Quasi-Stability of Partially Hyperbolic Diffeomorphisms

TL;DR

This paper investigates the stability properties of partially hyperbolic diffeomorphisms, establishing conditions under which they are topologically or structurally quasi-stable, and explores implications for the continuity of topological entropy.

Contribution

It proves that all partially hyperbolic diffeomorphisms are topologically quasi-stable, and that those with $C^1$ center foliation are also structurally quasi-stable, advancing understanding of their stability behavior.

Findings

All partially hyperbolic diffeomorphisms are topologically quasi-stable.

Partially hyperbolic diffeomorphisms with $C^1$ center foliation are structurally quasi-stable.

Continuity of topological entropy is established for certain cases with low-dimensional center foliation.

Abstract

A partially hyperbolic diffeomorphism $f$ is structurally quasi-stable if for any diffeomorphism $g$ $C^1$-close to $f$, there is a homeomorphism $\pi$ of $M$ such that $\pi\circ g$ and $f\circ\pi$ differ only by a motion $\tau$ along center directions. $f$ is topologically quasi-stable if for any homeomorphism $g$ $C^0$-close to $f$, the above holds for a continuous map $\pi$ instead of a homeomorphism. We show that any partially hyperbolic diffeomorphism $f$ is topologically quasi-stable, and if $f$ has $C^1$ center foliation $W^c_f$, then $f$ is structurally quasi-stable. As applications we obtain continuity of topological entropy for certain partially hyperbolic diffeomorphisms with one or two dimensional center foliation.