$C^1$-generic continuum-wise expansive surface diffeomorphisms

TL;DR

This paper identifies a large set of surface $C^1$ diffeomorphisms that are continuum-wise expansive without being expansive, and shows that in a dense set, expansiveness implies Anosov behavior.

Contribution

It demonstrates the existence of continuum-wise expansive diffeomorphisms that are not expansive and clarifies the relationship between expansiveness and Anosov systems in the $C^1$ topology.

Findings

Existence of a residual set of continuum-wise expansive but non-expansive surface diffeomorphisms.

An open dense set where expansiveness implies Anosov behavior.

Differentiates between continuum-wise expansiveness and expansiveness in surface dynamics.

Abstract

We exhibit a local residual set of surface $C^1$ diffeomorphisms that are continuum-wise expansive but are not expansive. We also exhibit an open and dense set of surface $C^1$ diffeomorphisms where expansiveness implies being Anosov.