Semiconjugacies between the Julia sets of geometrically finite rational maps II

Newton maps of complex exponential functions and parabolic surgery

“…In [GM93, page 16], Goldberg and Milnor formulated the following conjecture: for a polynomial p with a parabolic cycle there exists a small perturbation of p such that the immediate basin of the parabolic cycle of p is converted to the basins of some attracting cycles; and the perturbed polynomial on its Julia set is topologically conjugate to p when restricted to the Julia set. Affirmative answers to the conjecture for the case of geometrically finite rational functions were given by many people, including G. Cui, P. Haïssinsky, T. Kawahira, and Tan Lei (see [CT11,CT,Ha98,Kaw05,Kaw06]). We remark that the local dynamics near repelling and parabolic fixed points are never conjugate to each other.…”

Hyperbolic-parabolic deformations of rational maps

“…For any arc α in Q which joins the two horizontal sides of Q, either α is contained in Q i for some i ∈ {1, 2, 3} and joins the two horizontal sides of Q i , or α intersects the two vertical sides of either Q 12 or Q 23 . In the former case,…”

“…We prove this conjecture in the setting of geometrically finite rational maps. Refer [16,17,18] and [22,23] for proofs of the same conjecture in various settings, but with different methods.…”

Semiconjugacies between the Julia sets of geometrically finite rational maps

“…For a given geometrically finite rational map, the existence of such perturbations is guaranteed by [10]. Example 1.…”

“…In general, any geometrically finite rational map has such a perturbation. See [10]. For other partial solutions of this conjecture, see [3] and [7].…”