Nearly Euclidean Thurston Maps

We take an in-depth look at Thurston’s combinatorial characterization of rational functions for a particular class of maps we call nearly Euclidean Thurston maps. These are orientation-preserving branched maps f : S2 → S2 whose local degree at every critical point is 2 and which have exactly four postcritical points. These maps are simple enough to be tractable, but are complicated enough to have interesting dynamics. In this work, we take an in-depth look at Thurston’s combinatorial characterization of rational functions for a particular class of maps we call nearly Euclidean Thurston (NET) maps. Suppose f : S → S is an orientation-preserving branched map. Following Thurston, we define νf : S 2 → Z+ ∪ {∞} by νf (x) = { lcm(Df (x)) if Df (x) is finite, ∞ if Df (x) is infinite, where Df (x) = {n ∈ Z+ : there exists m ∈ Z+ and y ∈ S such that f◦m(y) = x and f◦m has degree n at y}. The points x ∈ S with νf (x) > 1 are called postcritical points, and the set of postcritical points is denoted by Pf . The map f is postcritically finite if Pf is finite. A Thurston map is an orientation-preserving branched map f : S → S which is postcritically finite. In this case, we denote by T the Teichmüller space of the orbifold (S, νf ). The map f induces a map Σf : T → T by pulling back complex structures. In a CBMS Conference in 1983, Thurston [11] addressed the problem of determining when a Thurston map f : S → S is equivalent to a rational map, where f ∼ g if there is an orientation-preserving homeomorphism h : S → S such that h(Pf) = Pg, (h ◦ f) ∣∣ Pf = (g ◦ h) ∣∣ Pf , and h ◦ f is isotopic, rel Pf , to g ◦ h. His main theorems were (1) that f is equivalent to a rational map exactly if Σf has a fixed point, and (2) if (S, νf ) is hyperbolic, then Σf has a fixed point exactly if there are no Thurston obstructions (these will be defined next). Thurston didn’t publish his proofs of the theorems, but proofs were given later by Douady and Hubbard in [3]. Now we define Thurston obstructions. A multicurve Γ is a finite collection of pairwise disjoint simple closed curves in S \ Pf such that each element of Γ is nontrivial, each element of Γ is nonperipheral, and distinct elements of Γ are not isotopic. A multicurve Γ is invariant or f -stable if each element of f−1(Γ) is either trivial, peripheral, or isotopic to an element of Γ. If Γ is an invariant multicurve, Received by the editors April 16, 2012. 2010 Mathematics Subject Classification. Primary 37F10, 37F20. c ©2012 American Mathematical Society Reverts to public domain 28 years from publication