Finite type coarse expanding conformal dynamics

TL;DR

This paper introduces the class of finite type coarse expanding conformal systems, characterizing certain non-invertible dynamical systems with finitely many iterates up to rescaling and bounded distortion, including specific rational maps and subdivision rules.

Contribution

It defines finite type systems and demonstrates that various classes, such as subhyperbolic rational maps and certain group actions, belong to this class, extending the understanding of their dynamics.

Findings

Finite type systems characterized by finitely many iterates up to rescaling.

Subhyperbolic rational maps are of finite type.

Finite subdivision rules with bounded valence and shrinking mesh are of finite type.

Abstract

We continue the study of non-invertible topological dynamical systems with expanding behavior. We introduce the class of {\em finite type} systems which are characterized by the condition that, up to rescaling and uniformly bounded distortion, there are only finitely many iterates. We show that subhyperbolic rational maps and finite subdivision rules (in the sense of Cannon, Floyd, Kenyon, and Parry) with bounded valence and mesh going to zero are of finite type. In addition, we show that the limit dynamical system associated to a selfsimilar, contracting, recurrent, level-transitive group action (in the sense of V. Nekrashevych) is of finite type. The proof makes essential use of an analog of the finiteness of cone types property enjoyed by hyperbolic groups.