Thurston obstructions and Ahlfors regular conformal dimension

Abstract

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 \[ \inf_{X \in \GGG(f)} \hdim(X) \geq Q(f)=\inf_Γ\{Q \geq 2: λ(f_{Γ,Q}) \geq 1\}.\] The infimum is over all multicurves $\Gamma \subset S^2-P_f$. The map $f_{\Gamma ,Q}:{\mathbb{R}}^{\Gamma }\to {\mathbb{R}}^{\Gamma }$ is defined by \[ f_{Γ, Q}(γ) =\sum_{[γ']\inΓ} \sum_{δ\sim γ'} °(f:δ\to γ)^{1-Q}[γ'],\] where the second sum is over all preimages $\delta$ of $\gamma$ freely homotopic to ${\gamma }^{\prime }$ 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.

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