An algebraic formulation of Thurston's characterization of rational functions

TL;DR

This paper provides an algebraic reformulation of Thurston's theorem on rational functions, applying it to analyze dynamics on homotopy classes and deriving finiteness results that inform the global dynamics in Teichmüller space.

Contribution

It introduces an algebraic approach to Thurston's characterization, linking it to dynamics on homotopy classes and establishing new finiteness results for the pullback map.

Findings

Finiteness results for dynamics on homotopy classes

New insights into the global dynamics of the pullback map

Algebraic formulation of Thurston's characterization

Abstract

Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic formulation of Thurston's characterization of rational functions. The techniques developed are applied to the analysis of the dynamics on the set of free homotopy classes of simple closed curves induced by a rational function. The resulting finiteness results yield new information on the global dynamics of the pullback map on Teichm\"uller space used in the proof of the characterization theorem.