Quasi-shadowing property for nonuniformly partially hyperbolic systems

TL;DR

This paper introduces a new quasi-shadowing property for nonuniformly partially hyperbolic systems, extending existing concepts and applying them to growth rates of periodic points related to ergodic measures.

Contribution

It establishes a novel quasi-shadowing property for nonuniformly partially hyperbolic sets and extends Katok's results on periodic orbit growth to all ergodic measures.

Findings

Quasi-shadowing property is established for nonuniformly partially hyperbolic sets.

Quasi-specification and quasi-closing properties are investigated.

Number of quasi-periodic points grows exponentially with metric entropy.

Abstract

In this paper, we establish a new quasi-shadowing property for any nonuiformly partially hyperbolic set of a $C^{1+\alpha}$ diffeomorphism, which is adaptive to the movement of the pseudo-orbit. Moreover, the quasi-specification property and quasi-closing property are also investigated. As an application of quasi-closing property, we extend Katok's reslut on the growth of periodoc orbits for hyperbolic ergodic measure to any ergodic measure: the number of quasi-periodic points grows exponentially at least the metric entropy.