Vector fields with stably limit shadowing
- PDF / 171,591 Bytes
- 6 Pages / 595.28 x 793.7 pts Page_size
- 57 Downloads / 166 Views
RESEARCH
Open Access
Vector fields with stably limit shadowing Manseob Lee* *
Correspondence: [email protected] Department of Mathematics, Mokwon University, Daejeon, 302-729, Korea
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: 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 diffeomorphism f : M → M has a C -neighborhood U (f ) such that every periodic point of g ∈ 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 the C -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 the C -interior of the limit shadowing property is equal to the set of transitive Anosov diffeomorphisms. Very recently, Ribeiro [] proved that for C -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 -interior of the set of all vector fields having the limit shadowing property, which extends the result of []. Let M be a closed n(≥ )-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 -vector fields on M endowed with the C -topology. Then every X ∈ X(M) generates a C -flow Xt : M × R → M; that is, a C -map such that Xt : M → M is a diffeomorphism satisfying X (x) = x and Xt+s (x) = Xt (Xs (x)) for all s, t ∈ R and x ∈ M. © 2013 Lee; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use,
Data Loading...
