Groups, Geometry, and Dynamics

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Volume 5, Issue 3, 2011, pp. 603–661
DOI: 10.4171/GGD/141

Published online: 2011-06-18

Finite type coarse expanding conformal dynamics

Peter Haïssinsky[1] and Constantin Dorin Dumitrașcu[2]

(1) Université Paul Sabatier, Toulouse, France
(2) Adrian College, USA

We continue the study of noninvertible topological dynamical systems with expanding behavior. We introduce the class of 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.

Keywords: Conformal dynamics, selfsimilar group, subdivision rule

Haïssinsky Peter, Dumitrașcu Constantin Dorin: Finite type coarse expanding conformal dynamics. Groups Geom. Dyn. 5 (2011), 603-661. doi: 10.4171/GGD/141