Anomalous cw-expansive homeomorphisms on compact surfaces of higher genus

TL;DR

This paper constructs novel cw-expansive homeomorphisms on higher genus surfaces, demonstrating complex local stable set structures and expanding the understanding of expansiveness properties in topological dynamics.

Contribution

It introduces new examples of cw-expansive homeomorphisms with connected but not locally connected stable sets on surfaces of higher genus, answering an open question.

Findings

Existence of cw-expansive homeomorphisms with connected but not locally connected stable sets.

Construction of 2-expansive but not expansive homeomorphisms on higher genus surfaces.

New cw2-expansive homeomorphisms on the sphere and torus that are not N-expansive for any N.

Abstract

In this paper, we construct cw-expansive homeomorphisms on compact surfaces of genus greater than or equal to zero with a fixed point whose local stable set is connected but not locally connected. This provides an affirmative answer to question posed by Artigue [3]. To achieve this, we generalize the construction from the example of Artigue, Pacifico and Vieitez [6], obtaining examples of homeomorphisms on compact surfaces of genus greater than or equal to two that are 2-expansive but not expansive. On the sphere and the torus, we construct new examples of cw2-expansive homeomorphisms that are not N -expansive for all N greater than or equal to one.