Expanding Thurston Maps

TL;DR

This paper investigates the dynamics of expanding Thurston maps on 2-spheres, exploring their associated fractal geometries via visual metrics, and characterizes when these maps are conjugate to rational maps or Lattès maps.

Contribution

It introduces a framework linking expanding Thurston maps with fractal geometry through visual metrics and provides criteria for conjugacy to rational and Lattès maps.

Findings

Expanding Thurston maps induce fractal geometries on the sphere.

Topological conjugacy to rational maps is characterized by quasisymmetric equivalence.

Existence and uniqueness of invariant Jordan curves containing postcritical points are established.

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.