Sarah Koch
Sarah Koch

Ph.D. (2008) Cornell University

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Dissertation
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Abstract: Given a Thurston map f : S^{2} → S^{2} with postcritical set P, C. McMullen proved that the graph of the Thurston pullback map, σ_{f} : Teich(S^{2},P) → Teich(S^{2},P), covers an algebraic subvariety of V_{f} ⊂ Mod(S^{2},P) × Mod(S^{2},P). In [2], L. Bartholdi and V. Nekrashevych examined three examples of Thurston maps f, where P = 4, identifying Mod(S^{2},P) with P^{1} – {0,1,∞}. They proved that for these three examples, the algebraic subvariety V_{f} ⊂ P^{1} × P^{1} is actually the graph of a function g : P^{1} → P^{1} such that g ° π ° σ_{f} = π, where π : Teich(S^{2},P) → P^{1} – {0,1,∞} is the universal covering map. We generalize the BartholdiNekrashevych construction to the case where P is arbitrary and prove that if f : S^{2} → 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^{P – 3} → P^{P – 3} such that g_{f} ° π ° σ_{f} = π. Moreover, the complement of the postcritical locus of g_{f} is Kobayashi hyperbolic.
We prove that if V_{f} ⊂ P^{P – 3} × P^{P – 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 σ_{f} : Teich(S^{2},P) → Teich(S^{2},P) 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 σ_{f} to understand the map g_{f} and vice versa.