Abstract

Abstract In this paper, we show that for generic C1, if a flow Xt has the shadowing property on a bi-Lyapunov stable homoclinic class, then it does not contain any singularity and it is hyperbolic.

Highlights

  • Let M be a compact smooth Riemannian manifold, and let Di (M) be the space of di eomorphisms ofM endowed with the C topology

  • In this paper, we show that for generic C, if a ow Xt has the shadowing property on a bi-Lyapunov stable homoclinic class, it does not contain any singularity and it is hyperbolic

  • A set of di eomorphisms is generic if it contains a countable intersection of dense open sets of Di (M)

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Summary

Introduction

Let M be a compact smooth Riemannian manifold, and let Di (M) be the space of di eomorphisms of. If a basic set contains a hyperbolic periodic point, it is a homoclinic class. Ahn et al [3] proved that for generic C , if a di eomorphism f has the shadowing property on a locally maximal homoclinic class, 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, 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, it is hyperbolic. From the results of Guchenheimer [18], the Lorenz attractor satis es the star condition, but it is not Ω-stable because the attractor contains a hyperbolic singular point. If a ow does not contain singularities and satis es the star condition, it is Ω stable (see [19])

Basic notions and main theorem
Manseob Lee
Proof of the Theorem
Full Text
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