# 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.

## Introduction

### 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.

### 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.

## 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 $$\epsilon >0$$,

1. (a)

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

2. (b)

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

3. (c)

the endpoints of $$\mathcal{I}$$ are hyperbolic, and

4. (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:

1. (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]).

2. (b)

$$H(p, f)=C(p, f)$$ (see [3]).

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

1. (i)

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

2. (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

1. (i)

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

2. (ii)

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

3. (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. □

## 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:

1. (a)

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

2. (b)

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

3. (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:

1. (a)

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

2. (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 $L= ( A O O B )$ 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

1. (a)

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

2. (b)

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

3. (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

1. (a)

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

2. (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

1. (a)

$$\mathcal{J}$$ is periodic with period T,

2. (b)

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

3. (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

1. (i)

the map g is defined by Y,

2. (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,

3. (iii)

$$0<\operatorname{diam}\mathcal{I}\leq \epsilon$$ and $$0<\operatorname{diam}\mathcal{D}\leq \epsilon$$,

4. (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

5. (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.

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## Acknowledgements

The author would like to thank the referees for many helpful comments.

## Funding

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2017R1A2B4001892).

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The author has contributed solely to the writing of this paper. He read and approved the manuscript.

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Correspondence to Manseob Lee.

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Lee, M. Continuum-wise expansive homoclinic classes for robust dynamical systems. Adv Differ Equ 2019, 333 (2019). https://doi.org/10.1186/s13662-019-2249-3

• Accepted:

• Published:

• 37C20
• 37D20
• 37C27

### Keywords

• Expansive
• Measure expansive
• Chain recurrent set
• Homoclinic class
• Generic
• Hyperbolic