Continuum-wise expansive homoclinic classes for robust dynamical systems

Abstract

In the study, we consider continuum-wise expansiveness for the homoclinic class of a kind of $C^{1}$-robustly expansive dynamical system. First, we show that if the homoclinic class $H(p, f)$, which contains a hyperbolic periodic point p, is R-robustly continuum-wise expansive, then it is hyperbolic. For a vector field, if the homoclinic class $H(\gamma , X)$ does not include singularities and is R-robustly continuum-wise expansive, then it is hyperbolic.

1 Introduction

1.1 Continuum-wise expansiveness for diffeomorphisms

Let M be a closed connected smooth Riemannian manifold. A point $x\in M$ is called a periodic point if there is $\pi (x)>0$ such that $f^{\pi (x)}(x)=x$, where $\pi (x)$ is the period of x. A periodic point p with period $\pi (p)>0$ is considered hyperbolic if the derivative $D_{p}f^{\pi (p)}$ has no eigenvalues with norm one. Let $\operatorname{Per}(f)=\{x\in M: x\mbox{ is a periodic point of }f\}$, and let $p\in \operatorname{Per}(f)$ be hyperbolic. Subsequently, there are $C^{r}$ ($r\geq 1$) sets $W^{s}(p)$ and $W^{u}(p)$, which are called the stable manifold of p and the unstable manifold of p, respectively, such that $f^{i\pi (p)}(x)\to p$ (as $i\to \infty$) for $x\in W^{s}(p)$ and $f^{-i\pi (p)}(x)\to p$ (as $i\to \infty$) for $x\in W^{u}(p)$.

Let $p, q\in \operatorname{Per}(f)$ be hyperbolic. We say that p and q are homoclinically related if $W^{s}(p)\pitchfork W^{u}(q) \neq\emptyset$ and $W^{u}(p)\pitchfork W^{s}(q)\neq\emptyset$, and in such a case, we write $p\sim q$. Let us denote $H(p, f)=\overline{ \{q\in \operatorname{Per}(f): p\sim q\}}$. It is known that $H(p, f)$ is a closed, f-invariant, and transitive set. Here a closed f-invariant set Λ is transitive if there is $x\in \varLambda$ such that $\omega (x)=\varLambda$, where $\omega (x)$ is the omega limit set of x.

According to the result of Samle [27], if a diffeomorphism f satisfies Axiom A, that is, the nonwandering set $\varOmega (f)=\overline{ \operatorname{Per}(f)}$ is hyperbolic, then this set can be written as the finite disjoint union of closed f-invariant sets that are homoclinic classes of a periodic point inside them. An interesting problem is the hyperbolicity of homoclinic classes under various $C^{1}$-perturbations of expansiveness (see [13, 22, 23, 25, 26, 29]).

Let d be the distance on M induced from a Riemannian metric $\|\cdot \|$ on the tangent bundle TM. A closed f-invariant set Λ (⊂M) is expansive for f if there is $e>0$ such that, for any distinct points $x, y\in \varLambda$, there is $n\in \mathbb{Z}$ such that $d(f^{n}(x), f^{n}(y))\geq e$.

Let $p\in \operatorname{Per}(f)$ be hyperbolic. Then there exist a $C^{1}$-neighborhood $\mathcal{U}(f)$ of f and a neighborhood U of p such that, for any $g\in \mathcal{U}(f)$, $p_{g}= \bigcap_{n\in \mathbb{Z}}g^{n}(U)$ is a unique hyperbolic periodic point of g, where $p_{g}$ is said to be the continuation of p.

We say that the homoclinic class $H(p, f)$ is $C^{1}$-robustly expansive if there is a $C^{1}$-neighborhood $\mathcal{U}(f)$ of f such that, for any $g\in \mathcal{U}(f)$, $H(p_{g}, g)$ is expansive, where $p_{g}$ is the continuation of p. Note that, in the definition, the expansive constant depends on $g\in \mathcal{U}(f)$.

A closed f-invariant set $\varLambda \subset M$ is hyperbolic if the tangent bundle $T_{\varLambda }M$ has a Df-invariant splitting $E^{s}\oplus E^{u}$ and there exist constants $C>0$ and $0<\lambda <1$ such that

$$ \bigl\Vert D_{x}f^{n}|_{E_{x}^{s}} \bigr\Vert \leq C\lambda ^{n} \quad \text{and} \quad \bigl\Vert D_{x}f^{-n}|_{E_{x}^{u}} \bigr\Vert \leq C\lambda ^{n} $$

for all $x\in \varLambda$ and $n\geq 0$.

Sambarino and Vieitez [25] proved that if the homoclinic class $H(p, f)$ is $C^{1}$-robustly expansive and germ expansive, then it is hyperbolic. Here $H(p, f)$ is germ expansive for f indicating that if there is $e>0$ such that, for any $x\in H(p,f)$, $y\in M$ if $d(f^{i}(x), f^{i}(y))< e$ for all $i\in \mathbb{Z}$, then $x=y$. We say that the homoclinic class $H(p, f)$ is $C^{1}$-stably expansive if there exist a $C^{1}$-neighborhood $\mathcal{U}(f)$ of f and a neighborhood U of $H(p, f)$ such that, for any $g\in \mathcal{U}(f)$, $\varLambda _{g}=\bigcap_{n\in \mathbb{Z}}g ^{n}(U)$ is expansive, where $\varLambda _{g}$ is the continuation of Λ. Lee and Lee [13] proved that if the homoclinic class $H(p, f)$ is $C^{1}$-stably expansive, then it is hyperbolic.

For obtaining the results, we use a general notion of expansiveness (continuum-wise expansive) and consider the hyperbolicity of the homoclinic class. Continuum-wise expansiveness is a general notion of expansiveness (see [11, Example 3.5]). A set A is nondegenerate if it is not reduced to a point. We say that $A\subset M$ is a nontrivial continuum if it is a compact connected nondegenerate subset of M.

Definition 1.1

Let $f:M\to M$ be a diffeomorphism. A closed f-invariant set Λ (⊂M) is said to be a continuum-wise expansive subset of f if there is a constant $e>0$ such that, for any nondegenerate subcontinuum $A\subset \varLambda$, there is $n\in \mathbb{Z}$ such that

$$ \operatorname{diam} f^{n}(A)\geq e, $$

where $\operatorname{diam} A = \sup \{d(x,y) : x, y \in A \}$ for any subset $A\subset \varLambda$.

Thus the constant e is called a continuum-wise expansive constant for f. In the definition a diffeomorphism f is continuum-wise expansive if $\varLambda =M$.

Das, Lee, and Lee [6] proved that if the homoclinic class $H(p, f)$ is $C^{1}$-robustly continuum-wise expansive and satisfies the chain condition, then $H(p, f)$ is hyperbolic. However, it is still an open question if the chain condition is omitted. Subsequently, we consider that the homoclinic class $H(p, f)$ is a type of $C^{1}$-robustly continuum-wise expansiveness. Let $\operatorname{Diff}(M)$ be the space of diffeomorphisms of M endowed with the $C^{1}$ topology. We call a subset $\mathcal{G}\subset \operatorname{Diff}(M)$ a residual subset if it contains a countable intersection of open and dense subsets of $\operatorname{Diff}(M)$. A dynamic property is called a $C^{1}$-generic property if it holds in a residual subset of $\operatorname{Diff}(M)$. Sambarino and Vieitez [26] proved that if the homoclinic class $H(p, f)$ is generically $C^{1}$-robustly expansive, then it is hyperbolic. Lee [17] proved that if a locally maximal homoclinic class $H(p, f)$ is homogeneous, then it is hyperbolic. Lee [16] proved that if a homoclinic class $H(p, f)$ is continuum-wise expansive, then it is hyperbolic. Using the $C^{1}$-generic condition, we define a type of $C^{1}$-robust expansiveness, which was introduced by Li [19].

Definition 1.2

Let p be a hyperbolic periodic point of f. We say that the homoclinic class $H(p, f)$ is R-robustly $\mathfrak{P}$ if there exist a $C^{1}$-neighborhood $\mathcal{U}(f)$ of f and a residual set $\mathcal{G}\subset \mathcal{U}(f)$ such that, for any $g\in \mathcal{G}$, $H(p_{g}, g)$ is $\mathfrak{P}$, where $p_{g}$ is the continuation of p.

In the definition, $\mathfrak{P}$ is replaced by various types of expansiveness. Accordingly, we introduce a general type of expansiveness proposed by Morales and Sirvent [20]. For a Borel probability measure μ on M, we consider that f is μ-expansive if there is $e>0$ such that $\mu (\varGamma _{e}(x))=0$ for all $x\in M$, where $\varGamma _{e}(x)=\{y\in M: d(f^{i}(x), f^{i}(y))\leq e\mbox{ for all } i\in \mathbb{Z}\}$. We say that f is measure expansive if it is μ-expansive for every nonatomic Borel probability measure μ on M. According to Artigue and Carrasco [2], we know the following:

$$\mbox{expansive} \Rightarrow \mbox{measure expansive} \Rightarrow \mbox{continuum-wise expansive}. $$

Lee [17] proved that if the homoclinic class $H(p, f)$ is R-robustly measure expansive, then it is hyperbolic. We can obtain the results for the R-robustly expansive homoclinic classes. According to these results, the following is a general result of [17].

Theorem A

Let p be a hyperbolic periodic point of f. If the homoclinic class $H(p, f)$ is R-robustly continuum-wise expansive, then $H(p, f)$ is hyperbolic.

1.2 Continuum-wise expansiveness for vector fields

Let M be defined as before, and let $\mathfrak{X}(M)$ denote the set of $C^{1}$-vector fields on M endowed with the $C^{1}$-topology. Thus every $X\in \mathfrak{X}(M)$ generates a $C^{1}$-flow $X_{t} : M \times \mathbb{R}\to M$, that is, a $C^{1}$-map such that $X_{t}:M \to M$ is a diffeomorphism satisfying (i) $X_{0}(x)=x$, (ii) $X_{t+s}(x)=X_{t}(X_{s}(x))$ for all $t,s\in \mathbb{R}$ and $x\in M$,, and (iii) it is generated by the vector field X if

$$ \frac{d}{dt} X_{t}(x)\bigg|_{t=t_{0}}=X \bigl(X_{t_{0}}(x)\bigr) $$

for all $x\in M$ and $t\in \mathbb{R}$. A point $\sigma \in M$ is singular if $X_{t}(\sigma )=\sigma$ for all $t\in \mathbb{R}$. We denote by $\operatorname{Sing}(X)$ the set of all singular points of X. For any $x\in M$, if x is not a singular point, then it is a regular point of X. Let $R_{X}$ be the set of all regular points of X. A periodic orbit of X is an orbit $\gamma = \operatorname{Orb}(p)$ such that $X_{T}(p)=p$ for some minimal $T>0$. We denote by $\operatorname{Per}(X)$ the set of all periodic orbits of X. A point $x\in M$ is a critical element if it is either a singular point or a periodic point of X. Let $\operatorname{Crit}(X)= \operatorname{Sing}(X)\cup \operatorname{Per}(X)$ be the set of all critical elements of X. Let $X_{t}$ be the flow of $X\in \mathfrak{X}(M)$. A closed $X_{t}$-invariant set Λ is considered hyperbolic for $X_{t}$ if there are constants $C>0$ and $\lambda >0$ and a splitting $T_{x}M=E^{s}_{x}\oplus \langle X(x) \rangle \oplus E^{u}_{x}$ such that the tangent flow $DX_{t}: TM \to TM$ leaves the invariant continuous splitting and

$$ \Vert DX_{t}|_{E^{s}_{x}} \Vert \leq Ce^{-\lambda t} \quad \text{and} \quad \Vert DX_{-t}|_{E^{u}_{x}} \Vert \leq Ce^{-\lambda t} $$

for $t>0$ and $x\in \varLambda$, where $\langle X(x)\rangle$ is the subspace generated by $X(x)$.

An increasing homeomorphism $h:\mathbb{R}\to \mathbb{R}$ with $h(0)=0$ is called a reparameterization. Let $\operatorname{Hom}(\mathbb{R})$ denote the set of all homeomorphisms of $\mathbb{R}$. Let $\operatorname{Rep}(\mathbb{R})=\{h\in \operatorname{Hom}( \mathbb{R}): h\mbox{ is a reparameterization} \}$. Bowen and Walters [4] introduced and studied expansiveness for vector fields. They showed that if a vector field X is expansive, then every singular point is isolated.

A closed invariant set $\varLambda \subset M$ is expansive of $X\in \mathfrak{X}(M)$ if, for every $\epsilon >0$, there exist $\delta >0$ and $h\in \operatorname{Hom}(\mathbb{R})$ such that, for any $x, y\in \varLambda$, if $d(X_{t}(x), X_{h(t)}(y))\leq \delta$ for all $t\in \mathbb{R}$, then $y\in X_{(-\epsilon , \epsilon )}(x)$. If $\varLambda =M$, then X is called expansive.

Regarding the notion of expansiveness, Arbieto, Codeiro, and Pacifico [1] introduced and studied a general notion of expansiveness for vector fields. They proved that if a vector field X is continuum-wise expansive, then every singular point is isolated. Here we explain continuum-wise expansiveness for vector fields in further detail. For a subset A of M, $C^{0}(A, \mathbb{R})$ denotes the set of real continuous maps defined on A. We define

$$ \begin{aligned} \mathcal{H}(A)&=\bigl\{ h:A\to \operatorname{Rep}(\mathbb{R}):\\ &\quad \text{there is } x _{h}\in A \text{ with } h(x_{h})=\mathit{id}, \text{ and } h(\cdot ) (t) \in C^{0}(A, \mathbb{R}) \text{ for all } t\in \mathbb{R}\bigr\} , \end{aligned} $$

and if $t\in \mathbb{R}$ and $h\in \mathcal{H}(A)$, then

$$ \mathcal{X}^{t}_{h}(A)=\bigl\{ X_{h(x)(t)}(x):x \in A\bigr\} . $$

For convenience, we set $h(x)(t)=h_{x}(t)$ for all $x\in A$ and $t\in \mathbb{R}$. Let Λ be a closed set of M. A set A is called nondegenerate if it is not reduced to a point. We say that $A\subset M$ is a continuum if it is a compact connected nondegenerate subset A of M.

Definition 1.3

Let $X\in \mathfrak{X}(M)$. We say that X is continuum-wise expansive if, for any $\epsilon >0$, there is $\delta >0$ such that if $A\subset M$ is a continuum and $h\in \mathcal{H}(A)$ satisfies

$$ \operatorname{diam}\bigl(\mathcal{X}^{t}_{h}(A)\bigr)< \delta \quad \text{for all } t \in \mathbb{R}, $$

then $A\subset X_{(-\epsilon , \epsilon )}(x)$ for some $x\in A$.

Let $\gamma \in \operatorname{Per}(X)$ be hyperbolic. We consider that the dimension of the stable manifold $W^{s}(\gamma )$ of γ is the index of γ, denoted by $\operatorname{index}(\gamma )$. The homoclinic class of X associated with a hyperbolic closed orbit γ, denoted by $H(\gamma , X)$, is defined as the closure of the transverse intersection of the stable and unstable manifolds of γ, that is,

$$ H(\gamma , X)=\overline{{W^{s}(\gamma )} \pitchfork {W^{u}(\gamma )}}, $$

where $W^{s}(\gamma )$ is the stable manifold of γ, and $W^{u}(\gamma )$ is the unstable manifold of γ. It is evident that it is closed, $X_{t}$-invariant, and transitive. Here, a closed invariant set Λ is transitive if there is $x\in \varLambda$ such that $\omega (x)=\varLambda$.

For two hyperbolic closed orbits γ and η of X, we say that γ and η are homoclinically related, denoted by ${\gamma }\sim {\eta }$, if

$$ W^{s}(\gamma ) \pitchfork W^{u}(\eta )\neq \emptyset \quad \text{and}\quad W^{s}(\eta )\pitchfork W^{u}(\gamma )\neq \emptyset . $$

If γ and η are homoclinically related, then $\operatorname{index}(\eta )=\operatorname{index}(\gamma )$. Let $\gamma \in \operatorname{Per}(X)$ be hyperbolic. Thus there exist a $C^{1}$-neighborhood $\mathcal{U}(X)$ of X and a neighborhood U of γ such that, for any $Y\in \mathcal{U}(X)$, there is a unique hyperbolic periodic orbit $\gamma _{Y}=\bigcap_{t\in \mathbb{R}}Y_{t}(U)$. The hyperbolic periodic orbit $\gamma _{Y}$ is called the continuation of γ with respect to Y.

We say that the homoclinic class $H(\gamma , X)$ is $C^{1}$-robustly expansive if there is a $C^{1}$-neighborhood $\mathcal{U}(X)$ of X such that, for any $Y\in \mathcal{U}(X)$, $H(\gamma _{Y}, Y)$ is expansive, where $\gamma _{Y}$ is the continuation of γ.

A subset $\mathcal{G}\subset \mathfrak{X}^{1}(M)$ is called a residual subset if it contains a countable intersection of the open and dense subsets of $\mathfrak{X}^{1}(M)$. A dynamic property is called a $C^{1}$-generic property if it holds in a residual subset of $\mathfrak{X}(M)$.

Lee and Park [18] proved that, for a $C^{1}$-generic X, if an isolated homoclinic class $H(\gamma , X)$ is expansive, then it is hyperbolic. Here, a closed $X_{t}$-invariant set Λ is isolated if there is a neighborhood U of Λ such that $\varLambda =\bigcap_{t\in \mathbb{R}}X_{t}(U)$. We consider that a closed invariant set Λ is germ expansive if, for any $\epsilon >0$, there is $\delta >0$ such that, for any $x\in \varLambda$ and $y\in M$, there is $h\in \operatorname{Hom}(\mathbb{R})$ such that if $d(X_{t}(x), X_{h(t)}(y))<\delta$ for all $t\in \mathbb{R}$, then $y\in X_{(-\epsilon , \epsilon )}(x)$. It is evident that, if Λ is expansive, then it is germ expansive. However, the converse is not true. Note that if Λ is isolated germ expansive, then Λ is expansive.

Gang [10] proved that if the homoclinic class $H(\gamma , X)$ is $C^{1}$-robustly expansive and $H(\gamma , X)$-germ expansive, then it is hyperbolic.

A vector field X has the shadowing property on Λ if, for any $\epsilon >0$, there exists $\delta >0$ such that, for any $(\delta , 1)$-pseudo orbit $\xi =\{(x_{i}, t_{i}): t_{i}\geq 1, i \in \mathbb{Z}\}\subset \varLambda$, there exist $y\in M$ and $h\in \operatorname{Hom}(\mathbb{R})$ satisfying

$$ d\bigl(X_{h(t)}(y), X_{t-s_{i}}(x_{i})\bigr)< \epsilon $$

for any $s_{i}\leq t< s_{i+1}$, where $s_{i}$ are defined as $s_{0}=0$, $s_{n}=\sum_{i=0}^{n-1}t_{i}$, and $s_{-n}=\sum_{i=-n}^{-1}t _{i}$, $n=1, 2, \ldots$ .

Lee, Lee, and Lee [14] proved that if the homoclinic class $H(\gamma , X)$ is $C^{1}$-robustly expansive and shadowable, then it is hyperbolic. According to the results, we consider the hyperbolicity of the homoclinic class $H(\gamma , X)$ under a type of $C^{1}$-robustly continuum-wise expansiveness.

Definition 1.4

Let $X\in \mathfrak{X}(M)$. We say that the homoclinic class $H(\gamma , X)$ is R-robustly continuum-wise expansive if there exist a $C^{1}$-neighborhood $\mathcal{U}(X)$ of X and a residual set $\mathcal{G}\subset \mathcal{U}(X)$ such that, for any $Y\in \mathcal{G}$, $H(\gamma _{Y}, Y)$ is continuum-wise expansive, where $\gamma _{Y}$ is the continuation of γ.

Using this definition, we have the following theorem.

Theorem B

Let $X\in \mathfrak{X}(M)$ and $H_{X}( \gamma )\cap \operatorname{Sing}(X)=\emptyset$. If the homoclinic class $H(\gamma , X)$ is R-robustly continuum-wise expansive, then it is hyperbolic for X.

2 Proof of Theorem A

Let M be defined as before, and let $f:M\to M$ be a diffeomorphism. For any $\delta >0$, a sequence $\{x_{i}\}_{i\in \mathbb{Z}}$ is called a δ-pseudo-orbit of f if $d(f(x_{i}), x_{i+1})< \delta$ for all $i\in \mathbb{Z}$. For a given $x, y\in M$, we write $x\rightsquigarrow y$ if for any $\delta >0$, there is a finite δ-pseudo-orbit $\{x_{i}\}_{i=0}^{n}$ ($n\geq 1$) of f such that $x_{0}=x$ and $x_{n}=y$. We write $x\leftrightsquigarrow y$ if $x\rightsquigarrow y$ and $y\rightsquigarrow x$. The set of points $\{x\in M:x\leftrightsquigarrow x\}$ is called the chain recurrent set of f and is denoted by $\mathcal{CR}(f)$. The chain recurrence class of f is the set of equivalent classes ↭ on $\mathcal{CR}(f)$. Let p be a hyperbolic periodic point of f. Denote $C(p, f)=\{x\in M : x\rightsquigarrow p \text{ and } p \rightsquigarrow x\}$, which is a closed invariant set.

It is known that $C(p, f)$ is a closed f-invariant set. Moreover, $H(p, f)\subset C(p, f)$. A closed small arc $\mathcal{I}$ of f is called a simply periodic curve if, for any $\epsilon >0$,

• (a) there is $k>0$ such that $f^{k}(\mathcal{I})=\mathcal{I}$,

• (b) $0< l(f^{i}(\mathcal{I}))<\epsilon$ for all $0\leq i< k$,

• (c) the endpoints of $\mathcal{I}$ are hyperbolic, and

• (d) $\mathcal{I}$ is normally hyperbolic,

where $l(A)$ denotes the length of A (see [29]). It is evident that $\mathcal{I}$ is not a point set.

Lemma 2.1

There is a residual set $\mathcal{G}_{1}\subset \operatorname{Diff}(M)$ such that, for any $f\in \mathcal{G}_{1}$, we have the following:

• (a) f is Kupka–Smale, that is, every periodic point of f is hyperbolic, and the stable and unstable manifolds are transversal intersections (see [24]).

• (b) $H(p, f)=C(p, f)$ (see [3]).

• (c) if, for any $C^{1}$-neighborhood $\mathcal{U}(f)$ of f, there is $g\in \mathcal{U}(f)$ such that g has a simply periodic curve $\mathcal{I}$, then f has a simply periodic curve $\mathcal{J}$ (see [29]).

The following lemma is important for a $C^{1}$ perturbation property, which is called Franks’ lemma.

Lemma 2.2

([8])

Let $\mathcal{U}(f)$ be a $C^{1}$-neighborhood of f. Then there exist $\epsilon >0$ and a $C^{1}$-neighborhood $\mathcal{U}_{0}(f)\subset \mathcal{U}(f)$ of f such that, for any $g\in \mathcal{U}_{0}(f)$, a set $\{x_{1}, x_{2}, \ldots , x_{N}\}$, a neighborhood U of $\{x_{1}, x_{2}, \ldots , x_{N}\}$, and a linear map $L_{i} : T_{x_{i}}M\rightarrow T_{g(x_{i})}M$ satisfying $\|L_{i}-D_{x_{i}}g\|\leq \epsilon$ for all $1\leq i\leq N$, there is $\widehat{g}\in \mathcal{U}(f)$ such that $\widehat{g}(x)=g(x)$ if $x\in \{x_{1}, x_{2}, \ldots , x_{N}\}\cup (M\setminus U)$ and $D_{x_{i}}\widehat{g}=L_{i}$ for all $1\leq i\leq N$.

For any hyperbolic $p\in \operatorname{Per}(f)$, we say that p is weakly hyperbolic if, for any $\eta >0$, there is an eigenvalue μ of $D_{p}f^{\pi (p)}$ such that

$$ (1-\eta )^{\pi (p)}< \vert \mu \vert < (1+\eta )^{\pi (p)}. $$

It is evident that if p is a weakly hyperbolic periodic point of f, then there is g $C^{1}$-close to f such that $p_{g}$ is not hyperbolic for g.

Lemma 2.3

Let $p\in \operatorname{Per}(f)$ be hyperbolic. If $q\in H(p, f) \cap \operatorname{Per}(f)$ with $q\sim p$ is weakly hyperbolic, then there is g $C^{1}$-close to f such that g has a simply periodic curve $\mathcal{L}\subset C(p_{g}, g)$.

Proof

Suppose that $q\in H(p, f)\cap \operatorname{Per}(f)$ with $q\sim p$ is weakly hyperbolic. According to Lemma 2.2, there is g $C^{1}$-close to f such that $p_{g}$ is not hyperbolic. Thus $D_{p_{g}}g^{\pi (p_{g})}$ has an eigenvalue μ such that $|\mu |=1$. For simplicity, we may assume that $p_{g}$ is a fixed point of g. Let $E_{p_{g}}$ be the vector space associated with the eigenvalue μ. For the proof, we consider the case of $\mu \in \mathbb{R}$. Consider a nonzero vector v associated with μ. According to Lemma 2.2, there is $g_{1}$ $C^{1}$-close to g such that

• (i) $g_{1}(p_{g})=g(p_{g})=p_{g}$, and

• (ii) $g_{1}(\operatorname{exp}_{p_{g}}(v))=\operatorname{exp}_{p_{g}}\circ D_{p _{g}}g\circ \operatorname{exp}_{p_{g}}^{-1}(\operatorname{exp}_{p}(v))= \operatorname{exp}_{p_{g}}(v)$.

For any small $\beta >0$, we set $E_{p_{g_{1}}}(\beta )=\{t\cdot v: - \beta /2\leq t\leq \beta /2\}$. Thus we have a closed small curve $\mathcal{J}$ such that

• (i) $\mathcal{J}=\operatorname{exp}_{p_{g_{1}}}(E_{p_{g_{1}}}(\beta ))$ with $\operatorname{diam}\mathcal{J}=\beta$,

• (ii) $g_{1}^{\pi (p_{g_{1}})}({\mathcal{J}})=\mathcal{J}$ is the identity map, and

• (iii) $\mathcal{J}$ is normally hyperbolic.

It is evident that the identity map is contained in $C(p_{g_{1}}, g _{1})$. As $g_{1}^{\pi (p_{g_{1}})}({\mathcal{J}})=\mathcal{J}$ is the identity map, by Lemma 2.2 again, there is h $C^{1}$-close to g such that h has a closed small curve $\mathcal{L}\subset C(p _{h}, h)$. Thus the curve $\mathcal{L}$ is such that $h^{\pi (p_{h})}( {\mathcal{L}})=\mathcal{L}$ is the identity map, $\operatorname{diam} \mathcal{L}=\beta$, $\mathcal{L}$ is normally hyperbolic, and the endpoints of $\mathcal{L}$ are hyperbolic. The closed small curve $\mathcal{L}$ is a simply periodic curve of h, which is contained in $C(p_{h}, h)$. □

Note that, by Lemma 2.3, there is g $C^{1}$-close to f such that g has a simply periodic curve $\mathcal{L}\subset C(p_{g}, g)$. However, the simply periodic curve $\mathcal{L}$ is not contained in $H(p_{g}, g)$ (see [25]). Let $\mathcal{WH}$ denote the set of all weakly hyperbolic periodic points of f.

Lemma 2.4

If the homoclinic class $H(p, f)$ is R-robustly continuum-wise expansive, then $H(p, f)\cap \mathcal{WH}=\emptyset$.

Proof

Suppose that $H(p, f)\cap \mathcal{WH}\neq\emptyset$. Thus there is $q\in H(p, f)\cap \operatorname{Per}(f)$ with $q\sim p$ such that q is weakly hyperbolic. As $H(p, f)$ is R-robustly continuum-wise expansive and $q\in H(p, f)\cap \operatorname{Per}(f)$ with $q\sim p$ such that q is weakly hyperbolic, there is $g\in \mathcal{G}_{1}\cap \mathcal{U}(f)$ such that $H(p_{g}, g)=C(p _{g}, g)$, and according to Lemma 2.3, there is $\beta >0$ such that g has a simply periodic curve $\mathcal{J}\subset C(p_{g}, g)$ with $\operatorname{diam}\mathcal{J}=\beta /4$. As $C(p_{g}, g)$ is continuum-wise expansive, $\mathcal{J}$ is continuum-wise expansive. According to [12, Proposition 2.6], g is continuum-wise expansive if and only if $g^{n}$ is continuum-wise expansive for any $n\in \mathbb{Z}\setminus \{0\}$. Consider $e=\beta$. By the definition of a simply periodic curve there is $k>0$ such that

$$ \operatorname{diam} g^{ki}(\mathcal{J})=\operatorname{diam} \mathcal{J}< e $$

for all $i\in \mathbb{Z}$. By the definition of continuum-wise expansivity, $\mathcal{J}$ should be a point. As $\mathcal{J}$ is a simply periodic curve, this is a contradiction. □

The following was proven by Wang [28]. He considered the Lyapunov exponents of the periodic point in the homoclinic class $H(p, f)$.

Lemma 2.5

There is a residual set $\mathcal{G}_{2}\subset \operatorname{Diff}(M)$ such that, for any $f\in \mathcal{G}_{2}$, if $H(p, f)$ is not hyperbolic, then there is $q\in H(p, f)\cap \operatorname{Per}(f)$ with $q\sim p$ such that q is a weakly hyperbolic periodic point.

Proof of Theorem A

Let $\mathcal{U}(f)$ be a $C^{1}$-neighborhood of f, and let $\mathcal{G}=\mathcal{G}_{1} \cap \mathcal{G}_{2}$. As $H(p, f)$ is R-robustly continuum-wise expansive, $H(p_{g}, g)$ is continuum-wise expansive for any $g\in \mathcal{G}\cap \mathcal{U}(f)$. Assume that there is $g\in \mathcal{G}\cap \mathcal{U}(f)$ such that $H(p_{g}, g)$ is not hyperbolic. As $g\in \mathcal{G}\cap \mathcal{U}(f)$, there is $q\in H(p_{g}, g)\cap \operatorname{Per}(g)=C(p_{g}, g)\cap \operatorname{Per}(g)$ with $q\sim p_{g}$ such that q is a weakly hyperbolic point. According to Lemma 2.4, this is a contradiction. Thus, if $H(p, f)$ is R-robustly continuum-wise expansive, then, for any $g\in \mathcal{G}\cap \mathcal{U}(f)$, $H(p_{g}, g)$ is hyperbolic, and hence $H(p, f)$ is hyperbolic. □

3 Proof of Theorem B

Let M be defined as before, and let $X\in \mathfrak{X}(M)$. We denote by $T_{p}M(\delta )$ the ball $\{v\in T_{p}M :\|v\|\leq \delta \}$. For every $x\in R_{X}$, let $N_{x}=\langle X(x)\rangle ^{\bot }\subset T _{x}M$, and let $N_{x}(\delta )$ be the δ ball in $N_{x}$. We set $N_{x,r}=N_{x}\cap T_{x}M(r)$ ($r>0$) and $\mathcal{N}_{x, r_{0}}= \operatorname{exp}(N_{x}(r_{0}))$ for $x\in M$.

Let $\operatorname{Sing}(X)=\emptyset$, and let $N=\bigcup_{x\in R _{X}}N_{x}$. We define the linear Poincaré flow

$$ P_{t}^{X}:=\pi _{x}\circ D_{x}X_{t}, $$

where $\pi _{x}:T_{x}M\to N_{x}$ (⊂N) is the natural projection along the direction of $X(x)$, and $D_{x}X_{t}$ is the derivative map of $X_{t}$. The following is an important result to prove hyperbolicity.

Remark 3.1

([7])

Let $\varLambda \subset M$ be a compact invariant set of $X_{t}$. Then Λ is a hyperbolic set of $X_{t}$ if and only if the linear Poincaré flow restriction on Λ has a hyperbolic splitting $N_{\varLambda }=N^{s}\oplus N^{u}$.

Let $X\in \mathfrak{X}(M)$, and suppose $p\in \gamma \in \operatorname{Per}(X)$ ($X_{T}(p)=p$), where $T>0$ is the prime period. If $f:\mathcal{N}_{p, r_{0}}\to \mathcal{N}_{p}$ is the Poincaré map ($r_{0}>0$), then $f(p)=p$. Accordingly, γ is hyperbolic if and only if p is a hyperbolic fixed point of f. The following is a vector field version of Franks’ lemma.

Lemma 3.2

([21])

Let $X\in \mathfrak{X}(M)$, $p\in \gamma \in \operatorname{Per}(X)$ ($X_{T}(p)=p$, $T>0$), and let $f:\mathcal{N} _{p, r_{0}}\to \mathcal{N}_{p}$ be the Poincaré map for some $r_{0}>0$. Let $\mathcal{U}(X)\subset \mathfrak{X}(M)$ be a $C^{1}$-neighborhood of X, and let $0< r\leq r_{0}$ be given. Then there exist $\delta _{0}>0$ and $0<\epsilon _{0}<r/2$ such that, for an isomorphism $L: N_{p}\to N_{p}$ with $\|L-D_{p}f\|<\delta _{0}$, there is $Y\in \mathcal{U}(X)$ having the following properties:

• (a) $Y(x)=X(x)$ if $x\notin F_{p}(X_{t}, r, T/2)$,

• (b) $p\in \gamma \in \operatorname{Per}(Y)$,

• (c)
  $$g(x)=\textstyle\begin{cases} \operatorname{exp}_{p}\circ L\circ \operatorname{exp}_{p}^{-1}(x) & \textit{if } x\in B_{\epsilon _{0}/4}(p)\cap \mathcal{N}_{p, r}, \\ f(x) & \textit{if } x\notin B_{\epsilon _{0}}(p)\cap \mathcal{N}_{p, r}, \end{cases} $$

where $B_{\epsilon }(x)$ is a closed ball in M center at $x\in M$ with radius $\epsilon >0$, $F_{p}(X_{t}, r, T/2)=\{X_{t}(y): y\in \mathcal{N}_{x, r} \textit{ and } 0\leq t\leq T\}$, and $g:\mathcal{N}_{p, r}\to \mathcal{N}_{p}$ is the Poincaré map defined by $Y_{t}$.

Remark 3.3

Let $\varLambda \subset M$ be a closed $X_{t}$-invariant set, and let Λ be continuum-wise expansive for X. If $\varLambda \cap \operatorname{Sing}(X)\neq\emptyset$, then $\varLambda \cap \operatorname{Sing}(X)$ is totally disconnected.

Proof

Suppose that $\varLambda \cap \operatorname{Sing}(X)$ is not totally disconnected. Thus there is a set $\mathcal{C}\subset \varLambda \cap \operatorname{Sing}(X)$ such that $\mathcal{C}$ is closed and connected, that is, a nontrivial continuum. Let $\epsilon >0$ be given. We assume that $\operatorname{diam}(\mathcal{C})<\epsilon$. As $\mathcal{C}\subset \varLambda \cap \operatorname{Sing}(X)$, $X_{t}( \mathcal{C})=\mathcal{C}$ for all $t\in \mathbb{R}$. Thus we know that

$$ \operatorname{diam}\bigl(X_{t}(\mathcal{C})\bigr)= \operatorname{diam}(\mathcal{C})< \epsilon $$

for all $t\in \mathbb{R}$. Thus $\mathcal{C}$ should be an orbit. This is a contradiction as $\mathcal{C}$ is a nontrivial continuum. □

For any $x,y \in M$, we write $x \rightharpoonup y$ if, for any $\delta >0$, there is a δ-pseudo-orbit $\{(x_{i}, t_{i}) : t _{i} \geq 1\}_{i=1}^{n}\subset M$ such that $x_{0}=x$ and $d(X_{t_{n-1}}(x _{n-1}), y)< \delta$. Similarly, $y \rightharpoonup x$. We can observe that x, y satisfy both conditions, and thus $x \rightleftharpoons y$. Thus we have an equivalence relation on the set $\mathcal{R}(X)$. Every equivalence class of ⇌ is called a recurrence class of X. Let γ be a hyperbolic periodic point of X. For some $p\in \gamma$, let $C(\gamma , X)=\{x\in M: x\rightleftharpoons p\mbox{ denote the chain recurrence class of }X\}$. According to the definition, we can observe that $C(\gamma , X)$ is closed and $X_{t}$-invariant and that $H(\gamma , X) \subset C(\gamma , X)$. Bonatti and Crovisier [3] showed that, for a $C^{1}$-vector field X, the chain recurrence class $C(\gamma , X)$ is the homoclinic class $H(\gamma , X)$, which is a version of the vector field of diffeomorphisms. Note that if a vector field X does not contain singularities, then the $C^{1}$-generic results of diffeomorphisms can be used for $C^{1}$ generic vector fields (see [5, 9]).

Lemma 3.4

There is a residual set $\mathcal{R}_{1}\subset \mathfrak{X}(M)$ such that every $X\in \mathcal{R}_{1}$ satisfies the following conditions:

• (a) X is Kupka–Smale, that is, every critical point is hyperbolic and its invariant manifolds intersect transversally (see [12]).

• (b) the chain recurrence class $C(\gamma , X)=H(\gamma , X)$ for any $\gamma \in \operatorname{Per}(X)$ (see [3]).

We say that a vector field X is a local star on $H(\gamma , X)$ if there is a $C^{1}$-neighborhood $\mathcal{U}(X)$ of X such that, for any $Y\in \mathcal{U}(X)$, every $\eta \in H(\gamma _{Y}, Y)\cap \operatorname{Crit}(Y)$ is hyperbolic, where $\gamma _{Y}$ is the continuation of Y. Let $\mathcal{G}^{*}(H( \gamma , Y))$ denote the set of all vector fields satisfying the local star on $H(\gamma , X)$.

Proposition 3.5

Let $H_{X}(\gamma )\cap \operatorname{Sing}(X)=\emptyset$, and let $\gamma \in \operatorname{Per}(X)$ be hyperbolic. If the homoclinic class $H(\gamma , X)$ is R-robustly continuum-wise expansive, then $X\in \mathcal{G}^{*}(H(\gamma , X))$.

Proof

Since $H_{X}(\gamma )\cap \operatorname{Sing}(X)$, we prove that if $H(\gamma , X)$ is R-robustly continuum-wise expansive, then every $\eta \in H_{X}(\gamma )\cap \operatorname{Per}(X)$ is hyperbolic. Suppose by contradiction that there exist $Y\in \mathcal{U}(X)$ and $\gamma \in H(\gamma _{Y}, Y)\cap \operatorname{Per}(Y)$ such that γ is not hyperbolic. Consider $p\in \gamma$ such that $Y_{T}(p)=p (T>0)$, and let $f:\mathcal{N} _{p, r}\to \mathcal{N}_{p}$ (for some $r>0$) be the Poincaré map associated with Y. As γ is not hyperbolic, p is not hyperbolic. Thus we assume that there is an eigenvalue λ of $D_{p}f$ such that $|\lambda |=1$. Let $\delta _{0}>0$ and $0<\epsilon _{0}<r/4$ be given by Lemma 3.2, and let $L:N_{p}\to N_{p}$ be a linear isomorphism with $\|L-D_{p}f\|<\delta _{0}$ such that $L=\left(\begin{array}{cc}A& O\\ O& B\end{array}\right)$ with respect to some splitting $N_{p}=G_{p}\oplus H_{p}(=E ^{s}_{p}\oplus E^{u}_{p})$, where $A:G_{p}\to G_{p}$ has an eigenvalue λ such that $\operatorname{dim} G_{p}=1$ if $\lambda \in \mathbb{R}$ or $\operatorname{dim} G_{p}=2$ if $\lambda \in \mathbb{C}$ and $B:H_{p}\to H_{p}$ is hyperbolic. According to Lemmas 3.2 and 3.4, there exists $Z\in \mathcal{R}_{1}$ $C^{1}$-close to Y ($Z\in \mathcal{U}(X)$) such that

• (a) $Z(x)=Y(x)$ if $x\notin F_{p}(Y, r_{0}, T)$,

• (b) $p\in \gamma \in \operatorname{Per}(Z)$, and

• (c)
  $$ g(x)=\textstyle\begin{cases} \operatorname{exp}_{p}\circ L\circ \operatorname{exp}_{p}^{-1}(x) & \text{if } x\in B_{\epsilon _{0}/4}(p)\cap \mathcal{N}_{p, r_{0}}, \\ f(x) & \text{if } x\notin B_{\epsilon _{0}}(p)\cap \mathcal{N}_{p, r _{0}}. \end{cases} $$

Here $g:\mathcal{N}_{p, r_{0}}\to \mathcal{N}_{p}$ is the Poincaré map associated with Z. Consider a nonzero vector $u\in G_{p}$ such that $\|u\|\leq \epsilon _{0}/8$. Then we have

$$ g\bigl(\operatorname{exp}_{p}(u)\bigr) = \operatorname{exp}_{p} \circ L\circ \operatorname{exp} ^{-1}_{p} \bigl( \operatorname{exp}_{p}(u)\bigr) = \operatorname{exp}_{p}(u). $$

Case 1. $\operatorname{dim} G_{p}=1$. We may assume that $\lambda =1$ for simplicity (the other case is similar). We set an arc $\mathcal{I}_{u}=\{su:0\leq s\leq 1\}$ and $\operatorname{exp}_{p}( \mathcal{I}_{u})=\mathcal{J}_{p}$. Then we know that

• (a) $\mathcal{J}_{p}\subset B_{\epsilon _{0}}(p)\cap \mathcal{N}_{p, r_{0}}$, and

• (b) $g|_{\mathcal{J}_{p}}:\mathcal{J}_{p}\to \mathcal{J}_{p}$ is the identity map.

Let $\operatorname{diam}(\mathcal{J}_{p})=\epsilon _{0}/2$. As $g|_{ \mathcal{J}_{p}}:\mathcal{J}_{p}\to \mathcal{J}_{p}$ is the identity map, according to Lemma 3.4, $\mathcal{J}_{p}\subset C( \gamma _{Z}, Z)$, and hence $g|_{\mathcal{J}_{p}}:\mathcal{J}_{p} \to \mathcal{J}_{p}$ is continuum-wise expansive. However, it is evident that the identity map $g|_{\mathcal{J}_{p}}$ is not continuum-wise expansive, a contradiction.

Case 2. $\operatorname{dim} G_{p}=2$. According to Lemma 3.2, we can find $Z\in \mathcal{R}_{1}\cap \mathcal{U}(X)$ such that $D_{p}g$ is a rational rotation. Thus there is $l\neq 0$ such that $D_{p}g^{l}$ has an eigenvalue of 1. As in the proof of case 1, we can derive a contradiction. □

We say that $p\in \gamma \in \operatorname{Per}(X)$ is a weakly hyperbolic periodic point if, for any $\delta >0$, there is an eigenvalue λ of $D_{p}f$ such that

$$ (1-\delta )\leq \lambda \leq (1 + \delta ), $$

where $f:\mathcal{N}_{p, r}\to \mathcal{N}_{p}$ is the Poincaré map associated with X. We introduce the concept of a vector field version of diffeomorphisms (see [29]). Let $\operatorname{Sing}(X)= \emptyset$. For any $\eta >0$, we consider that a $C^{1}$-curve $\mathcal{J}$ is η-simply periodic for X if

• (a) $\mathcal{J}$ is periodic with period T,

• (b) the length of $X_{t}(\mathcal{J})$ is less than η for any $0 \leq t \leq T$, and

• (c) $\mathcal{J}$ is normally hyperbolic.

Lemma 3.6

For any $X\in \mathcal{R}_{1}$, if $p\in \eta \in H(\gamma , X)\cap \operatorname{Per}(X)$ with $\eta \sim \gamma$ is a weakly hyperbolic periodic point, then, for any $C^{1}$-neighborhood $\mathcal{U}(X)$ of X, there is $Y\in \mathcal{R}_{1}\cap \mathcal{U}(X)$ such that f has an ϵ-simply periodic curve $\mathcal{J}\subset H( \gamma _{Y}, Y)$ for some $\epsilon >0$, where $f:\mathcal{N}_{p, r} \to \mathcal{N}_{p}$ is the Poincaré map defined by Y.

Proof

Let $X\in \mathcal{R}_{1}$, and let $\mathcal{U}(X)$ be a $C^{1}$-neighborhood of X. Suppose that $p\in \eta \in H(\gamma , X)\cap \operatorname{Per}(X)$ with $\eta \sim \gamma$ is a weakly hyperbolic periodic point. As $\eta \sim \gamma$, we consider two points $x\in W^{s}(\eta )\pitchfork W^{u}(\gamma )$ and $y\in W^{u}( \eta )\pitchfork W^{s}(\gamma )$. Consider $Y\in \mathcal{R}_{1} \cap \mathcal{U}(X)$; thus, we have $H(\gamma _{Y}, Y)=C(\gamma _{Y}, Y)$. Thus, as in the proof of [15, Proposition 4.1], there exist $\epsilon >0$ and the Poincaré map $g:\mathcal{N}_{p,r}\to \mathcal{N}_{p}$ associated with Y such that

• (i) the map g is defined by Y,

• (ii) g has a closed arc $\mathcal{I}$ or a disc $\mathcal{D}$ such that $g_{|_{\mathcal{I}}}:\mathcal{I}\to \mathcal{I}$ is the identity map, or $g_{|_{\mathcal{D}}}:\mathcal{D}\to \mathcal{D}$ is a rotation map,

• (iii) $0<\operatorname{diam}\mathcal{I}\leq \epsilon$ and $0<\operatorname{diam}\mathcal{D}\leq \epsilon$,

• (iv) $Y_{-t}(x)\to \gamma$ and $Y_{t}(y)\to \gamma$ as $t\to \infty$, and $g^{n}(x)\to \mathcal{J}$ (or $\mathcal{D}$) and $g^{n}(y)\to \mathcal{I}$ (or $\mathcal{D}$) as $n\to \infty$, and

• (v) $\mathcal{I}\subset C(\gamma _{Y}, Y)$ and $\mathcal{D} \subset C(\gamma _{Y}, Y)$.

As $H(\gamma _{Y}, Y)=C(\gamma _{Y}, Y)$, we have $\mathcal{I}\subset H( \gamma _{Y}, Y)$ and $\mathcal{D}\subset H(\gamma _{Y}, Y)$, and they are ϵ-simply periodic curves. □

Lemma 3.7

If the homoclinic class $H(\gamma , X)$ is continuum-wise expansive, then there is no η-simply periodic curve $\mathcal{J}\subset H( \gamma , X)$.

Proof

Assume that there is an η-simply periodic curve $\mathcal{J}\subset H(\gamma , X)$. Thus there is $T>0$ such that $X_{T}(\mathcal{J})=\mathcal{J}$ and $\operatorname{diam} (X_{t}( \mathcal{J}))\leq \eta$ for any $0 \leq t \leq T$. It is evident that the curve $\mathcal{J}$ is a nontrivial continuum. As $X_{T}( \mathcal{J})=\mathcal{J}$, $X_{T}(x)=x$ for all $x\in \mathcal{J}$. We define $h:\mathcal{J}\to \operatorname{Rep}(\mathbb{R})$ such that $h_{x}(t)=t$ for all $x\in \mathcal{J}$ and $t\in \mathbb{R}$. Thus, for all $t\in \mathbb{R}$, we have

$$\begin{aligned} \operatorname{diam}\bigl(\mathcal{X}^{t}_{h}( \mathcal{J})\bigr) &=\max \bigl\{ d\bigl(X_{h_{x}(t)}(x), X_{h_{y}(t)}(y)\bigr): x, y\in \mathcal{J}\bigr\} \\ &=\max \bigl\{ d\bigl(X_{t}(x), X_{t}(y)\bigr): x, y \in \mathcal{J}\bigr\} < \eta . \end{aligned}$$

If η is a continuum-wise expansive constant, then it is a contradiction as $\mathcal{J}$ contains no any single orbit of $x\in \mathcal{J}$. □

Lemma 3.8

Let $\gamma \in \operatorname{Per}(X)$ be hyperbolic. If the homoclinic class $H(\gamma , X)$ is R-robustly continuum-wise expansive, then, for any $\eta \in H(\gamma , X)\cap \operatorname{Per}(X)$ with $\eta \sim \gamma$, $p\in \eta$ is not a weakly hyperbolic periodic point.

Proof

Suppose by contradiction that there is a hyperbolic $\eta \in H(\gamma , X)\cap \operatorname{Per}(X)$ with $\eta \sim \gamma$ such that $p\in \eta$ is a weakly hyperbolic periodic point. According to Lemma 3.6, there is $Y\in \mathcal{R}_{1}\cap \mathcal{U}(X)$ such that f has an ϵ-simply periodic curve $\mathcal{J}\subset H(\gamma _{Y}, Y)$ for some $\epsilon >0$, where $f:\mathcal{N}_{p, r}\to \mathcal{N}_{p}$ is the Poincaré map defined by Y. As $H(\gamma , X)$ is R-robustly continuum-wise expansive, according to Lemma 3.7, this is a contradiction. □

Let $p\in \gamma$ be a hyperbolic periodic point of X with period $\pi (p)$, and let $f:\mathcal{N}_{p,r}\to \mathcal{N}_{p}$ be the Poincaré map with respect to X. Subsequently, if $\mu _{1}, \mu _{2}, \ldots , \mu _{d}$ are the eigenvalues of $D_{p}f$, then

$$ \lambda _{i}=\frac{1}{\pi (p)}\log \vert \mu _{i} \vert $$

for $i=1, 2, \ldots , d$ are called the Lyapunov exponents of p. Wang [28] proved that, for a $C^{1}$-generic nonsingular vector field $X\in \mathfrak{X}(M)$, if a homoclinic class $H(\gamma , X)$ is not hyperbolic, then there is a periodic orbit $\operatorname{Orb}(q)$ of f that is homoclinically related to $\operatorname{Orb}(p)$ and has a Lyapunov exponent arbitrarily close to 0, which is a vector field version of the result of Wang [28]. Note that if a hyperbolic periodic orbit γ has a Lyapunov exponent arbitrarily close to 0, then there is a point $p\in \gamma$ such that p is a weakly hyperbolic periodic point of X. Thus, we can rewrite the result of Wang [28] as follows.

Lemma 3.9

There is a residual set $\mathcal{R}_{2}\subset \mathfrak{X}(M)$ such that, for any $X\in \mathcal{R}_{2}$, if $H(\gamma , X)\cap \operatorname{Sing}(X)=\emptyset$ and $H(\gamma , X)$ is not hyperbolic, then there is $\eta \in H(\gamma , X)\cap \operatorname{Per}(X)$ with $\eta \sim \gamma$ such that $p\in \eta$ is a weakly hyperbolic periodic point of X.

Proof of Theorem B

As $H(\gamma , X)$ is continuum-wise expansive, $H(\gamma , X)\cap \operatorname{Sing}(X)=\emptyset$. To derive a contradiction, we assume that $H(\gamma , X)$ is not hyperbolic. Consider $Y\cap \mathcal{U}(X)\cap \mathcal{R}_{1}\cap \mathcal{R}_{2}$. Thus, according to Lemma 3.9, there is $\eta \in H(\gamma _{Y}, Y)\cap \operatorname{Per}(X)$ with $\eta \sim \gamma _{Y}$ such that $p\in \eta$ is a weakly hyperbolic periodic point. As $H(\gamma , X)$ is R-robustly measure expansive, according to Lemma 3.8, Y has no weakly hyperbolic periodic points, a contradiction. □

Remark 3.10

Let $\varphi \equiv X_{1} : M \to M$ be a diffeomorphism, and let $p\in \gamma \in \operatorname{Per}(X)$ with $X_{\pi (p)}(p)=p$. We set $X_{1}(p)=p_{1}$. Then we define the homoclinic class $H_{\varphi }(p _{1})$ that contains $p_{1}$. By assumption $H_{X}(\gamma )\cap \operatorname{Sing}(X)=\emptyset$. According to [1, Theorem 3.2], a vector field X is continuum-wise expansive if and only if a suspension map φ of X is continuum-wise expansive. Thus as in the proof of Theorem A, we have that the homoclinic class $H_{\varphi }(p_{1})$ is hyperbolic if $H_{\varphi }(p_{1})$ is R-robustly continuum-wise expansive.