UCLA

Let A⊆S^2 be a finite set of points on the 2-sphere. A Thurston map f: (S^2,A) to (S^2,A) induces an associated holomorphic pullback map σ_f: T_A to T_A, where T_A is the Teichmüller space of the marked sphere (S^2,A). By [KPS16] this pullback map is known to satisfy a family of functional identities of the form σ_f ○ g = g̃ ○ σ_f where g, g̃ in Aut(T_A). In the case of four marked points we have T_A=H, and such maps are known as polymorphic maps. In this dissertation we develop a general framework for studying the cusp dynamics of polymorphic maps of the upper half-plane H. Applying this framework to the Thurston pullback map, we obtain a new proof of Thurston’s characterization theorem in the special case of four marked points. We also obtain new progress on the finite curve attractor conjecture of [Pil22]. Specifically, we prove that if a Thurston map f: (S^2,A) to (S^2,A) is totally unobstructed in the sense that all of its Thurston multipliers are strictly less than one, then (f,A) has a finite curve attractor.