Inverse periodic shadowing properties

TL;DR

This paper investigates inverse periodic shadowing properties in discrete dynamical systems, establishing their relation to stability, hyperbolicity, and Axiom A conditions, thus linking shadowing behaviors to fundamental dynamical classifications.

Contribution

It characterizes the $C^1$-interior of diffeomorphisms with inverse periodic shadowing as exactly the $ ext{Omega}$-stable diffeomorphisms and links Lipschitz inverse periodic shadowing to hyperbolicity and Axiom A systems.

Findings

The $C^1$-interior of diffeomorphisms with inverse periodic shadowing equals $ ext{Omega}$-stable diffeomorphisms.

Lipschitz inverse periodic shadowing is equivalent to hyperbolicity of periodic points.

Diffeomorphisms with Lipschitz inverse periodic shadowing and dense periodic points in the nonwandering set are exactly Axiom A diffeomorphisms.

Abstract

We consider inverse periodic shadowing properties of discrete dynamical systems generated by diffeomorphisms of closed smooth manifolds. We show that the $C^1$-interior of the set of all diffeomorphisms having so-called inverse periodic shadowing property coincides with the set of $\Omega$-stable diffeomorphisms. The equivalence of Lipschitz inverse periodic shadowing property and hyperbolicity of the closure of all periodic points is proved. Besides, we prove that the set of all diffeomorphisms that have Lipschitz inverse periodic shadowing property and whose periodic points are dense in the nonwandering set coincides with the set of Axiom A diffeomorphisms.