128 research outputs found
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 be an expanding branched covering map of the sphere to
itself with finite postcritical set . Associated to 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 . The map
is defined by where the second sum is over all preimages
of freely homotopic to in , and is its Perron-Frobenius leading eigenvalue. This
generalizes Thurston's observation that if , then there is no
-invariant classical conformal structure.Comment: Minor revisions are mad
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