Expanding Thurston maps as quotients

Details Expanding Thurston maps as quotients Expanding Thurston maps as quotients

2009-10-11

arxiv.org/abs/0910.2003v1

Mathematics

Complex Variables

49 pages, 7 figures

Scientific paper

A \emph{Thurston map} is a branched covering map $f\colon S^2\to S^2$ that is \emph{postcritically finite}. \emph{Mating of polynomials}, introduced by Douady and Hubbard, is a method to \emph{geometrically} combine the Julia sets of two polynomials (and their dynamics) to form a rational map. We show that every \emph{expanding} Thurston map $f$ has an iterate $F=f^n$ that is obtained as the mating of two polynomials. One obtains a concise description of $F$ via \emph{critical portraits}. The proof is based on the construction of the invariant Peano curve from \cite{peano}. As another consequence we obtain a large number of fractal tilings of the plane and the hyperbolic plane.

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