An expansive homeomorphism of a 3-manifold with a local stable set that is not locally connected

TL;DR

This paper constructs a specific example of an expansive homeomorphism on a 3-manifold that has a fixed point with a local stable set which is not locally connected, challenging assumptions about stability structures.

Contribution

It introduces a novel example of an expansive homeomorphism with a non-locally connected local stable set on a 3-manifold, derived from a perturbation of a quasi-Anosov diffeomorphism.

Findings

Existence of an expansive homeomorphism with non-locally connected local stable set

Construction method via topological perturbation of quasi-Anosov diffeomorphism

Counterexample to assumptions about local stable set connectivity

Abstract

In this article we construct an expansive homeomorphism of a compact three-dimensional manifold with a fixed point whose local stable set is not locally connected. This homeomorphism is obtained as a topological perturbation of a quasi-Anosov diffeomorphism that is not Anosov.