Thurston's theorem and the Nielsen-Thurston classification via Teichm\"uller's theorem

TL;DR

This paper provides a unified, self-contained proof of the Nielsen-Thurston classification theorem by integrating concepts from mapping class groups and complex dynamics, specifically Thurston's characterization of rational maps.

Contribution

It offers a new proof that combines techniques from different areas, extending the understanding of the classification of surface homeomorphisms.

Findings

Unified proof of Nielsen-Thurston classification

Connection between mapping class groups and complex dynamics

Extensions of Thurston's characterization of rational maps

Abstract

We give a unified and self-contained proof of the Nielsen-Thurston classification theorem from the theory of mapping class groups and Thurston's characterization of rational maps from the theory of complex dynamics (plus various extensions of these). Our proof follows Bers' proof of the Nielsen-Thurston classification.