On cohomological $C^0$-(in)stability

TL;DR

This paper classifies cohomologically $C^0$-stable homeomorphisms, showing that only periodic homeomorphisms possess this stability, thus providing a complete characterization in this context.

Contribution

It provides a complete classification of cohomologically $C^0$-stable homeomorphisms, identifying periodic homeomorphisms as the only such class.

Findings

Periodic homeomorphisms are the only cohomologically $C^0$-stable homeomorphisms.

Complete classification of cohomologically $C^0$-stable homeomorphisms.

Clarifies the structure of $C^0$-stability in topological dynamics.

Abstract

After Katok, a homeomorphism $f\colon M\to M$ is said to be cohomologically $C^0$-stable when its space of real $C^0$-coboundaries is closed in $C^0(M)$. In this short note we completely classify cohomologically $C^0$-stable homeomorphisms, showing that periodic homeomorphisms are the only ones.