Vector fields with stably limit shadowing

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R E S E A R C H

Open Access

Vector ﬁelds with stably limit shadowing

Manseob Lee

*

*_{Correspondence:}

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

Abstract

LetXbe a vector ﬁeld on a closed smooth manifoldM. In this paper, we show that ifX

belongs to theC1_{-interior of the set of all vector ﬁelds having the limit shadowing} property, then it is transitive Anosov.

MSC: Primary 37C50; secondary 34D10

Keywords: hyperbolic; limit shadowing; shadowing; chain transitive; transitive; Anosov

1 Introduction

Discrete case dynamical system results can be extended to the case of continuous, but not always, in particular results which involve the hyperbolic structure. For instance, it is well known that if a diffeomorphismf :M→Mhas aC-neighborhoodU(f) such that every periodic point ofg∈U(f) is hyperbolic, then the non-wandering set(f) is hyperbolic. However, the result is not true for the case of vector fields (see []). The notion of the limit shadowing property was introduced and studied by Eirola, Nevanlinna, and Pilyugin [, ]. It is different from the shadowing property (see []). Various shadowing properties are used in the investigation of the orbit structure. For instance, Sakai [] and Robinson [] proved that a diffeomorphism is structurally stable if and only if it belongs to the set of all diffeomorphisms having the shadowing property. For vector fields, Lee and Sakai [] proved that if a vector field does not admit singularities, then theC_{-interior of the set of}

all vector fields having the shadowing property coincides with the set of structurally stable vector fields. Recently, in [], Pilyugin showed that a diffeomorphism belongs to the set of all diffeomorphisms having the limit shadowing property if and only if it is-stable, that is, Axiom A and the no-cycle condition. From this, Carvalho [] proved that theC_-interior

of the limit shadowing property is equal to the set of transitive Anosov diffeomorphisms. Very recently, Ribeiro [] proved that forC_{-generic vector fields, if a vector field has the}

limit shadowing property in a closed isolated set, then it is a transitive hyperbolic set. Moreover, if the closed set is the whole space, then it is a transitive Anosov ﬂow. In this result, we study theC_{-interior of the set of all vector ﬁelds having the limit shadowing}

property, which extends the result of [].

LetMbe a closedn(≥)-dimensional smooth Riemmanian manifold, and letdbe the distance onMinduced from a Riemannian metric · on the tangent bundleTM, and denote byX(M) the set ofC_{-vector ﬁelds on}_M_{endowed with the}_C_{-topology. Then}

everyX∈X(M) generates aC_-ﬂow_X

t:M×R→M; that is, aC-map such thatXt :

M→Mis a diﬀeomorphism satisfyingX(x) =xandXt+s(x) =Xt(Xs(x)) for alls,t∈Rand

x∈M.

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For anyδ> , a sequence{(xi,ti) :xi∈M,ti≥, and –∞ ≤a<i<b≤ ∞}is aδ-pseudo

orbit ofX(orδ-chain ofX) ifd(Xti(xi),xi+) <δfor anya≤i≤b– .

For the sequence{ti}i∈Z, we denote

Si=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

t+t+· · ·+ti– ifi> ,

 ifi= ,

–t––t––· · ·–ti ifi< .

We say thatXhas theshadowing propertyif for any> , there isδ>  satisfying the following property: given anyδ-pseudo orbit{(xi,ti) :ti≥,i∈Z}for alli∈Z, there is a

pointy∈Mand an increasing homeomorphismh:R→Rsuch that

dXh(t)(y),Xt–Si(xi)

<

for anyi∈Zand for anySi≤t<Si+.

A sequence{(xi,ti) :ti≥,i∈Z}is alimit pseudo orbitofXif forti≥,i∈Z,

dXti(xi),xi+

→ asi→ ±∞.

The following deﬁnition introduced by Yujinet al. [] is diﬀerent from the notion of Ribeiro []. We say thatXhas thelimit shadowing propertyif for any{(xi,ti) :ti≥,i∈Z},

there isy∈Mand an increasing homeomorphismh:R→Rwithh() =  such that for

Si≤t<Si+,

dXh(t)(y),Xt–Si(xi)

→ asi→ ±∞.

Denote byLSP(M) the set of all diﬀeomorphisms having the limit shadowing property. We say thatXhas theC_-_{stably limit shadowing property}_{if there is a}_C_{-neighborhood}

U(X) ofXsuch that for anyY∈U(X),Yhas the limit shadowing property.

Letbe anXt-invariant compact set. Theis calledhyperbolicforXtif there are

con-stants C> ,λ>  and a splitting TxM=Exs ⊕ X(x) ⊕Exusuch that the tangent ﬂow

DXt:TM→TMleaves invariant the continuous splitting and

DXt|Esx ≤Ce

–λt _and _DX

–t|Exu ≤Ce

–λt

fort>  andx∈. If=M, thenXis Anosov.

A pointx∈Mis callednon-wanderingofXt if for any neighborhoodUofx, there is

t≥ such thatXt(U)∩U=∅. The set of non-wandering points ofXis denoted by(X).

Then we know thatSing(X)∪P(X)⊂(X). HereSing(X) is the set of singularities ofX, andP(X) is the set of periodic orbits ofX. DeﬁneCrit(X) =Sing(X)∪P(X). Here, for any σ∈Crit(X),σis called thecritical elementofX. We say thatX∈X(M) istransitiveif there is a pointx∈Msuch thatωX(x) =M, whereωX(x) is the omega limit set ofx. The following

is the main result of this paper.

Theorem . Let X∈X(M).If X belongs to the C_{-interior of}_LSP₍_M_),_{then X is transitive}

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2 Proof of Theorem 1.1

LetMbe as before, and letX∈X(M). We say that a vector ﬁeldX∈X(M) isrobustly transi-tiveif there exists aC_{-neighborhood}_U₍_X_{) of}_X_{such that for all}_Y_∈_U₍_X_),_Y_{is transitive.}

For robustly transitive vector ﬁelds, Doering [] showed that if a compact manifoldM

is three-dimensional, then every robustly transitive vector field is Anosov. In [], Vivier proved that robustly transitive flows on the wholen-dimensional compact without bound-ary manifoldMmust have a dominated splitting for the linear Poincaré flow and have no singularities.

Theorem .[, Theorem ] If X is a robustly transitive vector ﬁeld,then X admits no

singular points.

We say that a vector ﬁeldX∈X(M) ishomogeneousonU⊂Mif there is aC

-neigh-borhoodU(X) ofXsuch that

(a) for allY∈U(X), there are no sinks nor sources inU, (b) every critical element ofYinY(U) =

t∈RYt(U)is hyperbolic, and

(c) the index of the continuation onU(X)of every critical point does not change. From the deﬁnition, we know the following theorem.

Theorem .[, Theorem .] Let X∈X(M)be a robustly transitive homogeneous ﬂow on an n(≥)-manifold M.Then X is Anosov.

Letbe a closedXt-invariant set. We say thatisattractingif=t≥Xt(U) for some

neighborhoodUofsatisfyingXt(U)⊂Ufor allt> . An attractor ofXis a transitive

attracting set ofXand arepelleris an attractor for –X. We say thatis aproper attractor

orproper repellerif∅ ==M.

Lemma .[, Proposition ] A vector ﬁeld X is chain transitive in an isolated setif

and only ifhas no proper attractor for X.

By Lemma ., if=M, then we omit the isolated condition. Then we have the follow-ing.

Lemma . If X has the limit shadowing property,then X has no proper attractor.

Proof To derive a contradiction, we may assume thatXhas a proper attractor. By deﬁni-tion,=∅and=M. Sinceis an attractor, there isη>  such that⊂Bη(), where

Bη() is theη-neighborhood of. SinceM\=∅, choosex∈M\Bη() andy∈. Then we can construct a limit pseudo orbit{(xi,ti) :ti≥,i∈Z}as follows: for alli∈Z,

(i)Xi(x) =xi,ti= ,i>  and (ii)Xi(y) =xi,ti= ,i≤. By the limit shadowing property,

there arez∈Mand an increasing homeomorphismh:R→Rwithh() =  such that for

i∈Z,d(Xh(t)(z),Xt–Si(xi))→ asi→ ±∞. Then we can ﬁnd –k∈Nsuch that

dXh(t)(z),Xt–Sk(x–k)

=d(Xh(t)(z),Xt–Sk

X–k(y)

< η

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forSk≤t<Sk+. Thus, for allSk≤t<Sk+,Xt(Xh(t)(z))∈Bη() for allt≥. SinceBη() is

We introduce the notion of chain transitive set, which is a weaker notion of transitive set. We say that a setischain transitiveif for anyx,y∈andδ> , there is aδ-pseudo orbit{x=x,x, . . . ,xn=y}withxi∈for anyti≥ and ≤i≤n. The following is a

version of the vector ﬁelds of the result of Gu [].

Lemma .[, Theorem .] Let X∈X(M).If X has the limit shadowing property,then X is transitive if and only if X is chain transitive.

By Lemma .,Xis transitive,Xdoes not contain sinks or sources. Thus, forγ∈P(X), we just consider saddle-type periodic orbits. Letγ be a hyperbolic closed orbit of a vector ﬁeldX∈X(M). We deﬁne thestableandunstable manifoldsofγ by

Denote byindex(γ) the dimension of the stable manifold ofγ.

Lemma . Letη,γ ∈P(X)be hyperbolic orbits.If X has the limit shadowing property,

then Ws(η)∩Wu(γ)=∅.

Proof Letη,γ ∈P(X) be hyperbolic orbits. Suppose thatXhas the limit shadowing prop-erty. Takep∈ηandq∈γ such thatXπ(p)(p) =pandXπ(q)(q) =q, whereπ(a) is the period

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Also, we can ﬁndk∈Zsuch that fork≤t<k+ ,

Xt–Sk(xk) =Xt–k

Xk–(q)

=Xt–(q).

Then, forSk≤t<Sk+, we haved(Xh(t)(y),Xt–Sk(xk)) =d(Xh(t)(y),Xt–(p)). Ift→ ∞, then

dXt(y),Xt(p)

=dXt(y),γ

→.

Thus OX(y)∩Wu(η)∩Ws(γ), and soWu(η)∩Ws(γ)=∅, whereOX(y) is the orbits

ofy.

A vector fieldX∈X(M) isKupka-Smaleif its critical orbits are all hyperbolic and their stable and unstable manifolds intersect transversely. It is well known that the Kupka-Smale vector field is a residual subset inX(M). Denote byKS(M) the set of all vector fields sat-isfying the Kupka-Smale.

Lemma .[, Lemma .] Let X∈X(M)be a Kupka-Smale vector ﬁeld,and letη,γ ∈

P(X)be hyperbolic orbits.IfdimWs(η) +dimWu(γ)≤dimM,then Ws(η)∩Wu(γ) =∅.

Lemma . Let X have the C_{-stably limit shadowing property}_,_{and let}_U₍_X₎_{be as in the}

deﬁnition.Then,for any Y∈U(X)and for any hyperbolicη,γ∈P(Y),index(η) =index(γ).

Proof Suppose that X has the C_{-stably limit shadowing property. Then there is a}

C_{-neighborhood} _U₍_X_{) of} _X _{such that for any} _Y _∈_U₍_X_), _Y _{has the limit shadowing}

property. Let η,γ ∈P(Y) be hyperbolic orbits. To derive a contradiction, we may as-sume thatindex(η)=index(γ). Then we know thatdimWs(η) +dimWu(γ) <dimMor dimWu_{(η) +}_dim_Ws_(γ_{) <}_dim_M_{. Assume that}_dim_Ws_{(η) +}_dim_Wu_(γ_{) <}_dim_M_{. Since}_X

has theC_{-stably limit shadowing property, we can choose}_Z_∈_V₍_Y₎_⊂_U₍_X₎_∩_KS₍_M₎

such thatdimWs_(η

Z) +dimWu(γZ) <dimM, whereηZandγZare the continuations ofη

andγ, respectively. SinceZ∈KS(M), by Lemma .,Ws(ηZ)∩Wu(γZ) =∅. SinceZhas

the limit shadowing property, by Lemma . this is a contradiction.

Lemma . [, Theorem .] LetU(X)be a C_{-neighborhood of X}_._If_γ _∈_P₍_X₎_{is not}

hyperbolic,then there is Y∈U(X)such that Y has two hyperbolic orbitsη,η∈P(Y)with

diﬀerent indices.

Proof of Theorem. LetX have theC_{-stably limit shadowing property. Then there is}

aC-neighborhoodU(X) of Xsuch that for anyY ∈U(X),Y has the limit shadowing property. To get the conclusion, it is enough to show that everyγ∈P(Y) is hyperbolic by Lemma .. By contradiction, we may assume thatγ ∈P(Y) is not hyperbolic. Then, by Lemma ., there isZ∈U(X) such thatZhas two hyperbolic periodic orbitsη,η∈P(Z)

with diﬀerent indices. SinceZhas the limit shadowing property, this is a contradiction by

Lemma ..

Competing interests

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Author’s contributions

The author carried out the proof of the theorem and approved the ﬁnal manuscript.

Acknowledgements

We thank the referees very much for their helpful comments and suggestions. The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007649).

Received: 10 June 2013 Accepted: 8 August 2013 Published: 21 August 2013 References

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