Continuum-wise expansive diffeomorphisms and conservative systems

Abstract

We prove that $C^1$-generically, continuum-wise expansive diffeomorphisms satisfy both Axiom A and the no-cycle condition. Moreover, (i) if a volume-preserving diffeomorphism belongs to the $C^1$-interior of the set of all continuum-wise expansive volume-preserving diffeomorphisms then it is Anosov, and (ii) $C^1$-generically, every continuum-wise expansive volume-preserving diffeomorphism is transitive Anosov.

MSC:37C20, 37D20.

1 Introduction

Let $Diff(M)$ be the space of diffeomorphisms of closed $C^{\mathrm{\infty }}$-manifolds M endowed with the $C^1$-topology, and let d denote the distance on M induced from a Riemannian metric $\parallel \cdot \parallel$ on the tangent bundle TM. In dynamical systems, expansivity is a useful notion to study of the stability. Roughly speaking, if two points stay near for future and past iterates, then they must be equal. We say that f is expansive if there is $e>0$ such that for any pair of distinct points $x,y\in M$, $d(f^n(x),f^n(y))>e$ for some $n\in \mathbb{Z}$. The number $e>0$ is called an expansive constant for f.

For a point $x\in M$, we say that x is a non-wandering point if for any neighborhood U of x, there is $n\in \mathbb{Z}$ such that $f^n(U)\cap U\ne \mathrm{\varnothing }$. Denote by $\mathrm{\Omega }(f)$ the set of all non-wandering points of f. It is clear $\overline{P(f)}\subset \mathrm{\Omega }(f)$, where $P(f)$ is the set of the periodic points of f, and $\overline{P(f)}$ is the closure of $P(f)$. We say that f satisfies Axiom A if $\mathrm{\Omega }(f)=\overline{P(f)}$ is hyperbolic. We say that f is quasi-Anosov if for any $v\in TM$ ($v\ne 0$) the set $\{\parallel D{f}^n(v)\parallel :n\in \mathbb{Z}\}$ is unbounded. It follows that f satisfies Axiom A.

For expansivity, in [1], Mañé showed that a diffeomorphism belongs to the $C^1$-interior of the set of all expansive diffeomorphisms if and only if f is quasi-Anosov.

In this paper, we study the notion of continuum-wise expansivity which was introduced by Kato in [2]. Let Λ be a closed set of M. A set Λ is nondegenerate if the set Λ is not reduced to one point. We say that $\mathrm{\Lambda }\subset M$ is a subcontinuum if it is a compact connected nondegenerate subset Λ of M. A diffeomorphism f on M is said to be continuum-wise expansive if there is a constant $e>0$ such that for any nondegenerate subcontinuum A there is an integer $n=n(A)$ such that $diam{f}^n(A)\ge e$, where $diamS=sup\{d(x,y):x,y\in S\}$ for any subset S of M. Such a constant α is called a continuum-wise expansive constant for f. Note that every expansive homeomorphism is continuum-wise expansive diffeomorphism, but its converse is not true (see [[3], Example 3.5]). For diffeomorphisms, we introduce an example. It is well known that ${\mathbf{\text{S}}}^2$ does not admit an expansive diffeomorphism, but it admits a continuum-wise expansive diffeomorphisms (see [4]).

2 Continuum-wise diffeomorphisms

Let M be as before, and let $f\in Diff(M)$. Denote by $\mathcal{E}(M)$ and $\mathcal{CWE}(M)$ the set of all expansive diffeomorphisms and the set of all continuum-wise expansive diffeomorphisms, respectively. Sakai [5] proved that $f\in \mathcal{CWE}(M)$ if and only if the diffeomorphism is quasi-Anosov. By Mañé’s result [1], we know the following.

Theorem 2.1 The $C^1$-interior of $\mathcal{CWE}(M)$ coincides with the $C^1$-interior of $\mathcal{E}(M)$.

We say that Λ is transitive set if there is a point $x\in \mathrm{\Lambda }$ such that ${\omega }_f(x)=\mathrm{\Lambda }$, where ${\omega }_f(x)$ is the ω-limit set of x. Let $\mathrm{\Lambda }\subset M$ be an f-invariant closed set. We say that Λ admits a dominated splitting if the tangent bundle $T_{\mathrm{\Lambda }}M$ has a continuous Df-invariant splitting $E\oplus F$ and there exist constants $C>0$ and $0<\lambda <1$ such that

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

for all $x\in \mathrm{\Lambda }$ and $n\ge 0$. Recently, Lee [6] showed that if a transitive set Λ is $C^1$-stably continuum-wise expansive then it admits a dominated splitting.

A subset $\mathcal{R}\subset Diff(M)$ is called residual if it contains a countable intersection of open and dense subsets of $Diff(M)$. A dynamic property is called $C^1$-generic if it holds in a residual subset of $Diff(M)$. We use the terminology for $C^1$-generic f to express there is a residual subset $\mathcal{R}\subset Diff(M)$, and $f\in \mathcal{R}$.

Recently, in [7], Arbieto proved that for $C^1$-generic $f\in Diff(M)$, f is expansive then f is Ω-stable, that is, obeys Axiom A and the no-cycle condition. We stated the above fact.

Theorem 2.2 For $C^1$-generic f, if f is expansive then f satisfies both Axiom A and the no-cycle condition.

In this spirit, we show that $C^1$-generically, every continuum-wise expansive diffeomorphism satisfies both Axiom A and the no-cycle condition. This is a generalization of the remarkable result in [7].

Theorem A For $C^1$-generic f, if f is continuum-wise expansive then f satisfies both Axiom  A and the no-cycle condition.

3 Continuum-wise volume-preserving diffeomorphisms

Let M be a closed $C^{\mathrm{\infty }}$ Riemannian manifold endowed with a volume form ω. Let μ denote the Lebesgue measure associated to ω, and let d denote the metric induced on M by the Riemannian structure. Denote by ${Diff}_{\mu }(M)$ the set of diffeomorphisms which preserves the Lebesgue measure μ endowed with the Whitney $C^1$-topology. Note that in volume-preserving diffeomorphisms, the non-wandering set $\mathrm{\Omega }(f)=M$ by recurrent theorem. We say that Λ is hyperbolic if the tangent bundle $T_{\mathrm{\Lambda }}M$ has a Df-invariant splitting $E^s\oplus E^u$ and there exist constants $C>0$ and $0<\lambda <1$ such that

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

for all $x\in \mathrm{\Lambda }$ and $n\ge 0$. Moreover, if $\mathrm{\Lambda }=M$ then f is Anosov. Note that f is Anosov then f is expansive, and so, f is continuum-wise expansive. In [8], Bessa et al. proved that a volume-preserving diffeomorphism belongs to the $C^1$-interior of the set of all expansive volume-preserving diffeomorphisms if and only if it is Anosov. For the another conservative cases, that is, geodesic flow and a Hamiltonian system, Bessa et al. have shown in [9] that if a Hamiltonian system belongs to the $C^2$-interior of the set of all expansive Hamiltonian systems then it is Anosov. And Ruggiero [10] showed that if a geodesic flow belongs to the $C^1$-interior of the set of all expansive geodesic vector fields then it is Anosov.

Let ${\mathcal{CWE}}_{\mu }(M)$ be the set of all continuum-wise expansive volume-preserving diffeomorphisms. In this paper, we study the continuum-wise expansive case, and if f belongs to the $C^1$-interior of ${\mathcal{CWE}}_{\mu }(M)$, then f is Anosov. Let $int{\mathcal{CWE}}_{\mu }(M)$ denote the $C^1$-interior of the set of all continuum-wise expansive volume preserving diffeomorphisms. In this paper, we prove the following theorem.

Theorem B The set ${\mathcal{AN}}_{\mu }(M)$ of Anosov diffeomorphisms in ${Diff}_{\mu }(M)$ coincides with the $C^1$-interior of the set of continuum-wise expansive diffeomorphisms in ${Diff}_{\mu }(M)$; that is, ${\mathcal{AN}}_{\mu }(M)=int{\mathcal{CWE}}_{\mu }(M)$.

In diffeomorphisms, Arbieto [7] proved that $C^1$-generically, if f is expansive then f is Ω-stable. It is well known that for a Ω-stable diffeomorphism, there is a diffeomorphism such that the diffeomorphism is not expansive. However, for volume-preserving diffeomorphisms, the phenomenon cannot happen since $\mathrm{\Omega }(f)=M$. In $dimM=2$, for $C^1$-generic f, if a $C^1$-neighborhood $\mathcal{U}(f)$ of f, there is $g\in \mathcal{U}(f)$ such that g has a periodic point $p_g$ with homoclinic tangency $q_g$ then f has a periodic point p with homoclinic tangency q. In fact, it is closely related to the conjecture of Smale (see [11]). Note that if $dimM=2$ then it does not exist normally hyperbolic. In this paper, we consider $dimM\ge 3$. Recently, Bessa et al. [8] proved that $dimM\ge 3$, for $C^1$-generic f, if $f\in {Diff}_{\mu }(M)$ is expansive then f is Anosov. For a Hamiltonian system, Lee [12] showed that $C^2$-generically, an expansive Hamiltonian system is Anosov. In this spirit, we study the continuum-wise expansiveness for generic view point. Then we have the following.

Theorem C For $C^1$-generic f, if f is continuum-wise expansive then it is transitive Anosov.

4 Proof of Theorem A

Let $dimM\ge 3$ and let $f\in Diff(M)$. We prepare several lemmas to arrive at Theorem A. The Franks lemma [13] will play an essential role in our proofs.

Lemma 4.1 Let $\mathcal{U}(f)$ be any given $C^1$-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 given $g\in {\mathcal{U}}_0(f)$, a finite set $\{x_1,x_2,\dots ,x_N\}$, a neighborhood U of $\{x_1,x_2,\dots ,x_N\}$ and linear maps $L_i:T_{x_i}M\to T_{g(x_i)}M$ satisfying $\parallel L_i-D_{x_i}g\parallel \le \epsilon$ for all $1\le i\le N$, there exists $\hat{g}\in \mathcal{U}(f)$ such that $\hat{g}(x)=g(x)$ if $x\in \{x_1,x_2,\dots ,x_N\}\cup (M\setminus U)$ and $D_{x_i}\hat{g}=L_i$ for all $1\le i\le N$.

Let p be a periodic point of f, and let $0<\delta <1$. We say p has a δ-weak eigenvalue if $D_p{f}^{\pi (p)}$ has an eigenvalue λ such that ${(1-\delta )}^{\pi (p)}<|\lambda |<{(1+\delta )}^{\pi (p)}$. The following lemma will also play a crucial role in our proof.

Lemma 4.2 [[7], Lemma 5.1]

There exists a residual set ${\mathcal{R}}_1\subset Diff(M)$ such that for any $f\in {\mathcal{R}}_1$,

• (1) for any $\delta >0$, if for any $C^1$-neighborhood $\mathcal{U}(f)$, there is $g\in \mathcal{U}(f)$ which has a hyperbolic $p_g\in P(g)$ with a δ-weak eigenvalue, then f has a hyperbolic point $p\in P(f)$ with a 2δ-weak eigenvalue;

• (2) for any $\delta >0$, if f has a hyperbolic point $q\in P(f)$ with a δ-weak eigenvalue, then f has a hyperbolic point $p\in P(f)$ with a δ-weak eigenvalue, whose eigenvalues are all real.

Remark 4.3 If f has a normally hyperbolic, then by Hirsh et al. [14] and Mañé [15], it is $C^1$-robust, that is, for any g $C^1$-close to f, g has a normally hyperbolic then f also has a normally hyperbolic (see also [16]).

Lemma 4.4 There exists a residual set ${\mathcal{R}}_2\subset Diff(M)$ such that for $f\in {\mathcal{R}}_2$ if f is continuum-wise expansive, then there exists $\delta >0$ such that f has no δ-weak eigenvalue.

Proof Let ${\mathcal{R}}_2={\mathcal{R}}_1$, and let $f\in {\mathcal{R}}_2$ be continuum-wise expansive for f. Suppose, by contradiction, that for any $\delta >0$ there is a periodic point p of f such that p has a δ-weak eigenvalue. Let $\epsilon >0$, and let $\mathcal{V}(f)\subset {\mathcal{U}}_0(f)$ be a $C^1$-neighborhood of f which is given by Lemma 4.1 with respect to ${\mathcal{U}}_0(f)$. Then there exist $g\in \mathcal{U}(f)$ and a non-hyperbolic periodic point q of g such that an eigenvalue λ of $D_q{g}^{\pi (q)}$ with $|\lambda |=1$, and $T_{q}M=E^c(q)\oplus E^s(q)\oplus E^u(q)$, where $E^{\sigma }(q)$, $\sigma =c,s,u$, are $D_q{g}^{\pi (q)}$-invariant subspaces corresponding to eigenvalues λ of $D_q{g}^{\pi (q)}$ for $|\lambda |=1$, $|\lambda |<1$, and $|\lambda |>1$, respectively. Let $\mathcal{W}(f)\subset \mathcal{V}(f)$ be the $C^1$ ${\epsilon }_0$-ball of f. Set $C={sup}_{x\in M}\{\parallel D_{x}g\parallel \}$. For $0<{\epsilon }_1<{\epsilon }_0$, we can obtain a linear automorphism $\mathcal{O}:T_{q}M\to T_{q}M$ such that

• (i) $\parallel \mathcal{O}-\mathrm{id}\parallel <\frac{{\epsilon }_1}{C}$,

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

• (iii) all eigenvalues of $\mathcal{O}\circ D_q{g}^{\pi (q)}$, say ${\mu }_j$, $j=1,2,\dots ,c$, are roots of unity.

Let F be the finite set $\{q,g(q),\dots ,g^{\pi (q)-1}(q)\}$. Define

$$L_j=\{\begin{array}{ll}{D}_{g^j(q)}g,& j=0,1,\dots ,\pi (q)-2,\\ \mathcal{O}\circ D_{g^{\pi (q)-1}(q)}g,& j=\pi (q)-1.\end{array}$$

Observe that $\parallel L_{\pi (q)-1}-D_{g^{\pi (q)-1}(q)}g\parallel \le \parallel \mathcal{O}-\mathrm{id}\parallel \cdot \parallel D_{g^{k-1}(q)}g\parallel <{\epsilon }_0$. Thus $\parallel L_j-D_{g^j(q)}g\parallel <{\epsilon }_0$ for all $j=0,1,\dots ,\pi (p)-1$. By Lemma 4.1, we can find a diffeomorphism $g_1\in \mathcal{W}(f)$ and ${\delta }_0>0$ such that

• (a) $B_{4{\delta }_0}(g^i(q))\cap B_{4{\delta }_0}(q)=\mathrm{\varnothing }$, $0\le i\ne j\le \pi (q)-1$,

• (b) $g_1=g$ on $F\cup (M-{\bigcup }_{j=0}^{\pi (q)-1}{B}_{4{\delta }_0}(g^j(q)))$,

• (c) $g_1={exp}_{g^{j+1}(q)}\circ L_j\circ {exp}_{g^j(q)}^{-1}$ on $B_{{\delta }_0}(g^j(q))$, $0\le j\le \pi (q)-1$.

Define

$$L=\mathcal{O}\circ D_q{g}^{\pi (q)}=\prod _{j=0}^{\pi (q)-1}{L}_j,$$

where $B_{\delta }(p)$ denotes the δ-neighborhood of p.

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

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

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

$${\left(g_1^{\pi (q)}\right)}^m=g_1^{m\pi (q)}={exp}_q\circ G^m\circ {exp}_q^{-1}$$

on ${exp}_q(T_{q}M(4{\delta }_1))$.

We write

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

where $E^{\sigma }(q,{\delta }_1)=E^{\sigma }(q)\cap T_{q}M({\delta }_1)$, $\sigma =c,s,u$. Then ${exp}_q(E^c(q,4{\delta }_1))$ is ${(g_1^k)}^m$-invariant. Since $f\in {\mathcal{R}}_1$, we assume that the eigenvalue $\lambda \in \mathbb{R}$.

Put ${exp}_q(E^c(q,4{\delta }_1))$ is an arc ${\mathcal{I}}_q$ centered at q. Observe that ${(g_1^k)}^m=\mathrm{id}$ on ${exp}_q(E^c(q,4{\delta }_1))$. By our construction, ${(g_1^k)}^m$ is the identity on the arc ${\mathcal{I}}_q$. It is clear that the small arc ${\mathcal{I}}_q$ is normally hyperbolic for $g_1$. By Remark 4.3, for any g $C^1$-close to f, if g has a normally hyperbolic then f has a normally hyperbolic, that is, it is $C^1$-robust. Then we know that f has a small arc ${\mathcal{J}}_q$ which centered at q with $f^{\pi (q)}({\mathcal{J}}_q)={\mathcal{J}}_q$. Note that if f is continuum-wise expansive then $f^k$ is continuum-wise expansive for any $k\in \mathbb{Z}$ (see [[2], Proposition 2.6]). Denote by $l(A)$ the length of A. Take $e=2l({\mathcal{J}}_q)$. Since ${\mathcal{J}}_q$ is $f^{\pi (q)}$-invariant, for all $n\in \mathbb{Z}$,

$$diam(f^n({\mathcal{J}}_q))<e.$$

This is a contradiction. □

We say that f satisfies star condition if there is a $C^1$-neighborhood $\mathcal{U}(f)$ such that for any $g\in \mathcal{U}(f)$, every $p\in P(g)$ is hyperbolic. We denote by $\mathcal{F}(M)$ the set of diffeomorphisms satisfying star condition.

Lemma 4.5 There is a residual set ${\mathcal{R}}_3\subset Diff(M)$ such that for any continuum-wise expansive map $f\in {\mathcal{R}}_3$, $f\in \mathcal{F}(M)$.

Proof Let ${\mathcal{R}}_3={\mathcal{R}}_2$, and let $f\in {\mathcal{R}}_3$ be continuum-wise expansive. Proof by contradiction, we may assume that $f\notin \mathcal{F}(M)$. Then by Lemma 5.1, there is g $C^1$-close to f and $p_g\in P(g)$ such that for any $\delta >0$, $p_g$ has a $\delta /2$-weak eigenvalue. By Lemma 4.2, $p\in P(f)$ has a δ-weak eigenvalue. This is a contradiction by Lemma 4.4. □

Proof of Theorem A Let $f\in {\mathcal{R}}_3$ be continuum-wise expansive. By Lemma 4.5, $f\in \mathcal{F}(M)$. Since $f\in \mathcal{F}(M)$, By Aoki [17] and Hayashi [18], we know that f satisfies both Axiom A and the no-cycle condition. Thus it is Ω-stable. □

5 Proof of Theorem B and Theorem C

Let M and let $f\in {Diff}_{\mu }(M)$ be as before. To prove our result, we use the Franks lemma, which is proved in [[19], Proposition 7.4].

Lemma 5.1 Let $f\in {Diff}_{\mu }^1(M)$, and be a $C^1$-neighborhood of f in ${Diff}_{\mu }^1(M)$. Then there exist a $C^1$-neighborhood ${\mathcal{U}}_0\subset \mathcal{U}$ of f and $\epsilon >0$ such that if $g\in {\mathcal{U}}_0$, any finite f-invariant set $E=\{x_1,\dots ,x_m\}$, any neighborhood U of E and any volume-preserving linear maps $L_j:T_{x_j}M\to T_{g(x_j)}M$ with $\parallel L_j-D_{x_j}g\parallel \le \epsilon$ for all $j=1,\dots ,m$, there is a conservative diffeomorphism $g_1\in \mathcal{U}$ coinciding with f on E and out of U, and $D_{x_j}{g}_1=L_j$ for all $j=1,\dots ,m$.

We denote by ${\mathcal{F}}_{\mu }(M)$ the set of diffeomorphisms $f\in {Diff}_{\mu }(M)$ which have a $C^1$-neighborhood $\mathcal{U}(f)\subset {Diff}_{\mu }(M)$ such that for any $g\in \mathcal{U}(f)$, every periodic point of g is hyperbolic.

Very recently, Arbieto and Catalan [20] proved that every volume-preserving diffeomorphism in ${\mathcal{F}}_{\mu }(M)$ is Anosov.

Theorem 5.2 [[20], Theorem 1.1]

Every diffeomorphism in ${\mathcal{F}}_{\mu }(M)$ is Anosov.

To prove Theorem B, it is enough to show that a continuum-wise expansive volume-preserving diffeomorphism $f\in {\mathcal{F}}_{\mu }(M)$.

Remark 5.3 Let $f\in {Diff}_{\mu }^1(M)$. From the Moser theorem (see [21]), we can find a smooth conservative change of coordinates ${\phi }_x:U(x)\to T_{x}M$ such that ${\phi }_x(x)=0$, where $U(x)$ is a small neighborhood of $x\in M$.

Lemma 5.4 If $f\in int{\mathcal{CWE}}_{\mu }(M)$, then $f\in {\mathcal{F}}_{\mu }(M)$.

Proof Take $f\in int{\mathcal{CWE}}_{\mu }(M)$, and $\mathcal{U}(f)$ a $C^1$-neighborhood of f. Let $\epsilon >0$ and $\mathcal{V}(f)\subset \mathcal{U}(f)$ be corresponding number and $C^1$-neighborhood given by Lemma 5.1. To derive a contradiction, suppose that there is a non-hyperbolic periodic point $p\in P(g)$ for some $g\in \mathcal{V}(f)$. To simplify the notation in the proof, we may assume that $g(p)=p$. Then there is at least one eigenvalue λ of $D_{p}g$ such that $|\lambda |=1$. By making use of Lemma 5.1, we linearize f at p with respect to Moser’s theorem, that is, by choosing $\alpha >0$ sufficiently small we construct $g_1$ $C^1$-nearby g such that

$$g_1(x)=\{\begin{array}{ll}{\phi }_p^{-1}\circ D_{p}g\circ {\phi }_p(x)& \text{if }x\in B_{\alpha }(p),\\ g(x)& \text{if }x\notin B_{4\alpha }(p).\end{array}$$

Then $g(p)=g_1(p)=p$. Thus $T_{p}M=E^c\oplus E^{\sigma }$, where $E_p^c$ associated to $\lambda =1$ and $E_p^{\sigma }$ associated to eigenvalues less than one and greater than one. Take $\eta =\alpha /4$. Then we define $E^c(\eta )\cap {\phi }_p(B_{\alpha }(p))=E^c(\eta )$.

Case 1. $dim{E}_p^c=1$.

Since p is non-hyperbolic for $g_1$, by our construction, we may assume that there is $l>0$ such that $D_p{g}_1^l(v)=v$ for any $v\in E_p^c(\eta )\cap {\phi }_p(B_{\alpha }(p))$. Take $v\in E_p^c(\eta )$ such that $\parallel v\parallel =\eta /4$. Then we can find a small arc ${\mathcal{I}}_p={\phi }_p^{-1}(\{tv:1\le t\le \eta /4\})\subset B_{\alpha }(p)$ such that

• (i) $g_1^i({\mathcal{I}}_p)\cap g_1^j({\mathcal{I}}_p)=\mathrm{\varnothing }$ if $0\le i\ne j\le l-1$,

• (ii) $g_1^l({\mathcal{I}}_p)={\mathcal{I}}_p$, that is, $g_1^l{|}_{{\mathcal{I}}_p}$ is the identity map,

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

For simplicity, we assume that $g_1^l=g_1$. Take $e=\eta$. Then for all $n\in \mathbb{Z}$,

$$diam(g_1^n({\mathcal{I}}_p))<e.$$

This is a contradiction.

Case 2. $dim{E}_p^c=2$.

In the proof of the second case, to avoid notational complexity, we consider the case $g(p)=p$. By Lemma 5.1, there is $\alpha >0$ and $h\in \mathcal{U}(f)$ such that $h(p)=g(p)=p$ and $h(x)={\phi }_p^{-1}\circ D_{p}g\circ {\phi }_p(x)$ if $x\in B_{\alpha }(p)$. With a small modification of $D_{p}g$, we may assume that there is $l>0$ such that $D_p{g}^l(v)=v$ for any $v\in E_p^c(\alpha )$ by Lemma 5.1. We can choose $v\in E_p^c(\alpha )$ such that $\parallel v\parallel =\alpha /4$ and we set ${\mathcal{D}}_p={\phi }_p^{-1}(\{tv:1\le t\le \alpha /4\})\subset B_{\alpha }(p)$. Then the disk ${\mathcal{D}}_p$ satisfies the following conditions:

• (i) $h^i({\mathcal{D}}_{p_h})\cap h^j({\mathcal{D}}_{p_h})=\mathrm{\varnothing }$ if $0\le i\ne j\le l-1$,

• (ii) $h^l({\mathcal{D}}_{p_h})={\mathcal{D}}_{p_h}$, that is, $h^l{|}_{{\mathcal{D}}_{p_h}}$ is the identity map,

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

As in the proof of the $dim{E}_p^c=1$, we can derive a contradiction. □

Proof of Theorem B Suppose that $f\in int{\mathcal{CWE}}_{\mu }(M)$. By Lemma 5.4, $f\in {\mathcal{F}}_{\mu }(M)$. Thus by Theorem 5.2, f is Anosov. □

Proof of Theorem C The proof of Theorem C is parallel the proof of Theorem A. Indeed, to prove Theorem A we use previous results - Lemmas 4.2, 4.4 and 4.5. Then we have a volume-preserving diffeomorphism $f\in {\mathcal{F}}_{\mu }(M)$. Thus f is Anosov. □

In diffeomorphisms, there is an open problem: are Anosov diffeomorphisms transitive? In [13] Franks and [22] Newhouse proved it for codimension one Anosov diffeomorphisms. It was announced in Xia in a talk, Anosov diffeomorphisms are transitive, an invited talk of the Rocky Mountain Conference on Dynamical Systems, May 12-14, 2008, that every Anosov diffeomorphism is transitive. It has not been published yet. Nevertheless, in the volume-preserving diffeomorphism, an Anosov diffeomorphism has the non-wandering set equal to the whole manifold M by the Poincaré theorem. By the shadowing property of the hyperbolic sets the periodic points are dense in M. And by the Smale spectral decomposition theorem, we have a single piece equal to M, and so, we have transitivity. Thus, the Anosov volume-preserving diffeomorphism is transitive, which is a direct consequence of classic hyperbolic dynamics. But in volume-preserving diffeomorphisms Bonatti and Crovisier proved that $C^1$-generically, a volume-preserving diffeomorphism is transitive.

Theorem 5.5 [[23], Theorem 1.3]

There is a residual set ${\mathcal{R}}_4\subset {Diff}_{\mu }(M)$ such that for any $f\in {\mathcal{R}}_4$, f is transitive and M is a unique homoclinic class.

We say that f is transitive if there is a point $x\in M$ such that $\omega (x)=M$, where $\omega (x)$ is the omega limit set.

Remark 5.6 In [[24], Theorem 1.3], Newhouse showed that $C^1$-generic volume-preserving diffeomorphisms in surfaces are Anosov or else the elliptical points, nonreal eigenvalues conjugated and of norm one, are dense.

By [8] and Theorem A, we have the following.

Corollary 5.7 There is a residual set $\mathcal{G}\subset {Diff}_{\mu }(M)$ such that for any $f\in \mathcal{G}$, the following are equivalents:

• (a) f is expansive,

• (b) f is transitive Anosov.

  Moreover, if $dimM\ge 3$ then

• (c) f is continuum-wise expansive,

• (d) f has the shadowing property,

• (e) f has the weak specification property.

Proof Let $f\in \mathcal{G}={\mathcal{R}}_3\cap {\mathcal{R}}_4$ is continuum-wise expansive. By Theorem A, f is Anosov. Since $f\in {\mathcal{R}}_4$, by Lemma 5.5, f is transitive. Thus if f is continuum-wise expansive, then f is transitive Anosov. By Bessa et al. [8], f is expansive, then f is Anosov, and so, f is transitive Anosov. If $dimM\ge 3$, then by Bessa et al. [8] if f has the shadowing property and f has the weak specification property, then f is Anosov and since $f\in {\mathcal{R}}_4$, also f is transitive. Thus f is transitive Anosov. □