Boundary values of the Thurston pullback map

Abstract

For any Thurston map with exactly four postcritical points, we present an algorithm to compute the Weil-Petersson boundary values of the corresponding Thurston pullback map. This procedure is carried out for the Thurston map f (z) = 3z 2 2z 3 +1 originally studied by Buff, et al. The dynamics of this boundary map are investigated and used to solve the analogue of Hubbard's Twisted Rabbit problem for f .

Nearly Euclidean Thurston maps

“…For NET maps, computation of the slope function σ f together with the mapping degree and number of preimages amounts to computation of φ f on powers of Dehn twists. This is what Lodge exploits in [5] to calculate the slope function for the NET map of Example 3. Table 1 shows that c = 2 and d = 5.…”

“…. , (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.…”

Quadratic Thurston maps with few postcritical points

“…This correspondence is crucial for enumerating the obstructed Q(4) * maps in §5, as well as computing the pullback on curves in §6 (see e.g. [3,21,12]). For any member of Q(4) * , the correspondence on moduli space restricts to a quadratic rational map.…”

“…Let f ∈ Q(4) * be a rational map. We will compute its pullback on curves in terms of the virtual endomorphism on moduli space using the naturality statement of [12,Theorem 2.6]. Let C be an essential curve in (S 2 , P f ) and denote by G the group π 1 ( C\{0, 1, ∞}, z 0 ); recall this is a free group generated by α and β.…”

An algebraic formulation of Thurston’s characterization of rational functions

“…One can give analytic conditions on this correspondence, similar in spirit to those -1055 -given in Corollary 7.2, which again imply that the pullback relation on curves has a finite global attractor. However, Lodge [11] has found examples of rational maps for which the pullback relation on curves has a finite global attractor, but for which the source of this finiteness is not a consequence of known general algebraic or analytic properties.…”