By Kevin M. Pilgrim
This paintings is a research-level monograph whose target is to improve a basic blend, decomposition, and constitution concept for branched coverings of the two-sphere to itself, considered as the combinatorial and topological items which come up within the class of yes holomorphic dynamical structures at the Riemann sphere. it truly is meant for researchers attracted to the type of these complicated one-dimensional dynamical structures 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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Extra resources for Combinations of Complex Dynamical Systems
One could also strengthen the requirement in (2) to holomorphic conjugacy, but otherwise keep the remaining requirements unmodiﬁed. g. Cui [CJS], Epstein-Keen-Tresser [EKT], and McMullen ([McM7], Appendix A). In general, then, when the postcritical set is inﬁnite, there may be various choices of regularity on the maps h0 , h1 near the points of Y which one might wish to require. To keep our theory of general applicability we leave the details unspeciﬁed, referring to whatever restrictions are imposed in (1)-(4) above as tameness assumptions.
When combining, the new dynamical system is produced by C 0 modiﬁcations of the original maps away from the set Q, which contains all accumulation points of Y since Y = Q Z is closed and Z is ﬁnite. 5 (see §4) asserts roughly that the combinatorial class of the map resulting from a combination depends only on the classes of the input data (suitably deﬁned). The proof of this theorem relies on two steps: gluing together maps yielding equivalences of input data, and C 0 modiﬁcations away from the sets Q and Q.
Cut the sphere apart along a set of annuli A0 in S 2 − Q whose core curves are the elements of Γ . For the moment, discard these annuli A0 . For each of the remaining pieces, cap the resulting holes by disks with preferred center points to obtain a collection S of spheres. Denote the collection of resulting center points of these disks by Z. Then Z is ﬁnite. The set Q yields a corresponding set in this union of spheres which we denote by Q. Set Y = Q Z. After a suitable extension, one ﬁnds that the original map F : (S 2 , Q) → (S 2 , Q) yields a map F S : (S, Y) → (S, Y) from a ﬁnite set of spheres to itself such that Y and Z are forward-invariant under F S .