Thurston obstructions and Ahlfors regular conformal dimension

Let $f: S^2 \to S^2$ be an expanding branched covering map of the sphere to itself with finite postcritical set $P_f$. Associated to $f$ is a canonical quasisymmetry class \GGG(f) of Ahlfors regular metrics on the sphere in which the dynamics is (non-classically) conformal. We show \inf_{X \in \GGG(f)} \hdim(X) \geq Q(f)=\inf_\Gamma \{Q \geq 2: \lambda(f_{\Gamma,Q}) \geq 1\}. The infimum is over all multicurves $\Gamma \subset S^2-P_f$. The map $f_{\Gamma,Q}: \R^\Gamma \to \R^\Gamma$ is defined by $$f_{\Gamma, Q}(\gamma) =\sum_{[\gamma']\in\Gamma} \sum_{\delta \sim \gamma'} \deg(f:\delta \to \gamma)^{1-Q}[\gamma'],$$ where the second sum is over all preimages $\delta$ of $\gamma$ freely homotopic to $\gamma'$ in $S^2-P_f$, and $\lambda(f_{\Gamma,Q})$ is its Perron-Frobenius leading eigenvalue. This generalizes Thurston's observation that if $Q(f)>2$, then there is no $f$-invariant classical conformal structure.Comment: Minor revisions are mad

Finite type coarse expanding conformal dynamics

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.Comment: Updated versio