By Kevin M. Pilgrim

ISBN-10: 3540201734

ISBN-13: 9783540201731

This paintings is a research-level monograph whose aim is to increase a normal mixture, decomposition, and constitution idea for branched coverings of the two-sphere to itself, considered as the combinatorial and topological items which come up within the type of sure holomorphic dynamical structures at the Riemann sphere. it really is meant for researchers attracted to the class of these advanced one-dimensional dynamical platforms that are in a few unfastened experience *tame*. this system is stimulated through the dictionary among the theories of iterated rational maps and Kleinian groups.

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This paintings is a research-level monograph whose aim is to enhance a basic blend, decomposition, and constitution conception for branched coverings of the two-sphere to itself, considered as the combinatorial and topological items which come up within the category of yes holomorphic dynamical platforms at the Riemann sphere.

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M. Pilgrim: LNM 1827, pp. 49–57, 2003. c Springer-Verlag Berlin Heidelberg 2003 50 3 Combinations ∓ • the map π A ◦ ρ ◦ (π S )−1 induces the involution b± 0 → b0 on connected components. The new sphere S 2 is the quotient space (S − B) ρ A0 ≈ S 2 . The projection maps π S , π A then deﬁne a continuous projection π : S 2 → T . 2 Critical gluing data At this point it is possible, after making some choices of extensions over U and C, to deﬁne a new continuous map F : S 2 → S 2 on the quotient space S0 ρ A0 ≈ S 2 .

We extend over Uu by sending (Uu , ∂Uu ) → (F∗ (Uu ), ∂F∗ (Uu )) by an arbitrary homeomorphism. Case Q ∩ Uu = ∅. By Axiom A–7 #4 this occurs only when Q ∩ Uu = {q}, and by Topological Compatibility Axiom A–13 in this case we have τ (q) ∈ F∗ (Uu ) ∩ Q. Using again the fact that Uu is a preimage of a disk in S under F S , and the fact that there is at most one critical point of F S in Uu , we have that deg(F S , ∂Uu ) = deg(F S , q) = ω(q). We extend over Uu by sending (Uu , q) → (F∗ (Uu ), τ (q)) via a covering ramiﬁed of degree deg(u) = ω(q) over τ (q).

Hence the image F (R) is a nondegenerate annulus, which is impossible. As another consequence, if Γ0 is a multicurve in S 2 − Q, then it can be shown that the annuli in A0 map by unramiﬁed coverings under F . This is convenient, since then in our “mapping tree” caricature of the dynamics, any folding caused by critical points will be concentrated in vertices of this tree. 6. 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 .

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