Quasi-Shadowing for Partially Hyperbolic Diffeomorphisms

TL;DR

This paper proves that all partially hyperbolic diffeomorphisms possess a quasi-shadowing property, enabling pseudo orbits to be closely traced with controlled motions along the center direction, with applications to stability and classical hyperbolic theorems.

Contribution

It establishes the quasi-shadowing property for all partially hyperbolic diffeomorphisms and extends classical hyperbolic results to this broader setting.

Findings

All partially hyperbolic diffeomorphisms have quasi-shadowing.

Topological quasi-stability under $C^0$-perturbation.

Extensions of hyperbolic theorems to systems with compact center foliation.

Abstract

A partially hyperbolic diffeomorphism $f$ has quasi-shadowing property if for any pseudo orbit ${x_k}_{k\in \mathbb{Z}}$, there is a sequence of points ${y_k}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_k)$ by a motion $\tau$ along the center direction. We show that any partially hyperbolic diffeomorphism has quasi-shadowing property, and if $f$ has $C^1$ center foliation then we can require $\tau$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^0$-perturbation. When $f$ has uniformly compact $C^1$ center foliation, we also give partially hyperbolic diffeomorphism versions of some theorems holden for uniformly hyperbolic systems, such as Anosov closing lemma, cloud lemma and spectral decomposition theorem.