Dessins d'enfants and Hubbard Trees

We show that the absolute Galois group acts faithfully on the set of Hubbard trees. Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane. We establish an explicit relationship between certain Hubbard trees and the trees known as ``dessins d'enfant'' introduced by Grothendieck.Comment: 27 pages, 8 PostScript figure

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