Finite type coarse expanding conformal dynamics

Abstract

Abstract. 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.

Uniformization of two-dimensional metric surfaces

“…In particular, several problems in the theory of hyperbolic groups can be interpreted as uniformization problems concerning boundaries of the groups in question, cf. [9], [10], [12], [14], [26], [37].…”

One hundred years of complex dynamics

“…The result indicated a classification of all postcritically finite branched coverings of the sphere, which have come to be known as Thurston maps, and there has been extensive work on these and on extensions to them [114][115][116][117]. Checking the necessary and sufficient conditions for a map to be Thurston equivalent to a rational map is nontrivial.…”

Mapping schemes realizable by obstructed topological polynomials

“…Usually, researchers use Thurston's characterization to construct complex polynomials using purely topological methods. However, Haïssinsky and Pilgrim [HP09] and Bonk and Meyer [BM] have used obstructed topological polynomials to define metrics on the sphere that are not quasisymmetric to the standard sphere. Such metrics interest analysts who seek geometric criteria for quasisymmetric equivalence to the standard sphere.…”