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Continuum-wise expansive homoclinic classes for robust dynamical systems

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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.

Correspondence to Manseob Lee.

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MSC

  • 37C20
  • 37D20
  • 37C27

Keywords

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