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 $Γ\subset S^2-P_f$. The map $f_{Γ,Q}: \R^Γ\to \R^Γ$ 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 $δ$ of $γ$ freely homotopic to $γ'$ in $S^2-P_f$, and $ λ(f_{Γ,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.