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Vector fields with stably limit shadowing

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Abstract

Let X be a vector field on a closed smooth manifold M. In this paper, we show that if X belongs to the C 1 -interior of the set of all vector fields having the limit shadowing property, then it is transitive Anosov.

MSC:37C50, 34D10.

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 diffeomorphism f:MM has a C 1 -neighborhood U(f) such that every periodic point of gU(f) is hyperbolic, then the non-wandering set Ω(f) is hyperbolic. However, the result is not true for the case of vector fields (see [1]). The notion of the limit shadowing property was introduced and studied by Eirola, Nevanlinna, and Pilyugin [2, 3]. It is different from the shadowing property (see [3]). Various shadowing properties are used in the investigation of the orbit structure. For instance, Sakai [4] and Robinson [5] 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 [6] proved that if a vector field does not admit singularities, then the C 1 -interior of the set of all vector fields having the shadowing property coincides with the set of structurally stable vector fields. Recently, in [7], Pilyugin showed that a diffeomorphism belongs to the set of all diffeomorphisms having the limit shadowing property if and only if it is Ω-stable, that is, Axiom A and the no-cycle condition. From this, Carvalho [8] proved that the C 1 -interior of the limit shadowing property is equal to the set of transitive Anosov diffeomorphisms. Very recently, Ribeiro [9] proved that for C 1 -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 the C 1 -interior of the set of all vector fields having the limit shadowing property, which extends the result of [8].

Let M be a closed n(3)-dimensional smooth Riemmanian manifold, and let d be the distance on M induced from a Riemannian metric on the tangent bundle TM, and denote by X(M) the set of C 1 -vector fields on M endowed with the C 1 -topology. Then every XX(M) generates a C 1 -flow X t :M×RM; that is, a C 1 -map such that X t :MM is a diffeomorphism satisfying X 0 (x)=x and X t + s (x)= X t ( X s (x)) for all s,tR and xM.

For any δ>0, a sequence {( x i , t i ): x i M, t i 1, and a<i<b} is a δ-pseudo orbit of X (or δ-chain of X) if d( X t i ( x i ), x i + 1 )<δ for any aib1.

For the sequence { t i } i Z , we denote

S i ={ t 0 + t 1 + + t i 1 if  i > 0 , 0 if  i = 0 , t 1 t 2 t i if  i < 0 .

We say that X has the shadowing property if for any ϵ>0, there is δ>0 satisfying the following property: given any δ-pseudo orbit {( x i , t i ): t i 1,iZ} for all iZ, there is a point yM and an increasing homeomorphism h:RR such that

d ( X h ( t ) ( y ) , X t S i ( x i ) ) <ϵ

for any iZ and for any S i t< S i + 1 .

A sequence {( x i , t i ): t i 1,iZ} is a limit pseudo orbit of X if for t i 1, iZ,

d ( X t i ( x i ) , x i + 1 ) 0as i±.

The following definition introduced by Yujin et al. [10] is different from the notion of Ribeiro [9]. We say that X has the limit shadowing property if for any {( x i , t i ): t i 1,iZ}, there is yM and an increasing homeomorphism h:RR with h(0)=0 such that for S i t< S i + 1 ,

d ( X h ( t ) ( y ) , X t S i ( x i ) ) 0as i±.

Denote by LSP(M) the set of all diffeomorphisms having the limit shadowing property. We say that X has the C 1 -stably limit shadowing property if there is a C 1 -neighborhood U(X) of X such that for any YU(X), Y has the limit shadowing property.

Let Λ be an X t -invariant compact set. The Λ is called hyperbolic for X t if there are constants C>0, λ>0 and a splitting T x M= E x s X(x) E x u such that the tangent flow D X t :TMTM leaves invariant the continuous splitting and

D X t | E x s C e λ t andD X t | E x u C e λ t

for t>0 and xΛ. If Λ=M, then X is Anosov.

A point xM is called non-wandering of X t if for any neighborhood U of x, there is t1 such that X t (U)U. The set of non-wandering points of X is denoted by Ω(X). Then we know that Sing(X)P(X)Ω(X). Here Sing(X) is the set of singularities of X, and P(X) is the set of periodic orbits of X. Define Crit(X)=Sing(X)P(X). Here, for any σCrit(X), σ is called the critical element of X. We say that XX(M) is transitive if there is a point xM such that ω X (x)=M, where ω X (x) is the omega limit set of x. The following is the main result of this paper.

Theorem 1.1 Let XX(M). If X belongs to the C 1 -interior of LSP(M), then X is transitive Anosov.

2 Proof of Theorem 1.1

Let M be as before, and let XX(M). We say that a vector field XX(M) is robustly transitive if there exists a C 1 -neighborhood U(X) of X such that for all YU(X), Y is transitive. For robustly transitive vector fields, Doering [11] showed that if a compact manifold M is three-dimensional, then every robustly transitive vector field is Anosov. In [12], Vivier proved that robustly transitive flows on the whole n-dimensional compact without boundary manifold M must have a dominated splitting for the linear Poincaré flow and have no singularities.

Theorem 2.1 [[12], Theorem 1]

If X is a robustly transitive vector field, then X admits no singular points.

We say that a vector field XX(M) is homogeneous on UM if there is a C 1 -neighborhood U(X) of X such that

  1. (a)

    for all YU(X), there are no sinks nor sources in U,

  2. (b)

    every critical element of Y in Λ Y (U)= t R Y t (U) is hyperbolic, and

  3. (c)

    the index of the continuation on U(X) of every critical point does not change.

From the definition, we know the following theorem.

Theorem 2.2 [[13], Theorem 4.1]

Let XX(M) be a robustly transitive homogeneous flow on an n(3)-manifold M. Then X is Anosov.

Let Λ be a closed X t -invariant set. We say that Λ is attracting if Λ= t 0 X t (U) for some neighborhood U of Λ satisfying X t ( U ¯ )U for all t>0. An attractor of X is a transitive attracting set of X and a repeller is an attractor for −X. We say that Λ is a proper attractor or proper repeller if ΛM.

Lemma 2.3 [[9], Proposition 3]

A vector field X is chain transitive in an isolated set Λ if and only if Λ has no proper attractor for X.

By Lemma 2.3, if Λ=M, then we omit the isolated condition. Then we have the following.

Lemma 2.4 If X has the limit shadowing property, then X has no proper attractor.

Proof To derive a contradiction, we may assume that X has a proper attractor Λ. By definition, Λ and ΛM. Since Λ is an attractor, there is η>0 such that Λ B η (Λ), where B η (Λ) is the η-neighborhood of Λ. Since MΛ, choose xM B η (Λ) and yΛ. Then we can construct a limit pseudo orbit {( x i , t i ): t i 1,iZ} as follows: for all iZ, (i) X i (x)= x i , t i =1, i>0 and (ii) X i (y)= x i , t i =1, i0. By the limit shadowing property, there are zM and an increasing homeomorphism h:RR with h(0)=0 such that for iZ, d( X h ( t ) (z), X t S i ( x i ))0 as i±. Then we can find kN such that

d ( X h ( t ) ( z ) , X t S k ( x k ) ) =d( X h ( t ) (z), X t S k ( X k ( y ) ) < η 16

for S k t< S k + 1 . Thus, for all S k t< S k + 1 , X t ( X h ( t ) (z)) B η (Λ) for all t0. Since B η (Λ) is an attracting neighborhood of Λ, for S k t< S k + 1 ,

t 0 X t ( X h ( t ) ( z ) ) t 0 ( B η ( Λ ) ) =Λ.

Since Λ is an X t -invariant set, we know that zΛ. Thus we have that

d ( X t ( z ) , X t ( x ) ) >ηfor all t>0.

This is a contradiction. □

We introduce the notion of chain transitive set, which is a weaker notion of transitive set. We say that a set Λ is chain transitive if for any x,yΛ and δ>0, there is a δ-pseudo orbit { x 0 =x, x 1 ,, x n =y} with x i Λ for any t i 1 and 0in. The following is a version of the vector fields of the result of Gu [14].

Lemma 2.5 [[14], Theorem 4.1]

Let XX(M). If X has the limit shadowing property, then X is transitive if and only if X is chain transitive.

By Lemma 2.5, X is transitive, X does not contain sinks or sources. Thus, for γP(X), we just consider saddle-type periodic orbits. Let γ be a hyperbolic closed orbit of a vector field XX(M). We define the stable and unstable manifolds of γ by

W s (γ)= { y M : ω X ( y ) = γ } and W u (γ)= { y M : α X ( y ) = γ } .

Denote by index(γ) the dimension of the stable manifold of γ.

Lemma 2.6 Let η,γP(X) be hyperbolic orbits. If X has the limit shadowing property, then W s (η) W u (γ).

Proof Let η,γP(X) be hyperbolic orbits. Suppose that X has the limit shadowing property. Take pη and qγ such that X π ( p ) (p)=p and X π ( q ) (q)=q, where π(a) is the period of a. For iZ, put X i (p)= x i , t i =1, i0, and X i (q)= x i + 1 , t i =1, i0. Then

{ ( x i , t i ) : t i = 1 , i Z } = { , x 1 , x 0 ( = p ) , x 1 ( = q ) , x 2 , }

is a limit pseudo orbit of X. By the limit shadowing property, there are yM and an increasing homeomorphism h:RR with h(0)=0 such that for S i t< S i + 1 , d( X h ( t ) (y), X t S i ( x i ))0 as i±. Then we can find jZ such that for jt<j+1,

X t S j ( x j )= X t ( j ) ( X j ( p ) ) = X t (p).

Then, for S j t< S j + 1 , we have d( X h ( t ) (y), X t S j ( x j ))=d( X h ( t ) (y), X t (p)). If t, then

d ( X t ( y ) , X t ( p ) ) =d ( X t ( y ) , η ) 0.

Also, we can find kZ such that for kt<k+1,

X t S k ( x k )= X t k ( X k 1 ( q ) ) = X t 1 (q).

Then, for S k t< S k + 1 , we have d( X h ( t ) (y), X t S k ( x k ))=d( X h ( t ) (y), X t 1 (p)). If t, then

d ( X t ( y ) , X t ( p ) ) =d ( X t ( y ) , γ ) 0.

Thus O X (y) W u (η) W s (γ), and so W u (η) W s (γ), where O X (y) is the orbits of y. □

A vector field XX(M) is Kupka-Smale if 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 in X(M). Denote by KS(M) the set of all vector fields satisfying the Kupka-Smale.

Lemma 2.7 [[15], Lemma 3.4]

Let XX(M) be a Kupka-Smale vector field, and let η,γP(X) be hyperbolic orbits. If dim W s (η)+dim W u (γ)dimM, then W s (η) W u (γ)=.

Lemma 2.8 Let X have the C 1 -stably limit shadowing property, and let U(X) be as in the definition. Then, for any YU(X) and for any hyperbolic η,γP(Y), index(η)=index(γ).

Proof Suppose that X has the C 1 -stably limit shadowing property. Then there is a C 1 -neighborhood U(X) of X such that for any YU(X), Y has the limit shadowing property. Let η,γP(Y) be hyperbolic orbits. To derive a contradiction, we may assume that index(η)index(γ). Then we know that dim W s (η)+dim W u (γ)<dimM or dim W u (η)+dim W s (γ)<dimM. Assume that dim W s (η)+dim W u (γ)<dimM. Since X has the C 1 -stably limit shadowing property, we can choose ZV(Y)U(X)KS(M) such that dim W s ( η Z )+dim W u ( γ Z )<dimM, where η Z and γ Z are the continuations of η and γ, respectively. Since ZKS(M), by Lemma 2.7, W s ( η Z ) W u ( γ Z )=. Since Z has the limit shadowing property, by Lemma 2.6 this is a contradiction. □

Lemma 2.9 [[15], Theorem 4.3]

Let U(X) be a C 1 -neighborhood of X. If γP(X) is not hyperbolic, then there is YU(X) such that Y has two hyperbolic orbits η 1 , η 2 P(Y) with different indices.

Proof of Theorem 1.1 Let X have the C 1 -stably limit shadowing property. Then there is a C 1 -neighborhood U(X) of X such that for any YU(X), Y has the limit shadowing property. To get the conclusion, it is enough to show that every γP(Y) is hyperbolic by Lemma 2.8. By contradiction, we may assume that γP(Y) is not hyperbolic. Then, by Lemma 2.9, there is ZU(X) such that Z has two hyperbolic periodic orbits η 1 , η 2 P(Z) with different indices. Since Z has the limit shadowing property, this is a contradiction by Lemma 2.8. □

Author’s contributions

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

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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).

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Correspondence to Manseob Lee.

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Keywords

  • hyperbolic
  • limit shadowing
  • shadowing
  • chain transitive
  • transitive
  • Anosov