zbMATH — the first resource for mathematics

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)

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
Full Text: DOI