Expanding Thurston maps as quotients

Abstract

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

58 pages, 11 figures

References in corpus (1)

• Dynamics in one complex variable: introductory lectures

Cited by in corpus (2)

• Combination Theorems in Groups, Geometry and Dynamics
• A characterization of Sierpinski carpet Rational maps