Lattes Maps and Combinatorial Expansion.

Abstract

A Lattes map f : C → C is a rational map that is obtained from a finite quotient of a conformal torus endomorphism. In this thesis, we give a characterization of Lattes maps by their combinatorial expansion behavior. More specifically, to any Thurston maps, which are branched covering maps over the 2-sphere with finite post-critical sets, there are natural cell-decompositions of the 2-sphere induced by the dynamics following Bonk and Meyer. We show that these cell decompositions give us a natural Gromov hyperbolic space, and we deduce new necessary and sufficient conditions for a Thurston map to be topologically conjugate to a Lattes map.