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Collet, Eckmann and Hölder. (English) Zbl 0916.30023
[L. Carleson, P. Jones and J.-C. Yoccoz, Bol. Soc. Bras. Mat. Nova Sér. 25, 1-30 (1994; Zbl 0804.30023)] determined the dynamical property (semihyperbolicity) necessary and sufficient for the Fatou components of a polynomial to be John domains. The paper under review examines the more general question of the conditions for Fatou components of a rational map to be Hölder domains. A rational function $$F$$ is said to be C-E (Collet-Eckmann) with constants $$C_1>0$$, $$\lambda_1>1$$ if for any critical point $$c$$ whose forward orbit does not contain any other critical point and belongs to or accumulates on the Julia set we have $$|(F^n)'(Fc) |>C_1 \lambda^n_1$$. One also says that a rational function $$F$$ satisfies the second Collet-Eckmann condition C-E $$2(z)$$ at $$z$$ with constants $$C_2>0$$, $$\lambda_2>1$$ if for any preimage $$y\in F^{-n}z$$ we have $$|(F^n)'(y) |> C_2\lambda_2^n$$. A periodic Fatou component $$G$$ of $$F$$ is called C-E if for any (equivalently some) point $$z\in G$$ away from the critical orbits we have $$|(F^n)'(y) |>C \lambda^n$$ for any preimage $$y\in F^{-n} z\cap G$$ with constants $$C>0$$, $$\lambda >1$$. The multiplicity of a critical point $$c$$ is the order of $$c$$ as a zero of $$F(z)-F(c)$$. For simplicity we assume that no critical point belongs to another critical orbit. The main theorem states that rational C-E maps have neither Siegel discs, Herman rings, nor parabolic or Cremer points. The Fatou components of such maps are Hölder domains. Further (i) C-E implies CE $$2(c)$$ for the critical points $$c$$ of maximal multiplicity $$\mu_m$$ whose backward orbits do not contain any other critical points. (ii) C-E implies CE $$2(z)$$ for all $$z$$ away from the critical orbits. (iii) An attracting or superattracting Fatou component is Hölder if and only if it is C-E. In particular polynomial C-E sets are locally connected if they are connected and their Hausdorff dimension is strictly less than two.
Reviewer: I.N.Baker (London)

##### MSC:
 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
Fatou components
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