Asymptotic orbital shadowing property for diffeomorphisms

Manseob Lee

doi:10.1515/math-2019-0002

Abstract

Abstract Let M be a closed smooth Riemannian manifold and let f : M → M be a diffeomorphism. We show that if f has the C1 robustly asymptotic orbital shadowing property then it is an Anosov diffeomorphism. Moreover, for a C1 generic diffeomorphism f, if f has the asymptotic orbital shadowing property then it is a transitive Anosov diffeomorphism. In particular, we apply our results to volume-preserving diffeomorphisms.

Highlights

Let M be a closed smooth Riemannian manifold with dimM ≥, and let Di (M) be the space of di eomorphisms of M endowed with the C topology
We show that if f has the C robustly asymptotic orbital shadowing property it is an Anosov di eomorphism
Robinson [34] and Sakai [37] proved that f belongs to the C interior of the set of all di eomorphisms having the shadowing property if and only if it is structurally stable

Summary

Introduction

Let M be a closed smooth Riemannian manifold with dimM ≥ , and let Di (M) be the space of di eomorphisms of M endowed with the C topology. Lee and Wen [20] proved that for a C generic f , if f has the shadowing property on a locally maximal chain transitive set it is hyperbolic. The remarkable result of this problem, Lee [29] proved that for C generic f of a two dimensional smooth manifold M, if f has the average shadowing property or the asymptotic average shadowing property it is Anosov. Since the asymptotic orbital shadowing property is a general notion of the limit shadowing property, such as those results before, we will prove the following. To prove Theorem B, we will show that for a C generic f , if f has the asymptotic orbital shadowing property it is a transitive Anosov di eomorphism

Proof of Theorem A

If λ

Proof of Theorem B

Volume preserving di eomorphisms

Esp is the eigenvalues smaller than