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A new link between Teichmueller theory and complex dynamics

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Given a Thurston map f:S2→S2 with postcritical set \po, C. McMullen proved that the graph of the Thurston pullback map, σf:\rabteich⟶\rabteich, covers an algebraic subvariety of Vf⊂\rabmod×\rabmod. In \cite{bn}, L. Bartholdi and V. Nekrashevych examined three examples of Thurston maps f, where |\po|=4, identifying \rabmod with \P1−{0,1,}. They proved that for these three examples, the algebraic subvariety Vf⊂\P1×\P1 is actually the graph of a function g:\P1→\P1 such that gπσf=π, where π:\rabteich⟶\P1−{0,1,} is the universal covering map. We generalize the Bartholdi-Nekrashevych construction to the case where |\po| is arbitrary and prove that if f:S2→S2 is a Thurston map of degree d whose ramification points are all periodic, then there is a postcritically finite endomorphism gf:\P|\po|−3⟶\P|\po|−3 such that gfπσf=π. Moreover, the complement of the postcritical locus of gf is Kobayashi hyperbolic.

We prove that if Vf⊂\P|\po|−3×\P|\po|−3 is the graph of such a map gf, so that the algebraic degree of gf is d, then gf is a completely postcritically finite endomorphism. Moreover, we prove in this case that the Thurston pullback map σf:\rabteich⟶\rabteich is a covering map of its image, and it is not surjective. We discuss the dynamics of the maps gf in the context of Thurston's topological characterization of rational maps, and use the map σf to understand the map gf and vice versa.

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2008-07-30T12:39:49Z

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Teichmueller theory; complex dynamics

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bibid: 6397238

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