A Type of the Shadowing Properties for Generic View Points

Manseob Lee

doi:10.3390/axioms7010018

Abstract

We show that if a C 1 generic diffeomorphism of a closed smooth two-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it is Anosov. Moreover, if a C 1 generic vector field of a closed smooth three-dimensional manifold has the average shadowing property or the asymptotic average shadowing property, then it satisfies singular Axiom A without cycles.

Highlights

The shadowing property is an important notion to study the stability systems in dynamical systems
Robinson [1] and Sakai [2] proved that a diffeomorphism f of a closed smooth manifold M
Abdenur and Díaz [4] suggested the following problem; the shadowing property and hyperbolicity are equivalent in C1 generic sense

Summary

Introduction

The shadowing property is an important notion to study the stability systems in dynamical systems. Park [10] proved that C1 generically, if a diffeomorphism f has the average shadowing property or the asymptotic average shadowing property with every periodic points has intersection, it is Anosov. Arbieto and Ribeiro [7] proved that if a vector field X of a closed smooth three-dimensional manifold has the C1 robustly average shadowing property or the asymptotic average shadowing property, it is Anosov. X has the average shadowing property or the asymptotic average shadowing property with every periodic orbits intersect each other, is Anosov From these results, we show that if a diffeomorphism f or a vector field X has the average shadowing property or the asymptotic average shadowing property, it is hyperbolic, in C1 generic sense

Diffeomorphisms

Findings

Vector Fields