Volume preserving diffeomorphisms with inverse shadowing

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R E S E A R C H

Open Access

Volume-preserving diffeomorphisms with

inverse shadowing

Manseob Lee

*

*_{Correspondence:}

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

Abstract

Letfbe a volume-preserving diﬀeomorphism of a closedC∞n-dimensional Riemannian manifoldM. In this paper, we prove the equivalence between the following conditions:

(a) fbelongs to theC1_{-interior of the set of volume-preserving diﬀeomorphisms}

which satisfy the inverse shadowing property with respect to the continuous methods,

(b) fbelongs to theC1-interior of the set of volume-preserving diﬀeomorphisms which satisfy the weak inverse shadowing property with respect to the continuous methods,

(c) fbelongs to theC1_{-interior of the set of volume-preserving diﬀeomorphisms}

which satisfy the orbital inverse shadowing property with respect to the continuous methods,

(d) fis Anosov.

MSC: Primary 37C50; secondary 34D10

Keywords: shadowing; inverse shadowing; weak inverse shadowing; orbital inverse shadowing; Anosov; volume-preserving

1 Introduction

LetMbe a closedC∞n-dimensional Riemannian manifold, and letDiff(M) be the space of diﬀeomorphisms ofMendowed with theC_{-topology. Denote by}_d_{the distance on}_M

induced from a Riemannian metric · on the tangent bundleTM. Letf :M→Mbe a diﬀeomorphism, and let⊂Mbe a closedf-invariant set.

Forδ> , a sequence of points{xi}bi=a(–∞ ≤a<b≤ ∞) inMis called aδ-pseudo orbit

off ifd(f(xi),xi+) <δfor alla≤i≤b– . We say thatf has theshadowing propertyon

if for any> , there isδ>  such that for anyδ-pseudo orbit{xi}i∈Z⊂off, there

isy∈Msuch that d(fi_(y),_x

i) < fori∈Z. Note that in this deﬁnition, the shadowing

pointy∈Mis not necessarily contained in. We say thatf belongs to theC_-interior

shadowing propertyif there is aC-neighborhoodU(f) off such that for anyg∈U(f),g has the shadowing property.

The shadowing property usually plays an important role in the investigation of stability theory and ergodic theory [].

Now, we introduce the notion of the inverse shadowing property which is a ‘dual’ notion of the shadowing property. The inverse shadowing property was introduced by Corless and Pilyugin in [], and the qualitative theory of dynamical systems with the property was

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developed by various authors (see [–]). In this paper, we introduce the various inverse shadowing properties.

LetMZbe the space of all two-sided sequencesξ={xn:n∈Z}with elementsxn∈M,

endowed with the product topology. For anyδ> , letf(δ) denote the set of allδ-pseudo

orbits off. A mappingϕ:M→f(δ)⊂MZis said to be aδ-methodforf ifϕ(x)=x,

and eachϕ(x) is aδ-pseudo orbit off throughx, whereϕ(x)denotes the th component

of ϕ(x). For convenience, writeϕ(x) for{ϕ(x)k}k∈Z. The set of allδ-methods forf will

be denoted byT(f,δ). We say thatϕis acontinuousδ-methodforf ifϕ is continuous.

The set of all continuousδ-methods forf will be denoted byTc(f,δ). Ifg:M→Mis

a homeomorphism with d(f,g) <δ, thenginduces a continuousδ-methodϕg forf by

deﬁning

ϕg(x) =

gn(x) :n∈Z.

LetTh(f,δ) denote the set of all continuousδ-methodsϕg forf which are induced by a

homeomorphismg:M→Mwithd(f,g) <δ, wheredis the usualC-metric. LetTd(f,δ)

denote the set of all continuousδ-methodsϕgforf which are induced byg∈Diff(M) with

d(f,g) <δ. Then, clearly, we know that

Td(f)⊂Th(f)⊂Tc(f)⊂T(f),

Tα(f) =

δ>Tα(f,δ),α= ,c,h,d. We say thatf has theinverse shadowing propertywith

respect to the classTα(f),α= ,c,h,d, if for any> , there existsδ>  such that for any

δ-methodϕ∈Tα(f,δ) and any pointx∈M, there is a pointy∈M, such that

dfk(x),ϕ(y)k

<, k∈Z.

We say that f has theweak inverse shadowing propertywith respect to the classTα(f),

α= ,c,h,d, if for any> , there existsδ>  such that for anyδ-methodϕ∈Tα(f,δ) and

any pointx∈M, there is a pointy∈Msuch that

ϕ(y)⊂B

Of(x)

,

whereB(A) ={x∈M:d(x,A)≤}. Note that iff ∈Diff(M) has the inverse shadowing

property with respect to the class Tα(f) (α= ,c,h,d), then by the deﬁnition, it clearly

has the weak inverse shadowing property with respect to the class Tα(f) (α= ,c,h,d).

We say thatf has theorbital inverse shadowing propertywith respect to the classTα(f),

α= ,c,h,d, if for any> , there is aδ>  such that for anyδ-methodϕ∈Tα(f,δ) and

any pointx∈M, there is a pointy∈Msuch that

Of(x)⊂B

ϕ(y) and ϕ(y)⊂B

Of(x)

.

Note that iff has the inverse shadowing property with respect to the classTα(f) (α=

,c,h,d), then it has the orbital inverse shadowing property with respect to the classTα(f)

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property with respect to the classTd(f). We say thatf belongs to theC-interior inverse

(weak inverse or orbital inverse) shadowing propertywith respect to the classTα(f),α=

,c,h,dif there is aC_{-neighborhood}_U_(f_{) of}_f_{such that for any}_g_∈_U_(f_),_g_{has the inverse}

(weak inverse or orbital inverse) shadowing property with respect to the classTα(f),α=

,c,h,d.

Lee [], showed that a diﬀeomorphism belongs to the C_{-interior inverse shadowing}

property with respect to the classTd(f) if and only if it is structurally stable. And Pilyugin

[] proved that a diﬀeomorphism belongs to theC-interior inverse shadowing property with respect to the classTc(f) if and only if it is structurally stable. Thus, we can restate

the above facts as follows.

Theorem . Let f ∈Diff(M).A diﬀeomorphism f belongs to the C-interior inverse shad-owing property with respect to the classTd(f) (resp.Tc(f))if and only if it is structurally

stable.

In [] Choi, Lee and Zhang showed that a diﬀeomorphism belongs to theC-interior weak inverse shadowing property with respect to the classTd(f) if and only if it satisﬁes

both Axiom A and the no-cycle condition. Moreover, they proved that a diﬀeomorphism belongs to theC_{-interior orbital inverse shadowing property with respect to the class}

Td(f) if and only if it satisﬁes both Axiom A and the strong transversal condition. From

the above facts, we get the following.

Theorem . Let f ∈Diff(M).If a diﬀeomorphism f belongs to the C_{-interior weak inverse}

shadowing property with respect to the classTd(f),then f satisﬁes both AxiomAand the

no-cycle condition.Moreover,if f belongs to the C_{-interior orbital inverse shadowing property}

with respect to the classTd(f),then it is structurally stable.

By the theorem, even though a diﬀeomorphism is contained in theC_{-interior of the set}

of diﬀeomorphisms possessing the weak inverse shadowing property with respect to the classTd(f), it does not necessarily satisfy the strong transversality condition.

A periodic pointpoff ishyperbolicifDfπ(p)_{has eigenvalues with absolute values}

dif-ferent from the one, whereπ(p) is the period ofp. Denote byF(M) the set off ∈Diff(M) such that there is aC-neighborhoodU(f) off such that for anyg∈U(f), everyp∈P(g) is hyperbolic, whereP(g) is the set of periodic points ofg. It is proved by Hayashi [] that f ∈F(M) if and only iff satisﬁes both Axiom A and the no-cycle condition.

Letbe a closedf ∈Diff(M)-invariant set. We say thatishyperbolicif the tangent bundle TMhas a Df-invariant splittingEs⊕Eu and there exist constantsC>  and

 <λ<  such that

Dxfn|Es

x≤Cλ

n _and _D

xf–n|Eu

x≤Cλ

n

for allx∈andn≥. If=M, then we say thatf is anAnosov diﬀeomorphism.

2 Statement of the results

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stable diffeomorphism is an Axiom A diffeomorphism. And in [], Palis extended this result to-stable diffeomorphisms.

LetMbe a compactC∞n-dimensional Riemannian manifold endowed with a volume formω. Letμdenote the measure associated withωthat we call the Lebesgue measure, and letddenote the metric induced by the Riemannian structure. Denote byDiffμ(M)

the set of diﬀeomorphisms which preserves the Lebesgue measureμendowed with the C-topology. In the volume-preserving case, the Axiom A condition is equivalent to the diﬀeomorphism being Anosov, since(f) =Mby the Poincaré recurrence theorem.

We deﬁne the setFμ(M) as the set of diﬀeomorphismsf ∈Diffμ(M) which has aC

-neighborhoodU(f)⊂Diffμ(M) such that for anyg∈U(f), every periodic point ofg is

hyperbolic. Note thatFμ(M)⊂F(M) (see [, Corollary .]).

Very recently, Arbieto and Catalan [] proved that if a volume-preserving diﬀeomor-phism is contained inFμ(M), then it is Anosov. Indeed, at ﬁrst they used the Mañé results

([, Proposition II.]). Then they showed thatP(f) is hyperbolic, whereP(f) is the set of periodic points off. And they proved that the nonwandering set(f) =P(f) by Pugh’s closing lemma. Finally, by the Poincaré recurrence theorem,(f) =M. From the above facts, we can restate the theorem as follows.

Theorem . Any diﬀeomorphism inFμ(M)is Anosov.

In [], Lee showed that if a volume-preserving diﬀeomorphism belongs to the C

-interior expansive orC-interior shadowing property, then it is Anosov. And [] proved that if a volume-preserving diﬀeomorphism belongs to theC_{-interior weak shadowing}

property orC_{-interior weak limit shadowing property, then it is Anosov. From these}

re-sults, we study the cases when a volume-preserving diﬀeomorphismf is inC-interior various inverse shadowing property with respect to the classTd(f), then it is Anosov. Let intISμ(M) denote the set of volume-preserving diﬀeomorphisms inDiffμ(M) satisfying

the inverse shadowing property with respect to the class Td, and letintWISμ(M)

(re-spect.,intOISμ(M)) denote the set of volume-preserving diﬀeomorphisms inDiffμ(M)

satisfying the weak inverse shadowing property with respect to the classTd(respect., the

orbital inverse shadowing property with respect to the classTd). From now, we only

con-sider the classTdwhen we mention the inverse shadowing property; that is, the ‘inverse

shadowing property’ implies the ‘inverse shadowing property with respect to the classTd’.

Now we are in a position to state the theorem of our paper.

Theorem . Let f ∈Diffμ(M).We have

intISμ(M) =intOISμ(M) =intWISμ(M) =ANμ(M),

whereANμ(M)is the set of Anosov volume-preserving diﬀeomorphisms inDiffμ(M).

3 Proof of Theorem 2.2

LetMbe a compactC∞ n-dimensional Riemannian manifold endowed with a volume formω, and letf∈Diffμ(M). To prove the results, we will give the following well-known

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

μ(M).Then there

exist a C_{-neighborhood}_U

⊂Uof f and> such that if g∈U,for 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 ≤ for all j= , . . . ,m,there is a conservative dif-feomorphism g∈U coinciding with f on E and out of U,and Dxjg=Ljfor all j= , . . . ,m.

Remark . From the Moser theorem (see []), there is a smooth conservative change of coordinatesϕx:U(x)→TxMsuch thatϕx(x) =

→

, whereU(x) is a small neighborhood ofx∈M.

Proposition . If f ∈intISμ(M),then every periodic point of f is hyperbolic.

Proof Takef ∈intISμ(M) andU(f) is aC-neighborhood off ∈intISμ(M). Let> 

andV(f)⊂U(f) be a corresponding number and aC-neighborhood given by Lemma ..

Suppose that there exists a nonhyperbolic periodic pointp∈P(g) for someg∈V(f). To simplify the notation of the proof, we may assume thatg(p) =p. Then there is at least one eigenvalueλofDpg such that|λ|= . Denote byEcpthe eigenspace corresponding toλ.

Then we see that ifλ∈R, thendimEc

p= , and ifλ∈C, thendimEcp= .

First, we considerdimE_pc= . For simplicity, we may assume thatλ=  (the other case is similar). By making use of Lemma ., we linearizeg atpwith respect to the Moser theorem; that is, by choosingα>  suﬃciently small, we constructg C-nearbyg such

that

g(x) =

ϕ_p–◦Dpg◦ϕp(x) ifx∈Bα(p),

g(x) ifx∈/Bα(p).

Theng(p) =g(p) =p. Since the eigenvalueλofDpgis one,Dpg(v) =vfor anyv∈Ecp(α).

Takev∈Ecp(α) such thatv=α/. Then we set

Iv={t·v: –≤t≤} ⊂ϕp

Bα(p)

.

Take=α/. Let  <δ<be the number of the inverse shadowing property ofg. Then

by our construction ofg,ϕ–p (Iv)⊂Bα(p). PutJp=ϕ

–

p (Iv). Then we see thatg(Jp) =Jp

and it is the identity map. For the aboveδ> , we can deﬁneTd(g)-method as follows.

For anyx∈M, we setϕp(x) = (v, . . . ,vn) andDpg(v, . . . ,vn) = (v,A(v, . . . ,vn)). HereAis

corresponding to|λ| = . Then we deﬁne

h(x) =ϕ–_p (v) +δ/,ϕp–

A(v, . . . ,vn)

=x+δ/,x

,

where ϕ–

p (A(v, . . . ,vn)) =x= (x, . . . ,xn). Clearly, d(g,h) <δ, andh∈Td(g). Letpbe

identiﬁed with = (, . . . , ). Then choosex= (x, , . . . , )∈Jp such that d(,x) = .

Then

dhi(),g_i(x)=d(,x) = .

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We take a point y= (y, , . . . , )∈ Jp such that d(y,x) < and hi(y) = (ϕ–p (w) +

iδ/,ϕ_p–(Ai_(w

, . . . ,wn))) = (y+iδ/,yi), whereϕp(y) = (w, . . . ,wn),y=ϕ–p (A(w, . . . ,wn))

andAcorresponding to|λ| = . Thend(gi_(x),_hi_{(y)) =}_d(x,_y

+iδ/) and for somek∈Z,

d(x,y+kδ/) >α/ =.

This is a contradiction.

Therefore, we can choose a pointy∈M\Jpsuch thatd(x,y) <. Sinced(gi(x),hi(y)) =

d((x, , . . . , ), (y+iδ/,yi)), we can ﬁndk∈Zsuch that

d(x, , . . . , ),

y+kδ/,yk

>,

whereyk=ϕ–

p (Ak(w, . . . ,wn)). This is a contradiction sincef ∈intISμ(M).

Finally, ifλ∈C, thendimEc

p= . To avoid the notational complexity, we may assume

thatg(p) =p. As in the ﬁrst case, by Lemma ., there areα>  andg∈V(f) such that

g(p) =g(p) =pand

g(x) =

ϕ–

p ◦Dpg◦ϕp(x) ifx∈Bα(p),

g(x) ifx∈/Bα(p).

With aC_{-small modiﬁcation of the map}_D

pg, we may suppose that there isl>  (the

minimum number) such thatDpgl(v) =vfor anyv∈ϕp(Bα(p))⊂TpM. Takev∈ϕp(Bα(p))

such thatv=α/, and set

Lp=ϕ–p

{t·v: ≤t≤ +α/}

.

ThenLpis an arc such that

• gi

(Lp)∩gj(Lp) =∅for≤i=j≤l– ,

• gl(Lp) =Lp, and

• gl

|Lpis the identity map.

Note thatghas the inverse shadowing property if and only ifgkhas the inverse shadowing

property for allk∈Z(see []). As in the ﬁrst case, we can show thatgdoes not have the

inverse shadowing property, which contradicts the fact thatg∈U(f). Thus, every periodic

point off ∈intISμ(M) is hyperbolic.

Proposition . If f ∈intWISμ(M),then every periodic point of f is hyperbolic.

Proof Takef ∈intWISμ(M), andU(f) is aC-neighborhood off ∈intWISμ(M). Let

>  and V(f)⊂U(f) be a corresponding number and a C-neighborhood given by

Lemma .. Suppose that there exists a nonhyperbolic periodic pointp∈P(g) for some g∈V(f). To simplify the notation of the proof, we may assume that g(p) =p. Then, as in the proof of Proposition ., we can takeα>  suﬃciently small and a smooth map ϕp:Bα(p)→TpM. Then we can make an arcJp⊂Bα(p) and for someg∈V(f). Take

= (lengthJp)/. Let  <δ< be the number of the weak inverse shadowing property

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Letpbe identiﬁed with= (, . . . , ). Then choose a pointx= (x, , . . . , )∈Jpsuch that

d(,x) = . Sinceghas the weak inverse shadowing property,

Oh()⊂B

Og(x)

.

However, for anyy∈Jp,

gi

(y) =y and hi(y) =

y+iδ/,yi

,

whereyi=ϕ_p–(Ai_(w

, . . . ,wn)) is as in the proof of Proposition .. Thus, it is easily seen

that

Oh()⊂B

Og(x)

.

Ify= (y, , . . . , )∈Jp\ {p}, then

hk(y) =y+kδ/,yk

= (y+kδ/, ),

whereyk=ϕ_p–(Ak(w, . . . ,wn)). Thus, we know that

Oh(y)⊂B

Og(x)

.

Finally, we can choose a pointy∈M\Jpsuch thatd(x,y) <. Then we know that

,

whereyk=ϕ–

p (Ak(w, . . . ,wn)). Therefore,

hl(y) /∈B(Jp),

for somel∈Z. Then

Oh(y)⊂B

Og(x)

=B(x).

Thus,gdoes not have the weak inverse shadowing property. This is a contradiction.

Ifλ∈C, then as in the proof of Proposition ., forg∈V(f), we can takel>  such

thatDpgl(v) =vfor anyv∈ϕp(Bα(p))⊂TpM. As in the previous argument, we get a

con-tradiction. Thus, every periodic point off ∈intWISμ(M) is hyperbolic. Consequently, if

f ∈intWISμ(M), thenf∈Fμ(M).

Proposition . If f ∈intOISμ(M),then every periodic point of f is hyperbolic.

Proof The proof is almost the same as that of Proposition .. Indeed, letf ∈intOISμ(M)

andU(f) be aC-neighborhood off ∈intOISμ(M). Let>  andV(f)⊂U(f) be a

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a nonhyperbolic periodic point p∈P(g) for someg∈V(f). To simplify the notation of the proof, we may assume thatg(p) =p. Then, as in the proof of Proposition ., we can takeα>  suﬃciently small and a smooth mapϕp:Bα(p)→TpM. Then we can make an

arcJp⊂Bα(p) and for someg∈V(f). Take= (lengthJp)/. Let  <δ<be the

num-ber of the orbital inverse shadowing property for someg. Then we can construct a map

h∈Diff(M) as in the proof of Proposition .. Letpbe identiﬁed with= (, . . . , ). Then choose a pointx= (x, , . . . , )∈Jpsuch thatd(,x) = . Sinceghas the orbital inverse

shadowing property,

Oh()⊂B

Og(x)

and Og(x)⊂B

Oh()

.

However, for anyy∈Jp,

g_i(y) =y and hi(y) =y+iδ/,yi

,

whereyi=ϕ_p–(Ai_(w

, . . . ,wn)) = (y, . . . ,yn) is as in the proof of Proposition .. Thus, it is

easily seen that

Oh()⊂B

Og(x)

.

This is a contradiction sincef ∈intOISμ(M).

Ify= (y, , . . . , )∈Jp\ {p}, then

= (y+kδ/, ),

whereyk=ϕ–

p (Ak(w, . . . ,wn)). Thus, we know that

Oh(y)⊂B

Og(x)

.

Finally, we can choose a pointy∈M\Jpsuch thatd(x,y) <. Then we know that

,

whereyk=ϕ–

p (Ak(w, . . . ,wn)). Therefore,

hl(y) /∈B(Jp),

for somel∈Z. Then

Oh(y)⊂B

Og(x)

=B(x).

Thus,gdoes not have the orbital inverse shadowing property. This is a contradiction.

Ifλ∈C, then as in the proof of Proposition ., forg∈V(f), we can takel>  such that

Dpgl(v) =vfor anyv∈ϕp(Bα(p))⊂TpM. As in the previous argument, in order to reach

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Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author wishes to express their appreciation to the referee for their careful reading of the manuscript and valuable suggestions. This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology, Korea (No. 2011-0007649).

Received: 21 May 2012 Accepted: 13 November 2012 Published: 28 November 2012

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