A classification of minimal sets for surface homeomorphisms

TL;DR

This paper extends the classification of minimal sets for surface homeomorphisms from the torus to hyperbolic surfaces, analyzing their complement components and using Nielsen-Thurston Theory to understand their structure.

Contribution

It provides a comprehensive classification of minimal sets for hyperbolic surface homeomorphisms, expanding previous work on the torus and incorporating non-wandering dynamics and Nielsen-Thurston Theory.

Findings

Classification of minimal sets based on complement components

Extension of torus results to hyperbolic surfaces

Analysis within non-wandering and Nielsen-Thurston frameworks

Abstract

We classify minimal sets of (closed and oriented) hyperbolic surface homeomorphisms by studying the connected components of their complement. This extends the classification given by F. Kwakkel, T.J\"ager and A. Passeggi in the torus. The given classification is studied in the non-wandering setting and in light of the Nielsen-Thurston Theory.