mathcal{U}(f)$ of f such that for any $g\in \mathcal{U}(f)$, g is positive $\mathfrak{P}$.

A point $x\in M$ is a singular if $D_{x}f:T_{x}M\to T_{f(x)}M$ is not injective. Denoted by $S_f$ the set of singular points of $f.$

Sakai proved in Reference [15] that if a differentiable map f is $C^1$ robustly positive expansive then $S_f=\varnothing$ and it is an expanding map. Lee et al. [17] proved that if f is $C^1$ robustly positive measure-expansive, then $S_f=\varnothing$ and it is expanding. Note that if a differentiable map f is expanding then it is expansive. According to these facts, we prove the following.

Theorem A

If a differentiable map $f:M\to M$ is $C^1$ robustly positive continuum-wise expansive (write $f\in \mathcal{RPCWE}$) then $S_f=\varnothing$ and it is expanding.

Let $D^1(M)$ be the set of differentiable maps $f:M\to M$. Note that $D^1(M)$ contains the set of diffeomorphisms ${\mathrm{Diff}}^1(M)$ on M and ${\mathrm{Diff}}^1(M)$ is open in $D^1(M)$. We say that a subset $\mathcal{G}\subset D^1(M)$ is residual if it contains a countable intersection of open and dense subsets of $D^1(M)$. Note that the countable intersection of residual subsets is a residual subset of $D^1(M)$. A property “P” holds generically if there exists a residual subset $\mathcal{G}\subset D^1(M)$ such that for any $f\in \mathcal{G},$ f has the “P”. Some times we write for $C^1$ generic $f\in D^1(M)$ which means that there exists a residual set $\mathcal{G}\subset D^1(M)$ such that for any $f\in \mathcal{G}.$ Arbieto [8] and Sakai [15] proved that, $C^1$ generically, a positively expansive map is expanding. Ahn et al. [16] proved that for a $C^1$ generic $f\in D^1(M)$, if $S_f=\varnothing$ and f is positively measure expansive, then it is expanding. Recently, Lee et al. [17] showed that, $C^1$ generically, if $f\in D^1(M)$ is positively measure-expansive then $S_f=\varnothing$ and f is expanding. According to these results, we consider $C^1$ generic positively continuum-wise expansive for $f\in D^1(M)$ and prove the following.

Theorem B

For $C^1$ generic $f\in D^1(M)$, if f is positively continuum-wise expansive then $S_f=\varnothing$ and it is expanding.

2. The Proof of Theorem A

The following proof is similar to Lemma 2.2 in Reference [19].

Lemma 1.

Let $C\subset M$ be compact and connected. A differentiable map $f\in \mathcal{PCWE}$ if and only if there is a constant $\delta >0$ such that for all $x\in M$, if a continuum $C\subset {\Gamma }_{\delta }^{+}(x)$ then C is a trivial continuum set.

Proof. 

Let $\delta >0$ be a continuum-wise expansive constant and C be compact and connected (that is, a continuum). Take $c=\delta /2.$. We assume that for any $x\in M$, if $C\subset {\Gamma }_c^{+}(x)$ then $\mathrm{diam}{f}^n(C)\le 2c$ for all $n\ge 0.$ Since f is positively continuum-wise expansive, C should be a trivial continuum set. Thus, if $f\in \mathcal{PCWE}$, then for all $x\in M$, if a continuum $C\subset {\Gamma }_c^{+}(x)$, then C is a trivial continuum set.

For the converse part, suppose that $f\in \mathcal{PCWE}$. Then, there is a constant $c>0$ such that $\mathrm{diam}{f}^n(C)\le c$ $\forall n\ge 0,$ where C is a continuum. Let $x\in C$ be given. Since $\mathrm{diam}{f}^n(C)\le c$, for all $y\in C$ we have

$$d(f^n(x),f^n(y))\le c\forall n\ge 0.$$

Thus, we know $y\in {\Gamma }_c(x).$ Since $y\in C$ and y is arbitrary, we have $C\subset {\Gamma }_c(x).$ Since a continuum $C\subset {\Gamma }_c(x)$, we have that C is a trivial continuum set. □

A periodic point $p\in P(f)$ is hyperbolic if $D_p{f}^{\pi (p)}:T_{p}M\to T_{p}M$ has no eigenvalue with a modulus equal to 0 or 1, where $\pi (p)$ is the period of $p.$ Then, $T_{p}M=E_p^s\oplus E_p^u$ of subspaces such that

(a)

$D_p{f}^{\pi (p)}(E_p^{\sigma })=E_p^{\sigma }$($\sigma =s,u)$, and

(b)

there exist constants $C>0$, and $\lambda \in (0,1)$ satisfies for all positive integer $n\in \mathbb{N}$,

• $\Vert D_p{f}^n(v)\Vert \le C{\lambda }^n\Vert v\Vert$ for any $v\in E_p^s$, and
• $\Vert D_p{f}^{-n}(v)\Vert \le C{\lambda }^n\Vert v\Vert$ for any $v\in E_p^u$

A hyperbolic point $p\in P(f)$ is a sink if $E_p^u=\left\{0\right\}$, a source if $E_p^s=\left\{0\right\}$, and a saddle if $E_p^s\ne \left\{0\right\}$ and $E_p^u\ne \left\{0\right\}$. Let $P_h(f)$ be the set of hyperbolic periodic points of $f.$ The dimension of the stable manifold $W^s(p)=\{x\in M:d(f^i(x),f^i(p))\to 0$ as $i\to \infty \}$ is written by the index of p, and denoted by $\mathrm{ind}(p)$. Then, we know $0\le \mathrm{ind}(p)\le \dim M.$ Let $P_i(f)$ be the set of all $p\in P_h(f)$ with $\mathrm{ind}(p)=i.$

Lemma 2.

If a differentiable map $f\in \mathcal{PCWE}$ then $P_i(f)=\varnothing$ for $1\le i\le \dim M.$

Proof. 

By contradiction, we assume that there is $i\in [1,\dim M]$ such that $P_i(f)\ne \varnothing$. Take $p\in P_i(f)$ and $\delta >0$. Then, we can find a local stable manifold $W_{\delta }^s(p)$ of p such that $W_{\delta }^s(p)\ne \varnothing .$ We can construct a continuum ${\mathcal{J}}_p\subset W_{\delta }^s(p)$ centered at p such that $\mathrm{diam}{\mathcal{J}}_p=\delta /4.$ Let ${\Gamma }_{\delta /2}^{+}(p)=\{y\in M:d(f^i(p),f^i(y))\le \delta /2$ $\forall i\ge 0\}$. Then, we know ${\mathcal{J}}_p\subset {\Gamma }_{\delta /2}^{+}(p).$ By Lemma 1, ${\mathcal{J}}_p$ should be a trivial continuum set. This is a contradiction since ${\mathcal{J}}_p$ is not a trivial continuum set. □

In Reference [17], the authors showed that there is a positively expansive differentiable map $f:S^1\to S^1$ such that $S_f\ne \varnothing .$ Thus, if f is positively measure-expansive then $S_f\ne \varnothing .$ But if f is $C^1$ robustly positive measure-expansive then $S_f=\varnothing .$ For that, we consider that f is $C^1$ robustly positive continuum-wise expansive.

The following is a version of differentiable maps of Franks’ lemma (see Lemma 2.1 in Reference [8]).

Lemma 3

([20]). Let $f:M\to M$ be a differentiable map and let $\mathcal{U}(f)$ be a $C^1$ neighborhood of $f.$ Then, there exists $\delta >0$ such that for a finite set $A=\{x_1,x_2,\dots ,x_n\}\subset M,$ a neighborhood U of A and a linear map $L_i:T_{x_i}M\to T_{f(x_i)}M$ satisfying $\parallel L_i-D_{x_i}f\parallel <\delta$ for $1\le i\le n,$ there exist ${\epsilon }_0>0$ and $g\in \mathcal{U}(f)$ having the following properties;

(a)

$g(x)=f(x)$ if $x\in A$, and

(b)

$g(x)={\exp }_{f(x_i)}\circ L_i\circ {\exp }_{x_i}^{-1}(x)$ if $x\in B_{{\epsilon }_0}(x_i)$ and $\forall i\in \{1,\dots ,n\}.$

It is clear that assertion (b) implies that

$$g(x)=f(x)ifx\in A$$

and that $D_{x_i}g=L_i,$$\forall i\in \{1,\dots ,n\}.$

Theorem 1.

If a differentiable map $f\in \mathcal{RPCWE}$ then $S_f=\varnothing .$

Proof. 

Suppose that there is $x\in S_f$. Then, by Lemma 3, we can take g$C^1$ close to f such that g has a closed connected small arc $B_{ϵ}(x)$ centered at x with radius $ϵ>0$, such that $\dim B_{ϵ}(x)=1$ and $g(B_{ϵ}(x))$ is one point. Take $\delta =2ϵ$. Let ${\Gamma }_{\delta }^{+}(x)=\{y\in M:d(g^i(x),g^i(y))\le \delta$$\forall i\ge 0\}$. It is clear $B_{ϵ}(x)\subset {\Gamma }_{\delta }^{+}(x).$ Since $g(B_{ϵ}(x))$ is one point, for any $y\in B_{ϵ}(x),$ we know that $\mathrm{diam}{g}^i(B_{ϵ}(x))\le \delta$ for all $i\ge 0.$ However, $B_{ϵ}(x)$ is not a trivial continuum set, by Lemma 1 this is a contradiction. □

Recall that a differentiable map $f:M\to M$ is star if every periodic point of $g(C^1$ nearby $f)$ is hyperbolic.

Lemma 4.

If a differentiable map $f\in \mathcal{RPCWE}$ then f is star.

Proof. 

Suppose that f is not star. Then, we can take g$C^1$ close to f such that g has a non-hyperbolic $p\in P(g).$ As Lemma 3, we can find $g_1$$C^1$ close to g ($g_1$$C^1$ close to f) such that $D_p{g}_1^{\pi (p)}$ has an eigenvalue $\lambda$ with $\left|\lambda \right|=1.$ For simplicity, we assume that $g_1^{\pi (p)}(p)=g_1(p)=p.$ Let $E_p^c$ be associated with $\lambda$. If $\lambda \in \mathbb{R}$ then $\dim E_p^c=1$, and if $\lambda \in \mathbb{C}$ then $\dim E_p^c=2.$

First, we consider $\dim E_p^c=1.$ Then, we assume that $\lambda =1$ (the other case can be proved similarly). By Lemma 3, there are $ϵ>0$ and h$C^1$ close to $g_1$ (also, $C^1$ close to f), having the following properties;

• $h(p)=g_1(p)=p,$
• $h(x)={\exp }_p\circ D_p{g}_1\circ {\exp }_p^{-1}(x)$ if $x\in B_{ϵ}(p),$ and
• $h(x)=g_1(x)$ if $x\notin B_{4ϵ}(p).$

Since $\lambda =1$, we can construct a closed connected small arc ${\mathcal{I}}_p\subset B_{ϵ}(p)\cap {\exp }_p(E_p^c(ϵ))$ with its center at p such that

• $\mathrm{diam}{\mathcal{I}}_p=ϵ/4,$
• $h({\mathcal{I}}_p)={\mathcal{I}}_p,$ and
• the map ${h|}_{{\mathcal{I}}_p}:{\mathcal{I}}_p\to {\mathcal{I}}_p$ which is the identity.

Take $\delta =ϵ/2$. Let ${\Gamma }_{\delta }^{+}(p)=\{x\in M:d(h^i(x),h^i(p))\le \delta$ $\forall i\ge 0\}$. Then, it is clear ${\mathcal{I}}_p\subset {\Gamma }_{\delta }(p),$ and $\mathrm{diam}{h}^i({\mathcal{I}}_p)=\mathrm{diam}{\mathcal{I}}_p$ for all $i\ge 0.$ Since $f\in \mathcal{RPCWE}$, according to Lemma 1, ${\mathcal{I}}_p$ has to be just a trivial continuum set. This is a contradiction since ${\mathcal{I}}_p$ is not a trivial continuum set.

Finally, we consider $\dim E_p^c=2.$ For convenience, we assume that $g^{\pi (p)}(p)=g(p)=p.$ As Lemma 3, we can find $ϵ>0$ and $g_1\in \mathcal{U}(f)$, which has the following properties;

• $g_1(p)=g(p)=p,$
• $g_1(x)={\exp }_p\circ D_{p}g\circ {\exp }_p^{-1}(x)$ if $x\in B_{ϵ}(p),$ and
• $g_1(x)=g(x)$ if $x\notin B_{4ϵ}(p).$

For any $v\in E_p^c(ϵ)$, there is $l>0$ such that $D_p{g}^l(v)=v$. Take $u\in E_p^c(ϵ)$ such that $\parallel u\parallel =ϵ/2$. As in the previous arguments, we can construct a closed connected small arc ${\mathcal{J}}_p\subset B_{ϵ}(p)\cap {\exp }_p(E_p^c(ϵ))$ such that

• $\mathrm{diam}{\mathcal{J}}_p=ϵ/4$,
• $g_1^l({\mathcal{J}}_p)={\mathcal{J}}_p$, and
• $g_1^l{|}_{{\mathcal{J}}_p}:{\mathcal{J}}_p\to {\mathcal{J}}_p$ is the identity map.

As in the proof of the first case, take $\delta =ϵ/2$. Let ${\Gamma }_{\delta }^{+}(p)=\{x\in M:d(g_1^{li}(x),g_1^{li}(p)\le \delta$$\forall i\ge 0\}.$ It is clear that ${\mathcal{J}}_p\subset {\Gamma }_{\delta }^{+}(p).$ Then, by Lemma 1, ${\mathcal{J}}_p$ must be a trivial continuum set but it is not possible since ${\mathcal{J}}_p$ is a closed connected small arc. Thus, if $f\in \mathcal{RPCWE}$ then f is star. □

The differentiable maps $f,g:M\to M$ are conjugate if there is a homeomorphism $h:M\to M$ such that $f\circ h=h\circ g.$ We say that a differentiable map f is structurally stable if there is a $C^1$ neighborhood $\mathcal{U}(f)$ of $f\in D^1(M)$ such that for any $g\in \mathcal{U}(f),$g is conjugate to f. A differentiable map f is Ω stable if there is a $C^1$ neighborhood $\mathcal{U}(f)$ of $f\in D^1(M)$ such that for any $g\in \mathcal{U}(f),$${g|}_{\Omega (g)}$ is conjugate to ${f|}_{\Omega (f)}$, where $\Omega (f)$ denotes the nonwandering points of $f.$ Przytycki proved in Reference [21] that if f is an Anosov differentiable map then it is not an Anosov diffeomorphism or expandings which are not structurally stable. Moreover, assume that f is Axiom A (i.e., $\overline{P(f)}=\Omega (f)$ is hyperbolic) and has no singular points in the nonwandering set $\Omega (f)$. Then f is $\Omega$ stable if and only if f is strong Axiom A and has no cycles ( see Reference [22]). Here, f is strong Axiom A means that f is Axiom A and $\Omega (f)$ is the disjoint union ${\Lambda }_1\cup {\Lambda }_2$ of two closed f invariant sets.

According to the above results of a diffeomorphism $f\in {\mathrm{Diff}}^1(M)$, one can consider the case of a differentiable $f\in D^1(M)$ which is an extension of a diffeomorphism. For instance, a diffeomorphism $f\in \mathrm{Diff}(M)$ is said to be star if we can choose a $C^1$ neighborhood $\mathcal{U}(f)$ of f such that every periodic point of g is hyperbolic, for all $g\in \mathcal{U}(f)$.

If a diffeomorphism f is star then f is Axiom A and has no cycles (see References [23,24]). Aoki et al. Theorem A in Reference [25] proved that if a differentiable map f is star and the nonwandering set $\Omega (f)\cap S_f\subset \{p\in P(f):p$ is a sink } then f is Axiom A and has no cycles.

Theorem 2.

Let $f\in D^1(M).$ If $f\in \mathcal{RPCWE}$ then f is Axiom A and has no cycles.

Proof. 

Suppose that $f\in \mathcal{RPCWE}$. As Lemma 4, f is star. By Theorem 1, we know $S_f=\varnothing ,$ and so, $\Omega (f)\cap S_f=\varnothing$. By Lemma 2, there do not exist sinks in $P(f)$, that is, $\{p\in P(f):p$ is a sink $\}=\varnothing$. Thus, by Theorem A in Reference [25], f is Axiom A and has no cycles. □

Proof of Theorem A.

Suppose that $f\in \mathcal{RPCWE}$. Then, by Lemma 2, Theorem 2 and Proposition 2.7 in [17], $\Omega (f)=\overline{P_0(f)}$ is hyperbolic and $\overline{P_0(f)}$ is expanding. Then, by Lemma 2.8 in Reference [17], $M=\overline{P_0(f)}$. Thus, f is expanding. □

3. The Proof of Theorem B

Denote by $\mathcal{KS}$ the set of Kupka–Smale $C^1$ maps of $M.$ By Shub [26], $\mathcal{KS}$ is a residual set of $D^1(M).$ If $f\in \mathcal{KS}$ then every $p\in P(f)$ is hyperbolic. Then, we can see the following.

Lemma 5.

Let $f\in \mathcal{KS}$. If $f\in \mathcal{PCWE}$ then $P(f)=P_0(f).$

Proof. 

Let $f\in \mathcal{PCWE}$. Suppose, by contradiction, that $P_i(f)\ne \varnothing$ for some $1\le i\le \dim M.$ Take $p\in P_i(f)$ and $\delta >0$. Then, we can define a local stable manifold $W_{\delta }^s(p)$ of p such that $W_{\delta }^s(p)\ne \varnothing .$ We can construct a closed connected small arc ${\mathcal{J}}_p\subset W_{\delta }^s(p)$ with its center at p such that $\mathrm{diam}{\mathcal{J}}_p=\delta /4.$ Let ${\Gamma }_{\delta }^{+}(p)=\{x\in M:d(f^i(x),f^i(p))\le \delta$ for all $i\ge 0\}.$ Then, it is clear ${\mathcal{J}}_p\subset {\Gamma }_{\delta }^{+}(p).$ Since $f\in \mathcal{PCWE}$, by Lemma 1, ${\mathcal{J}}_p$ must be a trivial continuum set. This is a contradiction since ${\mathcal{J}}_p$ is not a trivial continuum set. Thus, every $p\in P(f)$ is a source so that $P(f)=P_0(f).$ □

Lemma 6.

Lemma 8 in [15]. There exists a residual set ${\mathcal{G}}_1\subset D^1(M)$ such that for given $f\in {\mathcal{G}}_1$, if for any $C^1$ neighborhood $\mathcal{U}(f)$ of f there exist $g\in \mathcal{U}(f)$ and $p\in P_h(g)$ with $\mathrm{ind}(p)=i(0\le i\le \dim M)$, then there is $p^{\prime }\in P_h(f)$ with $\mathrm{ind}(p^{\prime })=i.$

Lemma 7.

There exists a residual subset ${\mathcal{G}}_2\subset D^1(M)$ such that for a given $f\in {\mathcal{G}}_2$, if $f\in \mathcal{PCWE}$ then $S_f\cap \overline{P_0(f)}=\varnothing .$

Proof. 

Let $f\in {\mathcal{G}}_2=\mathcal{KS}\cap {\mathcal{G}}_1$ and $f\in \mathcal{PCWE}$. Suppose, by contradiction, that $S_f\cap \overline{P_0(f)}\ne \varnothing .$ Since $S_f\cap \overline{P_0(f)}\ne \varnothing$, we can choose a point $x\in S_f\cap \overline{P_0(f)}.$ Then, we can find a sequence of periodic points $\left\{p_n\right\}\subset P_0(f)$ with period $\pi (p_n)$ such that $p_n\to x$ as $n\to \infty .$ As Lemma 3, there exists g$C^1$ close to f such that $g^{\pi (p_n)}(p_n)=p_n$ and $p_n\in S_g.$ Again using Lemma 3, there exists $g_1$$C^1$ closed to g such that $g_1$$C^1$ is close to f, $g_1^{\pi (p_n)}(p_n)=p_n$, and $\mathrm{ind}(p_n)=i(1\le i\le \dim M).$ Since $f\in {\mathcal{G}}_1$, by Lemma 6, f has a hyperbolic saddle periodic point q with $\mathrm{index}(q)=i(1\le i\le \dim M).$ This is a contradiction by Lemma 2. □

For a $\delta >0$, a point $p\in P(f)(f^{\pi (p)}(p)=p)$ said to be a δ-hyperbolic (see Reference [27]) if for an eigenvalue of $D{f}^{\pi (p)}(p)$, we can take an eigenvalue $\lambda$ of $D{f}^{\pi (p)}(p)$ such that

$${(1-\delta )}^{\pi (p)}<\left|\lambda \right|<{(1+\delta )}^{\pi (p)}.$$

Lemma 8.

There exists a residual subset ${\mathcal{G}}_3\subset D^1(M)$ such that for a given $f\in {\mathcal{G}}_3$, if $f\in \mathcal{PCWE}$, then we can take $\delta >0$ such that f has no δ-hyperbolic.

Proof. 

Let $f\in {\mathcal{G}}_3=\mathcal{KS}\cap {\mathcal{G}}_1\cap {\mathcal{G}}_2,$ and let $f\in \mathcal{PCWE}$. Since $f\in \mathcal{KS}\cap {\mathcal{G}}_1\cap {\mathcal{G}}_2$, by Lemma 2 and Lemma 7, we know $S_f\cap \overline{P_0(f)}=\varnothing .$ Assume that for any $\delta >0$, there is a $p\in P_h(f)$ with a $\delta$-hyperbolic. By Lemma 3, we can take g$C^1$ close to f such that p has an eigenvalue with modulus one. Again using Lemma 3, there exists $g_1$$C^1$ close to g ($g_1$$C^1$ close to f) such that $g_1$ has a saddle $q\in P_h(g_1)$ with $\mathrm{ind}(q)=i(1\le i\le \dim M),$ where $P_h(g_1)$ is the set of all hyperbolic periodic points of $g_1.$ Since $f\in {\mathcal{G}}_1$, f has a saddle $q^{\prime }\in P_h(f)$ with $\mathrm{ind}(q^{\prime })=i(1\le i\le \dim M).$ This is a contradiction by Lemma 2. □

Lemma 9.

Lemma 7 in Reference [15]. There exists a residual subset ${\mathcal{G}}_4\subset D^1(M)$ such that for a given $f\in {\mathcal{G}}_4$ and $\delta >0$, if any $C^1$ neighborhood $\mathcal{U}(f)$ of f there exist $g\in \mathcal{U}(f)$ and $p\in P_h(g)$ with a δ-hyperbolic, then we can find $p^{\prime }\in P_h(f)$ with a $2\delta$-hyperbolic.

Lemma 10.

There exists a residual subset ${\mathcal{G}}_5\subset D^1(M)$ such that for a given $f\in {\mathcal{G}}_5$, if $f\in \mathcal{PCWE}$ then f is star.

Proof. 

Let $f\in {\mathcal{G}}_5={\mathcal{G}}_3\cap {\mathcal{G}}_4$ and $f\in \mathcal{PCWE}$. Suppose that f is not star. Then, as Lemma 3, we can take g$C^1$ close to f such that g has a $q\in P_h(g)$ with a $\delta /2$-hyperbolic for some $\delta >0.$ Since $f\in {\mathcal{G}}_4$, f has a hyperbolic periodic point $p^{\prime }$ with a $\delta$-hyperbolic. This is a contradiction by Lemma 8. □

The following is a differentiable version of closing Lemma under the generic sense (see Theorem 1 in Reference [28]). Then we set $\mathcal{CL}$ is the residual subset in $D^1(M)$ such that for any $f\in \mathcal{CL}$, $\Omega (f)=\overline{P}(f).$

Proof of Theorem B.

Let $f\in \mathcal{G}={\mathcal{G}}_5\cap \mathcal{CL}$ and $f\in \mathcal{PCWE}$. It is enough to show that $M=\overline{P_0(f)}.$ By Lemmas 5 and 7, $P(f)=P_0(f)$ and $S_f\cap \overline{P_0(f)}=\varnothing .$ Since $f\in \mathcal{CL}$, $\Omega (f)=\overline{P(f)}.$ According to Lemma 10, f is star, and so $\{\Omega (f)\setminus \overline{P(f)}\}\cap S_f=\varnothing$. Thus we have $\Omega (f)=\overline{P(f)}=\overline{P_0(f)}$ is hyperbolic. As Proposition 2.7 in Reference [17], we have that $\overline{P_0(f)}$ is expanding. Then, as in the proof of Lemma 3.8 in Reference [17], we have $M=\overline{P_0(f)}.$ □