# Expanding Thurston Maps

Bonk, Mario and Meyer, Daniel (2017) Expanding Thurston Maps. Mathematical Surveys and Monographs, 225 . American Mathematical Society,Providence, Rhode Island.

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## Abstract

We study the dynamics of Thurston maps under iteration. These are branched covering maps $f$ of 2-spheres $S^2$ with a finite set $\mathop{post}(f)$ of postcritical points. We also assume that the maps are expanding in a suitable sense. Every expanding Thurston map $f\: S^2 \to S^2$ gives rise to a type of fractal geometry on the underlying sphere $S^2$. This geometry is represented by a class of \emph{visual metrics} $\varrho$ that are associated with the map. Many dynamical properties of the map are encoded in the geometry of the corresponding {\em visual sphere}, meaning $S^2$ equipped with a visual metric $\varrho$. For example, we will see that an expanding Thurston map is topologically conjugate to a rational map if and only if $(S^2, \varrho)$ is quasisymmetrically equivalent to the Riemann sphere $\widehat{\mathbf{C}}$. We also obtain existence and uniqueness results for $f$-invariant Jordan curves $\mathcal{C}\subset S^2$ containing the set $\mathop{post}(f)$. Furthermore, we obtain several characterizations of Latt\`{e}s maps.

Item Type: Book 492 pages, 51 figures math.DS, math.CV, math.MG, 37F20 Symplectic Admin 22 Dec 2017 10:22 16 Apr 2021 11:43 https://livrepository.liverpool.ac.uk/id/eprint/3014259