Functional Analysis

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By Kevin M. Pilgrim

This paintings is a research-level monograph whose aim is to increase a basic blend, 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 category of definite holomorphic dynamical structures at the Riemann sphere. it's meant for researchers drawn to the category of these advanced one-dimensional dynamical structures that are in a few unfastened experience. this system is influenced through the dictionary among the theories of iterated rational maps and Kleinian teams.

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Below, by deg(∂c) we mean the value of the function deg on either boundary component of c; by Axiom A–4 1(b) above this is independent of which vertex is chosen. 2. Global homogeneity. For all y ∈ f (X), deg(x) + f (x)=y deg(u) + y∈f∗ (u) deg(∂c) = d, y∈f∗ (c) and for all b0 ∈ f (B1 ) = B0 , deg(b) + b∈B1 ,f (b)=b0 deg(u) + b0 ∈f∗ (u) deg(∂c) = d. b0 ∈f∗ (c) It may be that there is some redundancy in Axioms A–4 #1(a,b) and A–4 #2. A–5 Orientations. We assume we have chosen arbitrarily a decomposition B0 = B0− B0+ + such that each annular subedge a0 is bounded by a pair (b− 0 , b0 ).

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.

Even for polynomial maps with totally disconnected Julia sets, such a classification may be very difficult. Emerson [Eme] associates to such a map an infinite tree, equipped with a self-map, which imitates the deployment of annuli in the complement of the Julia set bounded by level sets of the Green’s function; in particular he shows that uncountably many combinatorially inequivalent such trees can arise among maps of a given degree. e. to rational maps for which the intersection of the Julia and postcritical sets is finite.

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