A Type of the Shadowing Properties for Generic View Points

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.

Keywords

• average shadowing
• asymptotic average shadowing
• generic
• chain transitive
• Axiom A
• singular Axiom A
• Anosov