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 $\text{GGG}(f)$ of Ahlfors regular metrics on the sphere in which the dynamics is (non-classically) conformal. We show $$\underset{X\in \text{GGG}(f)}{\inf }\text{hdim}(X)\ge Q(f)=\underset{\mathrm{\Gamma }}{\inf }\{Q\ge 2:\lambda (f_{\mathrm{\Gamma },Q})\ge 1\}\mathrm{.}$$ The infimum is over all multicurves $\mathrm{\Gamma }\subset S^2-P_f$. The map $f_{\mathrm{\Gamma },Q}:{\mathbb{R}}^{\mathrm{\Gamma }}\to {\mathbb{R}}^{\mathrm{\Gamma }}$ is defined by $$f_{\mathrm{\Gamma },Q}(\gamma )=\sum _{[{\gamma }^{\mathrm{\prime }}]\in \mathrm{\Gamma }}\sum _{\delta \sim {\gamma }^{\mathrm{\prime }}}\mathrm{deg}(f:\delta \to \gamma {)}^{1-Q}[{\gamma }^{\mathrm{\prime }}],$$ where the second sum is over all preimages $\delta$ of $\gamma$ freely homotopic to ${\gamma }^{\mathrm{\prime }}$ in $S^2-P_f$, and $\lambda (f_{\mathrm{\Gamma },Q})$ 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.