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Continuumwise expansive diffeomorphisms and conservative systems
Journal of Inequalities and Applications volume 2014, Article number: 379 (2014)
Abstract
We prove that ${C}^{1}$generically, continuumwise expansive diffeomorphisms satisfy both Axiom A and the nocycle condition. Moreover, (i) if a volumepreserving diffeomorphism belongs to the ${C}^{1}$interior of the set of all continuumwise expansive volumepreserving diffeomorphisms then it is Anosov, and (ii) ${C}^{1}$generically, every continuumwise expansive volumepreserving 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 nonwandering 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 nonwandering 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 quasiAnosov 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 quasiAnosov.
In this paper, we study the notion of continuumwise 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 continuumwise 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 continuumwise expansive constant for f. Note that every expansive homeomorphism is continuumwise 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 continuumwise expansive diffeomorphisms (see [4]).
2 Continuumwise 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 continuumwise expansive diffeomorphisms, respectively. Sakai [5] proved that $f\in \mathcal{CWE}(M)$ if and only if the diffeomorphism is quasiAnosov. 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 ωlimit set of x. Let $\mathrm{\Lambda}\subset M$ be an finvariant closed set. We say that Λ admits a dominated splitting if the tangent bundle ${T}_{\mathrm{\Lambda}}M$ has a continuous Dfinvariant splitting $E\oplus F$ and there exist constants $C>0$ and $0<\lambda <1$ such that
for all $x\in \mathrm{\Lambda}$ and $n\ge 0$. Recently, Lee [6] showed that if a transitive set Λ is ${C}^{1}$stably continuumwise 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 Ωstable, that is, obeys Axiom A and the nocycle 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 nocycle condition.
In this spirit, we show that ${C}^{1}$generically, every continuumwise expansive diffeomorphism satisfies both Axiom A and the nocycle condition. This is a generalization of the remarkable result in [7].
Theorem A For ${C}^{1}$generic f, if f is continuumwise expansive then f satisfies both Axiom A and the nocycle condition.
3 Continuumwise volumepreserving 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 volumepreserving diffeomorphisms, the nonwandering set $\mathrm{\Omega}(f)=M$ by recurrent theorem. We say that Λ is hyperbolic if the tangent bundle ${T}_{\mathrm{\Lambda}}M$ has a Dfinvariant splitting ${E}^{s}\oplus {E}^{u}$ and there exist constants $C>0$ and $0<\lambda <1$ such that
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 continuumwise expansive. In [8], Bessa et al. proved that a volumepreserving diffeomorphism belongs to the ${C}^{1}$interior of the set of all expansive volumepreserving 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 continuumwise expansive volumepreserving diffeomorphisms. In this paper, we study the continuumwise 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 continuumwise 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 continuumwise 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 Ωstable. It is well known that for a Ωstable diffeomorphism, there is a diffeomorphism such that the diffeomorphism is not expansive. However, for volumepreserving 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 continuumwise expansiveness for generic view point. Then we have the following.
Theorem C For ${C}^{1}$generic f, if f is continuumwise 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 $\stackrel{\u02c6}{g}\in \mathcal{U}(f)$ such that $\stackrel{\u02c6}{g}(x)=g(x)$ if $x\in \{{x}_{1},{x}_{2},\dots ,{x}_{N}\}\cup (M\setminus U)$ and ${D}_{{x}_{i}}\stackrel{\u02c6}{g}={L}_{i}$ 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 δweak 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 δweak eigenvalue, then f has a hyperbolic point $p\in P(f)$ with a 2δweak eigenvalue;

(2)
for any $\delta >0$, if f has a hyperbolic point $q\in P(f)$ with a δweak eigenvalue, then f has a hyperbolic point $p\in P(f)$ with a δweak 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 continuumwise expansive, then there exists $\delta >0$ such that f has no δweak eigenvalue.
Proof Let ${\mathcal{R}}_{2}={\mathcal{R}}_{1}$, and let $f\in {\mathcal{R}}_{2}$ be continuumwise expansive for f. Suppose, by contradiction, that for any $\delta >0$ there is a periodic point p of f such that p has a δweak 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 nonhyperbolic 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
Observe that $\parallel {L}_{\pi (q)1}{D}_{{g}^{\pi (q)1}(q)}g\parallel \le \parallel \mathcal{O}\mathrm{id}\parallel \cdot \parallel {D}_{{g}^{k1}(q)}g\parallel <{\epsilon}_{0}$. 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
where ${B}_{\delta}(p)$ denotes the δneighborhood 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
where ${T}_{q}M({\delta}_{1})=\{v\in {T}_{q}M\parallel v\parallel \le {\delta}_{1}\}$. Then by (c) we have
on ${exp}_{q}({T}_{q}M(4{\delta}_{1}))$.
We write
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 continuumwise expansive then ${f}^{k}$ is continuumwise 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 $n\in \mathbb{Z}$,
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 continuumwise 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 continuumwise 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 δweak eigenvalue. This is a contradiction by Lemma 4.4. □
Proof of Theorem A Let $f\in {\mathcal{R}}_{3}$ be continuumwise 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 nocycle condition. Thus it is Ωstable. □
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 finvariant set $E=\{{x}_{1},\dots ,{x}_{m}\}$, any neighborhood U of E and any volumepreserving 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 volumepreserving 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 continuumwise expansive volumepreserving 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 nonhyperbolic 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
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 nonhyperbolic 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 $0\le i\ne j\le l1$,

(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 $n\in \mathbb{Z}$,
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 $0\le i\ne j\le l1$,

(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 volumepreserving 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 1214, 2008, that every Anosov diffeomorphism is transitive. It has not been published yet. Nevertheless, in the volumepreserving diffeomorphism, an Anosov diffeomorphism has the nonwandering 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 volumepreserving diffeomorphism is transitive, which is a direct consequence of classic hyperbolic dynamics. But in volumepreserving diffeomorphisms Bonatti and Crovisier proved that ${C}^{1}$generically, a volumepreserving 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 volumepreserving 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 continuumwise 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 continuumwise expansive. By Theorem A, f is Anosov. Since $f\in {\mathcal{R}}_{4}$, by Lemma 5.5, f is transitive. Thus if f is continuumwise 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. □
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Acknowledgements
The author would like to thank the referee for his/her useful comments, detailed corrections and suggestions. This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (No. 2014R1A1A1A05002124).
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Lee, M. Continuumwise expansive diffeomorphisms and conservative systems. J Inequal Appl 2014, 379 (2014). https://doi.org/10.1186/1029242X2014379
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Keywords
 Axiom A
 expansive
 continuumwise expansive
 Anosov
 transitive
 generic property