By Kevin M. Pilgrim
This paintings is a research-level monograph whose aim is to improve a normal 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 class of convinced holomorphic dynamical structures at the Riemann sphere. it truly is meant for researchers drawn to the category of these complicated one-dimensional dynamical structures that are in a few unfastened feel. this system is inspired by way of the dictionary among the theories of iterated rational maps and Kleinian teams.
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The topological and critical gluing data must satisfy certain additional axioms in order for the resulting map F : S 2 → S 2 to be a well-deﬁned embellished branched covering. 2. 1 Topological gluing data Recall that we have bijections πS πA ∂B −→ B0 ←− ∂A0 . Furthermore, B0 = B − B + and we have a bijective correspondence B − ↔ B+. e. 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 the the the set Y itself, map F near Y, and maps H0 , H1 occurring in the deﬁnition of combinatorial equivalence, isotopy joining H0 and H1 . A typical restriction in (1) is that Y have only ﬁnitely many accumulation points. g. e. one for which the intersection of the Julia set and postcritical set is ﬁnite. This requirement alone is usually too weak to be used for reasonable topological characterizations of rational functions with inﬁnite postcritical set. For instance, if the maps H0 , H1 in the deﬁnition of combinatorial equivalence are 1 required only to be homeomorphisms, then the map F(z) = z 2 − 10 having an attracting ﬁxed point is combinatorially equivalent to the map G(z) = z 2 − 34 having a parabolic ﬁxed point.
Let the ﬁber of f n over u0 be denoted f −n (u0 ). Let G = π1 (U0 , u0 ) denote the fundamental group of U0 based at u0 . By path-lifting, for each n, there is a transitive right action of G on the inverse image f −n (u0 ) of u0 under f n . The quotient of G by the kernel of this action is the monodromy group of f n , denoted Mon(f n ). If vn ∈ f −n (u0 ) and g ∈ G, then clearly (fn (vn ))g = fn (vng ). Hence for each n ≥ 2 the groups Mon(f n ) are imprimitive, and there is a surjective homomorphism Mon(f n ) → Mon(f n−1 ).
Combinations of Complex Dynamical Systems by Kevin M. Pilgrim