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On the quasiconformal surgery of rational functions. (English) Zbl 0621.58030
In the paper remarkable results concerning the conformal dynamics on the Riemann sphere are obtained. It is known that a rational map \(f: {\bar {\mathbb{C}}}\to {\bar {\mathbb{C}}}\) of degree ḏ may have only a finite number of nonrepelling cycles (Fatou, Julia). Fatou conjectured that the number of such cycles does not exceed 2d-2. In the paper this conjecture is proved and, moreover, it is shown that the estimate 2d-2 can be achieved for any ḏ.
A related problem concerns the number of cycles of domains in the set of normality. The classification and finiteness theorem for them is due to Sullivan. In the paper the best estimates of their numbers are obtained. Let \(n_{AB}\), \(n_{PB}\), \(n_{SD}\) and \(n_{HR}\) be the number of attractive basins, parabolic basins, cycles of Siegel disks and Herman rings correspondingly. Then \(n_{AB}+n_{PB}+n_{SD}+n_{HR}\leq 2d- 2\), \(n_{HR}\leq d-2\) (the first estimate is natural, the second is surprising). The main tool of the proofs is quasiconformal surgery of holomorphic maps introduced into the subject by Douady, Hubbard and Sullivan. The most delicate part of the proof concerns the surgery of Herman rings.
Reviewer: M.Lyubich

MSC:
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
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