Expanding Thurston Maps

Details Expanding Thurston Maps Expanding Thurston Maps

2010-09-19

arxiv.org/abs/1009.3647v1

Mathematics

Dynamical Systems

242 pages, 25 figures

Scientific paper

We study the dynamics of Thurston maps under iteration. These are branched covering maps $f$ of 2-spheres $S^2$ with a finite set $\post(f)$ of postcritical points. We also assume that the maps are expanding in a suitable sense. Every expanding Thurston map is a factor of a shift operator. This link to symbolic dynamics suggests our ultimate goal of finding a combinatorial description of the dynamics of an expanding Thurston map in terms of finite data. Relevant for this problem are existence and uniqueness results for $f$-invariant Jordan curves $\CC\sub S^2$ containing the set $\post(f)$. For every sufficiently high iterate $f^n$ of an expanding Thurston map such an invariant Jordan curve always exists. If the sphere $S^2$ is equipped with a "visual" metric $d$ adapted to the dynamics of $f$, then an $f$-invariant Jordan curve $\CC$ with $\post(f)\sub \CC$ is a quasicircle. The geometry of the metric space $(S^2,d)$ encodes many dynamical properties of $f$. For example, $f\:S^2\ra S^2$ is topologically conjugate to a rational map if and only if $(S^2,d)$ is quasisymmetrically equivalent to the Riemann sphere $\CDach$. Establishing a framework for proving these and other results for expanding Thurston maps is the main purpose of this work.

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