On Thurston's pullback map

Nearly Euclidean Thurston maps

“…Figure 8 shows the mapping scheme of f . This is the rational function which appears in the proof of statement 2 of Theorem 1.1 of [1].…”

“…We define the modulus of this family of curves just as we defined mod τ ( p q ). Because lengths of curves do not change when pulling back from S τ to T τ but area doubles, this new modulus is 1 2 mod τ ( p q ). Let f be a NET map.…”

“…. , (0, 5), (1,5) and h maps the image of (2,4) to the image of (2,5) and the image of (2, −1) to the image of (2, 0). Let f = h • g. The action of f is indicated in Figure 4.…”

“…A (right-handed) Dehn twist about γ is a homeomorphism from S to S which is the identity map outside of A and on A it has the form g • h • g −1 : A → A, where h(re 2πiθ ) = re 2πi(θ−r) . Thus for a fixed θ, as r traverses the closed interval [1,2] in either direction, h(re 2πiθ ) bends to the right as it winds once around the annulus {z ∈ C : 1 ≤ |z| ≤ 2}. Now let γ be a simple closed curve in S 2 \ P 2 , and let t : S 2 \ P 2 → S 2 \ P 2 be a right-handed Dehn twist about γ.…”

“…Everything in this paper arose from studying this example. The second example in Section 3 shows that the rational function which appears in the proof of statement 2 of Theorem 1.1 of [1] is a NET map.…”