Nearly Euclidean Thurston maps

Abstract

Abstract. 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 : S 2 → S 2 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.

Modular groups, Hurwitz classes and dynamic portraits of NET maps

“…2} so that f • π 1 = π 2 , as in Figure 1 in Section 1 of [4]. We may always, and usually do, assume Λ 2 = Z 2 .…”

“…The postcritical set of f is P 2 = π 2 (Λ 2 ). Statement 2 of Lemma 1.3 of [4] states that f −1 (P 2 ) contains exactly four points which are not critical points of f . This set of four points is P 1 = π 1 (Λ 1 ).…”

“…Recall that f is Levy-free if there is no simple closed curve γ ⊆ S 2 − P f which is essential, not peripheral and having a lift δ homotopic to γ which f maps to γ with degree 1. Using Theorem 4.1 of [4], we can see that the degree with which f maps δ to γ is divisible by n > 1. So f is Levy-free.…”

“…to lie in the domain of the virtual multi-endomorphism (Lemma 3.3). To calculate its image ψ, we exploit two facts: (i) the linear part of an element of Mod(S 2 , P f ) is uniquely determined by its value at a pair of distinct extended rationals p/q, r/s ∈ Q ∪ {1/0} =: Q, and (ii) the slope function µ f is algorithmically computable [4]. Section 7 illustrates this in a complicated example.…”

“…The NETmap program translates algebraic information about the virtual endomorphism into a rather complete geometric description about σ f ; cf. [4]. The data on the website tabulates this geometric information.…”