By Kevin M. Pilgrim
This paintings is a research-level monograph whose objective is to boost a basic mix, decomposition, and constitution idea for branched coverings of the two-sphere to itself, considered as the combinatorial and topological gadgets which come up within the category of convinced holomorphic dynamical structures at the Riemann sphere. it truly is meant for researchers attracted to the category of these advanced one-dimensional dynamical structures that are in a few free feel tame. this system is prompted through the dictionary among the theories of iterated rational maps and Kleinian groups.
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Extra resources for Combinations of Complex Dynamical Systems
When F is rational) then the corresponding IMG satisﬁes an additional property of being what is termed “contracting”. In this case, a general mechanism for identifying the Julia set as the boundary at inﬁnity of a certain Gromov hyperbolic graph is given, and in a manner which permits one to, in principle, algorithmically reconstruct the dynamics of F from purely combinatorial information. 3 Cannon’s conjecture The investigations of Cannon, Floyd, and Parry [CFP1], [CFP2], [CFP3] into connections between subdivision rules and postcritically ﬁnite rational maps are motivated in part by a desire to prove Cannon’s conjecture Let G be a Gromov hyperbolic group whose boundary at inﬁnity is homeomorphic to the two-sphere.
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 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 .
F is a homeomorphism on each edge of T 4. f |a0,i is a homeomorphism on each annular edge a0,i . 5. e. the image of each edge a1,j ∈ A1 is an annular edge a0,i ∈ A0 . This need not be surjective onto A0 . However, note that every annular edge contains at least one subedge in A1 . ) 6. f maps an edge joining an x-vertex and a u-vertex homeomorphically onto an edge joining an x-vertex to a b0 -vertex. In particular, any folding of T by f must be concentrated in the X vertices. Deﬁnition (image subtrees): Recall that u ∈ U is adjacent to exactly one x vertex.