  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.
