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By Kevin M. Pilgrim

This paintings is a research-level monograph whose aim is to increase a normal mix, decomposition, and constitution conception for branched coverings of the two-sphere to itself, considered as the combinatorial and topological gadgets which come up within the class of definite holomorphic dynamical structures at the Riemann sphere. it truly is meant for researchers attracted to the type of these complicated one-dimensional dynamical platforms that are in a few unfastened feel tame. this system is prompted via the dictionary among the theories of iterated rational maps and Kleinian groups.

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Quadratic rational maps. It turns out that the moduli space of M¨ obius conjugacy classes of quadratic rational maps is biholomorphic to C2 [Mil3]. Rees’ program [Ree3], [Ree4], [Ree5] yields detailed results on the combinatorial structure of the one-complex dimensional loci Vn ⊂ C2 of maps with a critical point of period n. The approach is by analogy with the theory for quadratic polynomials and proceeds using laminations. The last cited work also relates the homotopy types of various spaces of rational maps and their combinatorial analogs.

Thus, in general, it is convenient to reformulate the definitions of multicurve and combinatorial equivalence so as to replace the postcritical set PF 32 1 Introduction by its full preimage QF . However, we have already seen in the previous subsection how the process of decomposition leads naturally to consideration of maps F S defined on a set S of spheres. 12 (Embellished map of spheres). An embellished map of spheres is a triple (F, S, Y) where • • • S Y F – – – is a finite set of spheres, ⊂ S is closed, : (S, Y) → (S, Y) is a (tame) map of pairs such that F is an orientation-preserving branched covering, Crit(F) ⊂ Y, where Crit(F) denotes the set of critical points of F, F −1 (F(Y)) = Y.

A component of A0 is denoted by Aa0,i and the set of such components is denoted by A0 . Thus, A0 = Aa0,i . 2). e. Q is closed, forward-invariant, contains the critical points, and coincides with the preimage of its image. M. Pilgrim: LNM 1827, pp. 37–48, 2003. c Springer-Verlag Berlin Heidelberg 2003 38 2 Preliminaries • Let A1 be the union of preimages of elements of A0 which are essential in A0 . Note that each component of A1 is an annulus mapping under F as an unramified covering onto its image.

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