Characterizations of contracting Hurwitz bisets

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 Teichmüller 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.