Lipschitz perturbations of expansive systems

TL;DR

This paper extends results from smooth dynamical systems to Lipschitz homeomorphisms, exploring robustness of properties like expansiveness and stability under Lipschitz perturbations, and compares Lipschitz and $C^1$ topologies.

Contribution

It introduces the study of dynamical properties under Lipschitz perturbations and analyzes the relationship between Lipschitz and $C^1$ topologies on manifolds.

Findings

Robust expansiveness under Lipschitz perturbations

Structural stability in Lipschitz category

Comparison between Lipschitz and $C^1$ topologies

Abstract

We extend some known results from smooth dynamical systems to the category of Lipschitz homeomorphisms of compact metric spaces. We consider dynamical properties as robust expansiveness and structural stability allowing Lipschitz perturbations with respect to a hyperbolic metric. We also study the relationship between Lipschitz topologies and the $C^1$ topology on smooth manifolds.