The ergodic shadowing property from the robust and generic view point

Manseob Lee

doi:10.1186/1687-1847-2014-170

Abstract

In this paper, we discuss that if a diffeomorphisms has the -stably ergodic shadowing property in a closed set, then it is a hyperbolic elementary set. Moreover, -generically: if a diffeomorphism has the ergodic shadowing property in a locally maximal closed set, then it is a hyperbolic basic set. MSC:34D30, 37C20.

Highlights

1 Introduction Let M be a closed C∞ manifold, and let Diff(M) be the space of diffeomorphisms of M endowed with the C -topology
We say that f has the shadowing property in if for every > there is δ > such that, for any δ-pseudo orbit {xi}bi=a ⊂ of f (–∞ ≤ a < b ≤ ∞), there is a point y ∈ such that d(f i(y), xi) < for all a ≤ i ≤ b
The shadowing property usually plays an important role in the investigation of stability theory and ergodic theory

Summary

Introduction

Let M be a closed C∞ manifold, and let Diff(M) be the space of diffeomorphisms of M endowed with the C -topology. If f has the C -stably ergodic shadowing property in , it is a hyperbolic elementary set. We consider that C -generically: f has the ergodic shadowing property in a locally maximal closed set.

Results

Conclusion