Homoclinic classes with shadowing

Abstract

We show that for C 1 generic diffeomorphisms, an isolated homoclinic class is shadow-able if and only if it is a hyperbolic basic set.

Shadowing, expansiveness and specification for C1-conservative systems

“…We point out that a manifestation of the phenomenon of genericity of non shadowing away from hyperbolicity was considered in [1,2,23] for dissipative systems. With respect to the analog results for non expansive ones we refer the results [3,38].…”

Orbital shadowing property on chain transitive sets for generic diffeomorphisms

“…Unfortunately, this question still is open. For the problem, there are partial results [4,9,11]. Ahn et al [4] proved that for C 1 generic diffeomorphism f, if f has the shadowing property on a locally maximal homoclinic class then it is hyperbolic.…”

“…For the problem, there are partial results [4,9,11]. Ahn et al [4] proved that for C 1 generic diffeomorphism f, if f has the shadowing property on a locally maximal homoclinic class then it is hyperbolic. Lee and Wen [11] proved that for C 1 generic diffeomorphism f, if f has the shadowing property on a locally maximal chain transitive set then it is hyperbolic.…”

Lyapunov stable homoclinic classes for smooth vector fields

“…Let us give a short review of related results. Ahn et al [3] proved that for generic C , if a di eomorphism f has the shadowing property on a locally maximal homoclinic class, then it is hyperbolic. Lee [4] proved that for generic C , if a di eomorphism f has the limit shadowing property on a locally maximal homoclinic class, then it is hyperbolic.…”

“…Arbieto et al [5] proved that for generic C , if a bi-Lyapunov stable homoclinic class is homogeneous and has the shadowing property, then it is hyperbolic. See [3,4,[6][7][8][9][10][11][12][13][14][15] for related results.…”