A new link between Teichmueller theory and complex dynamics

Abstract

Given a Thurston map $f:S^2\to S^2$ with postcritical set $\po$, C. McMullen proved that the graph of the Thurston pullback map, $\sigma_f:\rabteich\longrightarrow\rabteich$, covers an algebraic subvariety of $V_f\subset\rabmod\times\rabmod$. In \cite{bn}, L. Bartholdi and V. Nekrashevych examined three examples of Thurston maps $f$, where $|\po|=4$, identifying $\rabmod$ with $\P^1-\{0,1,\infty\}$. They proved that for these three examples, the algebraic subvariety $V_f\subset\P^1\times\P^1$ is actually the graph of a function $g:\P^1\to\P^1$ such that $g\circ \pi\circ\sigma_f=\pi$, where $\pi:\rabteich\longrightarrow\P^1-\{0,1,\infty\}$ is the universal covering map. We generalize the Bartholdi-Nekrashevych construction to the case where $|\po|$ is arbitrary and prove that if $f:S^2\to S^2$ is a Thurston map of degree $d$ whose ramification points are all periodic, then there is a postcritically finite endomorphism $g_f:\P^{|\po|-3}\longrightarrow\P^{|\po|-3}$ such that $g_f\circ \pi\circ\sigma_f=\pi$. Moreover, the complement of the postcritical locus of $g_f$ is Kobayashi hyperbolic.

We prove that if $V_f\subset \P^{|\po|-3}\times\P^{|\po|-3}$ is the graph of such a map $g_f$, so that the algebraic degree of $g_f$ is $d$, then $g_f$ is a completely postcritically finite endomorphism. Moreover, we prove in this case that the Thurston pullback map $\sigma_f:\rabteich\longrightarrow\rabteich$ is a covering map of its image, and it is not surjective. We discuss the dynamics of the maps $g_f$ in the context of Thurston's topological characterization of rational maps, and use the map $\sigma_f$ to understand the map $g_f$ and vice versa.