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Julia and John. (English) Zbl 0804.30023
Let $$P$$ be a polynomial of degree $$\geq 2$$ with Julia set $$J$$ and basin at infinity $$A_ \infty$$ $$(J = \partial A_ \infty)$$. Denote by $$B(x,\varepsilon)$$ the disc $$| z - x | < \varepsilon$$ and by $$B_ n (x,\varepsilon)$$ any connected component of $$P^{-n} (B(x, \varepsilon))$$. Then $$P^ n$$ is a proper map of $$B_ n (x, \varepsilon)$$ onto $$B(x, \varepsilon)$$. $$P$$ is called semi-hyperbolic, if there exist $$\varepsilon > 0$$ and $$D \in \mathbb{N}$$ such that, for any $$x \in J$$ and $$n \in \mathbb{N}$$, $$P^ n$$ has degree $$d_ n (B_ n (x, \varepsilon)) \leq D$$ on $$B_ n (x, \varepsilon)$$. Note that subhyperbolic polynomials are semi-hyperbolic, and $$D=1$$ in the hyperbolic case. In the paper under review several important statements are proved, all of which are equivalent to semi-hyperbolicity:
(a) $$A_ \infty$$ is a John domain [and any bounded stable domain is a quasidisc].
(b) There exist $$\varepsilon > 0$$, $$0 < \theta < 1$$, and $$D \in \mathbb{N}$$ such that for $$x \in J$$ and $$n \in \mathbb{N}$$ $d_ n (B_ n (x, \varepsilon)) \leq D \qquad \text{and} \qquad \text{diam} B_ n (x, \varepsilon) = O (\theta^ n).$ (c) If $$x \in J$$, $$r>0$$ and $$n=n(x,r)$$ is the first integer such that $$P^ n (J \cap B (x,r)) = J$$, then $$P^ n$$ has degree $$\leq D_ 0$$ on $$B(x,2r)$$, where $$D_ 0 \in \mathbb{N}$$ is independent of $$x$$ and $$r$$.
(d) $$P$$ has no parabolic periodic points, and for all $$c \in J$$ with $$P'(c) = 0$$, it holds that $$c \notin \omega (c) = \overline {\cup_{n \geq 1} P^ n (c)}$$.
It has been previously shown by R. Mañé that (d) implies semi- hyperbolicity [Bol. Soc. Bras. Mat., Nova Ser. 24, 1-11 (1993; Zbl 0781.30023)]. [Reviewers remark: It seems more naturally to take (d) as definition of semi-hyperbolicity].

##### MSC:
 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
##### Keywords:
John domain; Julia set; semi-hyperbolic
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##### References:
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