Characterizations of contracting Hurwitz bisets

TL;DR

This paper explores the relationships between contraction properties of three associated iterated function systems in the context of critically finite branched self-covers, establishing their equivalence.

Contribution

It demonstrates that contraction in any one of the three systems implies contraction in the others, revealing a fundamental connection among these systems.

Findings

Contraction in the mapping class group system implies contraction in Teichmüller space.

Contraction in the Teichmüller space system implies contraction in the vector space system.

Contraction properties are equivalent across the three systems.

Abstract

A critically finite branched self-cover $f: (S^2, P) \to (S^2, P)$ determines naturally three iterated function systems: one on the pure mapping class group of the sphere marked at $P$, one on the Teichm\"uller space of the sphere marked at $P$, and one on a finite-dimensional real vector space. We show that contraction for any one of these systems implies contraction for the others.