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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
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