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## A new link between Teichmueller theory and complex dynamics

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**Author**

Koch, Sarah

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**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.

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**Date Issued**

2008-07-30#####
**Subject**

Teichmueller theory; complex dynamics