Continuum wise expansive diffeomorphisms and conservative systems

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R E S E A R C H

Open Access

Continuum-wise expansive diffeomorphisms

and conservative systems

Manseob Lee

*

*_{Correspondence:}

[email protected] Department of Mathematics, Mokwon University, Daejeon, 302-729, Korea

Abstract

We prove thatC1_{-generically, continuum-wise expansive diffeomorphisms satisfy} both Axiom A and the no-cycle condition. Moreover, (i) if a volume-preserving diffeomorphism belongs to theC1_{-interior of the set of all continuum-wise expansive} volume-preserving diffeomorphisms then it is Anosov, and (ii)C1_{-generically, every} continuum-wise expansive volume-preserving diffeomorphism is transitive Anosov.

MSC: 37C20; 37D20

Keywords: Axiom A; expansive; continuum-wise expansive; Anosov; transitive; generic property

1 Introduction

LetDiff(M) be the space of diﬀeomorphisms of closedC∞-manifoldsMendowed with theC_{-topology, and let}_d_{denote the distance on}_M_{induced from a Riemannian metric}

· on the tangent bundleTM. 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 thatf isexpansiveif there ise>  such that for any pair of distinct pointsx,y∈M,d(fn₍_x_),_fn₍_y_{)) >}_e_{for some}_n_∈_Z_{. The number}_e_{>  is called an}

expansive constantforf.

For a pointx∈M, we say thatxis anon-wandering pointif for any neighborhoodU ofx, there isn∈Zsuch thatfn₍_U₎_∩_U₌_∅_{. Denote by}₍_f_{) the set of all non-wandering}

points off. It is clearP(f)⊂(f), whereP(f) is the set of the periodic points off, andP(f) is the closure ofP(f). We say thatf satisﬁesAxiomA if(f) =P(f) is hyperbolic. We say thatf isquasi-Anosovif for anyv∈TM(v= ) the set{Dfn(v):n∈Z}is unbounded. It follows thatf satisﬁes Axiom A.

For expansivity, in [], Mañé showed that a diﬀeomorphism belongs to theC_-interior of the set of all expansive diﬀeomorphisms if and only iff is quasi-Anosov.

In this paper, we study the notion of continuum-wise expansivity which was introduced by Kato in []. Letbe a closed set ofM. A setisnondegenerateif the setis not reduced to one point. We say that⊂Mis asubcontinuumif it is a compact connected nondegenerate subsetofM. A diﬀeomorphismf onMis said to becontinuum-wise ex-pansiveif there is a constante>  such that for any nondegenerate subcontinuumAthere is an integern=n(A) such thatdiamfn(A)≥e, wherediamS=sup{d(x,y) :x,y∈S}for any subsetSofM. Such a constantαis called acontinuum-wise expansive constantforf. Note that every expansive homeomorphism is continuum-wise expansive diﬀeomorphism, but

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its converse is not true (see [, Example .]). For diffeomorphisms, we introduce an ex-ample. It is well known thatSdoes not admit an expansive diffeomorphism, but it admits a continuum-wise expansive diffeomorphisms (see []).

2 Continuum-wise diffeomorphisms

LetMbe as before, and letf ∈Diff(M). Denote byE(M) andCWE(M) the set of all ex-pansive diffeomorphisms and the set of all continuum-wise exex-pansive diffeomorphisms, respectively. Sakai [] proved thatf ∈CWE(M) if and only if the diffeomorphism is quasi-Anosov. By Mañé’s result [], we know the following.

Theorem . The C_{-interior of}_CWE₍_M₎_{coincides with the C}_{-interior of}_E₍_M_).

We say thatistransitive setif there is a pointx∈such thatωf(x) =, whereωf(x)

is theω-limit set ofx. Let⊂Mbe anf-invariant closed set. We say thatadmits a dominated splitting if the tangent bundleTMhas a continuousDf-invariant splitting E⊕Fand there exist constantsC>  and  <λ<  such that

Dxfn|E(x)·Dxf–n|F(fn₍_x₎₎≤Cλn

for allx∈andn≥. Recently, Lee [] showed that if a transitive setisC_-stably continuum-wise expansive then it admits a dominated splitting.

A subsetR⊂Diff(M) is calledresidualif it contains a countable intersection of open and dense subsets ofDiff(M). A dynamic property is calledC_-_generic_{if it holds in a} resid-ual subset ofDiff(M). We use the terminologyfor C_-_{generic f}_{to express}_{there is a residual} subsetR⊂Diff(M),and f ∈R.

Recently, in [], Arbieto proved that forC_-generic_f _∈_Diff(_M_),_f _{is expansive then}_f _is

-stable, that is, obeys Axiom A and the no-cycle condition. We stated the above fact.

Theorem . For C_{-generic f}_, _{if f is expansive then f satisﬁes both Axiom}_A _{and the}

no-cycle condition.

In this spirit, we show thatC-generically, every continuum-wise expansive diﬀeomor-phism satisﬁes both Axiom A and the no-cycle condition. This is a generalization of the remarkable result in [].

Theorem A For C-generic f,if f is continuum-wise expansive then f satisﬁes both

Ax-iomAand the no-cycle condition.

3 Continuum-wise volume-preserving diffeomorphisms

LetMbe a closedC∞Riemannian manifold endowed with a volume formω. Letμdenote the Lebesgue measure associated toω, and letddenote the metric induced onMby the Riemannian structure. Denote byDiffμ(M) the set of diﬀeomorphisms which preserves the Lebesgue measureμendowed with the WhitneyC-topology. Note that in volume-preserving diﬀeomorphisms, the non-wandering set(f) =Mby recurrent theorem. We say thatishyperbolicif the tangent bundleTMhas aDf-invariant splittingEs⊕Eu and there exist constantsC>  and  <λ<  such that

Dxfn|Es x≤Cλ

n _and _D xf–n|Eu

x≤Cλ

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for allx∈andn≥. Moreover, if=Mthenfis Anosov. Note thatfis Anosov thenfis expansive, and so,fis continuum-wise expansive. In [], Bessaet al.proved that a volume-preserving diffeomorphism belongs to theC-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, Bessaet al. have shown in [] that if a Hamiltonian system belongs to theC_{-interior of the set of all expansive Hamiltonian} systems then it is Anosov. And Ruggiero [] showed that if a geodesic flow belongs to the C_{-interior of the set of all expansive geodesic vector fields then it is Anosov.}

LetCWEμ(M) be the set of all continuum-wise expansive volume-preserving diﬀeomor-phisms. In this paper, we study the continuum-wise expansive case, and iff belongs to the C_{-interior of}_CWE_μ(_M_{), then}_f_{is Anosov. Let}_int_CWE_μ(_M_{) denote the}_C_{-interior of the} set of all continuum-wise expansive volume preserving diﬀeomorphisms. In this paper, we prove the following theorem.

Theorem B The setANμ(M)of Anosov diﬀeomorphisms inDiffμ(M)coincides with the

C_{-interior of the set of continuum-wise expansive diﬀeomorphisms in}_Diff_μ(_M_);_{that is}_,

ANμ(M) =intCWEμ(M).

In diﬀeomorphisms, Arbieto [] proved thatC_{-generically, if}_f _{is expansive then}_f _is

-stable. It is well known that for a-stable diffeomorphism, there is a diffeomorphism such that the diffeomorphism is not expansive. However, for volume-preserving diffeo-morphisms, the phenomenon cannot happen since(f) =M. IndimM= , forC-generic f, if aC_{-neighborhood}_U₍_f_{) of}_f_{, there is}_g_∈_U₍_f_{) such that}_g_{has a periodic point}_p

gwith

homoclinic tangencyqgthenf has a periodic pointpwith homoclinic tangencyq. In fact,

it is closely related to the conjecture of Smale (see []). Note that ifdimM=  then it does not exist normally hyperbolic. In this paper, we considerdimM≥. Recently, Bessaet al. [] proved thatdimM≥, forC_-generic_f_{, if}_f _∈_Diffμ(_M_{) is expansive then}_f _{is Anosov.} For a Hamiltonian system, Lee [] showed thatC_{-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_{-generic f}_,_{if f is continuum-wise expansive then it is transitive Anosov}_.

4 Proof of Theorem A

LetdimM≥ and letf ∈Diff(M). We prepare several lemmas to arrive at Theorem A. The Franks lemma [] will play an essential role in our proofs.

Lemma . LetU(f)be any given C-neighborhood of f.Then there existε> and a C

-neighborhood U(f)⊂U(f)of f such that for given g∈U(f),a ﬁnite set {x,x, . . . ,xN},

a neighborhood U of{x,x, . . . ,xN}and linear maps Li:TxiM→Tg(xi)M satisfyingLi–

Dxig ≤εfor all≤i≤N,there existsgˆ∈U(f)such thatgˆ(x) =g(x)if x∈ {x,x, . . . ,xN} ∪

(M\U)and Dxigˆ=Lifor all≤i≤N.

Letpbe a periodic point off, and let  <δ< . We sayphas aδ-weak eigenvalueifDpfπ(p)

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Lemma .[, Lemma .] There exists a residual setR⊂Diff(M)such that for any f ∈R,

() for anyδ> ,if for anyC-neighborhoodU(f),there isg∈U(f)which has a

hyperbolicpg∈P(g)with aδ-weak eigenvalue,thenf has a hyperbolic pointp∈P(f)

with aδ-weak eigenvalue;

() for anyδ> ,iff has a hyperbolic pointq∈P(f)with aδ-weak eigenvalue,thenf has a hyperbolic pointp∈P(f)with aδ-weak eigenvalue,whose eigenvalues are all real.

Remark . Iff has a normally hyperbolic, then by Hirshet al.[] and Mañé [], it is

C_{-robust, that is, for any}_{g C}_{-close to}_f_,_g _{has a normally hyperbolic then}_f _{also has a} normally hyperbolic (see also []).

Lemma . There exists a residual set R ⊂ Diff(M) such that for f ∈ R if f is

continuum-wise expansive,then there existsδ> such that f has noδ-weak eigenvalue.

Proof LetR=R, and letf ∈Rbe continuum-wise expansive forf. Suppose, by con-tradiction, that for any δ>  there is a periodic pointpoff such thatphas aδ-weak eigenvalue. Letε> , and letV(f)⊂U(f) be aC-neighborhood off which is given by Lemma . with respect toU(f). Then there existg∈U(f) and a non-hyperbolic periodic pointqofgsuch that an eigenvalueλofDqgπ(q)with|λ|= , andTqM=Ec(q)⊕Es(q)⊕

Eu₍_q_{), where}_Eσ₍_q_),_σ₌_c_,_s_,_u_{, are}_D

qgπ(q)-invariant subspaces corresponding to

eigenval-ues λofDqgπ(q)for|λ|= ,|λ|< , and|λ|> , respectively. LetW(f)⊂V(f) be theC

ε-ball off. SetC=sup_x_∈_M{Dxg}. For  <ε<ε, we can obtain a linear automorphism

O:TqM→TqMsuch that

(i) O– id<ε

C,

(ii) OkeepsEσ _{invariant, where}_σ₌_c_,_s_,_u_,

(iii) all eigenvalues ofO◦Dqgπ(q), sayμj,j= , , . . . ,c, are roots of unity.

LetFbe the ﬁnite set{q,g(q), . . . ,gπ(q)–₍_q₎_}_{. Deﬁne}

Lj=

D_gj₍_q₎g, j= , , . . . ,π(q) – ,

O◦D_gπ(q)–₍_q₎g, j=π(q) – .

Observe thatLπ(q)––Dgπ(q)–₍_q₎g ≤ O– id · D_gk–₍_q₎g<ε. ThusLj–Dgj₍_q₎g<ε for allj= , , . . . ,π(p) – . By Lemma ., we can ﬁnd a diﬀeomorphismg∈W(f) and

δ>  such that (a) Bδ(g

i₍_q₎₎_∩_B

δ(q) =∅,≤i=j≤π(q) – ,

(b) g=gonF∪(M–

π(q)–

j= Bδ(g

j₍_q₎₎₎_,

(c) g=expgj+₍_q₎◦L_j◦exp–

gj(q)onBδ(gj(q)),≤j≤π(q) – .

Deﬁne

L=O◦Dqgπ(q)=

π(q)–

j= Lj,

whereBδ(p) denotes theδ-neighborhood ofp. Then by (iii) we can ﬁndm>  such thatLm_|

Ec₍_q₎= id|_Ec₍_q₎. Choose a smallδ_satisfying

 < δ<δsuch that

LmkTqM(δ)

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whereTqM(δ) ={v∈TqM|v ≤δ}. Then by (c) we have

g_π(q)m=g_mπ(q)=exp_q◦Gm◦exp–_q onexp_q(TqM(δ)).

We write

TqM(δ) =Ec(q,δ)⊕Es(q,δ)⊕Eu(q,δ), whereEσ₍_q_,_δ_{) =}_Eσ₍_q₎_∩_T

qM(δ),σ=c,s,u. Thenexpq(Ec(q, δ)) is (gk)m-invariant. Since f ∈R, we assume that the eigenvalueλ∈R.

Putexp_q(Ec(q, δ)) is an arcIqcentered atq. Observe that (gk)m= id onexpq(Ec(q, δ)).

By our construction, (gk)mis the identity on the arcIq. It is clear that the small arcIqis

normally hyperbolic for g. By Remark ., for anyg C-close tof, ifg has a normally hyperbolic thenf has a normally hyperbolic, that is, it isC_{-robust. Then we know that}_f has a small arcJqwhich centered atqwithfπ(q)(Jq) =Jq. Note that iff is continuum-wise

expansive thenfk _{is continuum-wise expansive for any}_k_∈_Z_{(see [, Proposition .]).}

Denote byl(A) the length ofA. Takee= l(Jq). SinceJqisfπ(q)-invariant, for alln∈Z,

diamfn(Jq)

<e.

This is a contradiction.

We say thatf satisﬁesstar conditionif there is aC_{-neighborhood}_U₍_f_{) such that for} anyg∈U(f), everyp∈P(g) is hyperbolic. We denote byF(M) the set of diﬀeomorphisms satisfying star condition.

Lemma . There is a residual setR⊂Diff(M)such that for any continuum-wise

ex-pansive map f ∈R,f ∈F(M).

Proof LetR=R, and letf ∈Rbe continuum-wise expansive. Proof by contradiction, we may assume thatf ∈/F(M). Then by Lemma ., there isg C_{-close to}_f _and_p

g∈P(g)

such that for anyδ> ,pghas aδ/-weak eigenvalue. By Lemma .,p∈P(f) has aδ-weak

eigenvalue. This is a contradiction by Lemma ..

Proof of TheoremA Letf ∈Rbe continuum-wise expansive. By Lemma .,f ∈F(M). Sincef ∈F(M), By Aoki [] and Hayashi [], we know thatf satisﬁes both Axiom A and

the no-cycle condition. Thus it is-stable.

5 Proof of Theorem B and Theorem C

LetMand letf ∈Diffμ(M) be as before. To prove our result, we use the Franks lemma, which is proved in [, Proposition .].

Lemma . Let f ∈Diff_μ(M),andU be a C_{-neighborhood of f in}_Diff

μ(M).Then there exist a C_{-neighborhood}_U

⊂U of f andε> such that if g∈U,any ﬁnite f -invariant set E={x, . . . ,xm}, any neighborhood U of E and any volume-preserving linear maps

Lj:TxjM→Tg(xj)M withLj–Dxjg ≤εfor all j= , . . . ,m,there is a conservative

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We denote by Fμ(M) the set of diﬀeomorphisms f ∈ Diffμ(M) which have a C -neighborhoodU(f)⊂Diffμ(M) such that for anyg∈U(f), every periodic point ofg is hyperbolic.

Very recently, Arbieto and Catalan [] proved that every volume-preserving diﬀeomor-phism inFμ(M) is Anosov.

Theorem .[, Theorem .] Every diﬀeomorphism inFμ(M)is Anosov.

To prove Theorem B, it is enough to show that a continuum-wise expansive volume-preserving diﬀeomorphismf ∈Fμ(M).

Remark . Letf ∈Diff_μ(M). From the Moser theorem (see []), we can ﬁnd a smooth

conservative change of coordinatesϕx:U(x)→TxMsuch thatϕx(x) = , whereU(x) is a

small neighborhood ofx∈M.

Lemma . If f ∈intCWEμ(M),then f ∈Fμ(M).

Proof Takef ∈intCWEμ(M), and U(f) aC-neighborhood off. Let ε>  and V(f)⊂

U(f) be corresponding number andC_{-neighborhood given by Lemma .. To derive a} contradiction, suppose that there is a non-hyperbolic periodic point p∈P(g) for some g∈V(f). To simplify the notation in the proof, we may assume thatg(p) =p. Then there is at least one eigenvalue λofDpg such that|λ|= . By making use of Lemma ., we

linearize f at pwith respect to Moser’s theorem, that is, by choosingα>  suﬃciently small we constructgC-nearbygsuch that

g(x) =

ϕ_p–◦Dpg◦ϕp(x) ifx∈Bα(p), g(x) ifx∈/Bα(p).

Theng(p) =g(p) =p. ThusTpM=Ec⊕Eσ, whereEpcassociated toλ=  andEσp associated

to eigenvalues less than one and greater than one. Takeη=α/. Then we deﬁneEc₍_η₎_∩

ϕp(Bα(p)) =Ec(η).

Case.dimEc p= .

Sincepis non-hyperbolic forg, by our construction, we may assume that there isl>  such thatDpgl(v) =vfor anyv∈Ecp(η)∩ϕp(Bα(p)). Takev∈Epc(η) such thatv=η/.

Then we can ﬁnd a small arcIp=ϕp–({tv: ≤t≤η/})⊂Bα(p) such that

(i) gi

(Ip)∩gj(Ip) =∅if≤i=j≤l– ,

(ii) gl

(Ip) =Ip, that is,gl|Ipis the identity map,

(iii) Ipis normally hyperbolic.

For simplicity, we assume thatg_l=g. Takee=η. Then for alln∈Z, diamgn_(Ip)

<e.

This is a contradiction. Case.dimEc

p= .

In the proof of the second case, to avoid notational complexity, we consider the case g(p) =p. By Lemma ., there isα>  andh∈U(f) such thath(p) =g(p) =pandh(x) =

ϕ–

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isl>  such thatDpgl(v) =vfor anyv∈Ecp(α) by Lemma .. We can choosev∈Ecp(α) such

thatv=α/ and we setDp=ϕp–({tv: ≤t≤α/})⊂Bα(p). Then the diskDp satisﬁes

the following conditions: (i) hi(Dph)∩h

j₍_D

ph) =∅if≤i=j≤l– ,

(ii) hl₍_D

ph) =Dph, that is,h

l_|

Dph is the identity map,

(iii) Dpis normally hyperbolic.

As in the proof of thedimEc_p= , we can derive a contradiction. Proof of TheoremB Suppose thatf ∈intCWEμ(M). By Lemma .,f ∈Fμ(M). Thus by

Theorem .,f is Anosov.

Proof of TheoremC The proof of Theorem C is parallel the proof of Theorem A. Indeed, to prove Theorem A we use previous results - Lemmas ., . and .. Then we have a volume-preserving diﬀeomorphismf ∈Fμ(M). Thusf is Anosov.

In diffeomorphisms, there is an open problem: are Anosov diffeomorphisms transi-tive? In [] Franks and [] Newhouse proved it for codimension one Anosov diffeo-morphisms. It was announced in Xia in a talk, Anosov diffeomorphisms are transitive, an invited talk of the Rocky Mountain Conference on Dynamical Systems, May -, , that every Anosov diffeomorphism is transitive. It has not been published yet. Nev-ertheless, in the volume-preserving diffeomorphism, an Anosov diffeomorphism has the non-wandering set equal to the whole manifoldMby the Poincaré theorem. By the shad-owing property of the hyperbolic sets the periodic points are dense inM. And by the Smale spectral decomposition theorem, we have a single piece equal toM, and so, we have transi-tivity. 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 thatC_{-generically, a volume-preserving diffeomorphism is} transitive.

Theorem .[, Theorem .] There is a residual setR⊂Diffμ(M)such that for any

f ∈R,f is transitive and M is a unique homoclinic class.

We say thatf istransitiveif there is a pointx∈Msuch thatω(x) =M, whereω(x) is the omega limit set.

Remark . In [, Theorem .], Newhouse showed thatC-generic volume-preserving

diﬀeomorphisms in surfaces are Anosov or else the elliptical points, nonreal eigenvalues conjugated and of norm one, are dense.

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

Corollary . There is a residual setG⊂Diffμ(M)such that for any f ∈G,the following

are equivalents: (a) f is expansive, (b) f is transitive Anosov. Moreover,ifdimM≥then

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(d) f has the shadowing property, (e) f has the weak speciﬁcation property.

Proof Letf ∈G=R∩Ris continuum-wise expansive. By Theorem A,f is Anosov. Sincef ∈R, by Lemma .,f is transitive. Thus iff is continuum-wise expansive, then f is transitive Anosov. By Bessaet al.[], f is expansive, thenf is Anosov, and so,f is transitive Anosov. IfdimM≥, then by Bessaet al.[] iff has the shadowing property andf has the weak speciﬁcation property, thenf is Anosov and sincef ∈R, alsof is

transitive. Thusf is transitive Anosov.

Competing interests

The author declares to have no competing interests.

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).

Received: 9 May 2014 Accepted: 28 July 2014 Published:01 Oct 2014

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