Periodic shadowing and $\Omega$-stability

TL;DR

This paper proves the equivalence of three properties related to shadowing and stability in diffeomorphisms, establishing a connection between periodic shadowing, Lipschitz shadowing, and $ ext{Omega}$-stability.

Contribution

It demonstrates that the $C^1$-interior of diffeomorphisms with periodic shadowing coincides with those having Lipschitz periodic shadowing and being $ ext{Omega}$-stable.

Findings

Equivalence of $C^1$-interior of periodic shadowing and $ ext{Omega}$-stability.

Lipschitz periodic shadowing characterizes the same class of diffeomorphisms.

Provides a unified understanding of shadowing properties and stability in dynamical systems.

Abstract

We show that the following three properties of a diffeomorphism $f$ of a smooth closed manifold are equivalent: (i) $f$ belongs to the $C^1$-interior of the set of diffeomorphisms having periodic shadowing property; (ii) $f$ has Lipschitz periodic shadowing property; (iii) $f$ is $\Omega$-stable.