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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].

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
Full Text: DOI
[1] [B] A. Beurling.The collected works of Arne Beurling. Birkhäuser I (1989). · Zbl 0732.01042
[2] [C,G] L. Carleson and T. W. Gamelin.Complex dynamics. Springer-Verlag, Universitext: Tracts in Mathematics (1993). · Zbl 0782.30022
[3] [C,J] L. Carleson and P.W. Jones.On coefficient problems for univalent functions and conformal dimensions. Duke Math. J. 66 (1992), 169-206. · Zbl 0765.30005 · doi:10.1215/S0012-7094-92-06605-1
[4] [D,H] A. Douady and J. Hubbard.Études dynamique des polynômes complexes. Publ. Math. Orsay (1984-02, 1985-04). · Zbl 0552.30018
[5] [H] M. Herman, personal communication.
[6] [J] P.W. Jones.On removable sets for Sobolev spaces in the plane, in Conference in honor of E.M. Stein, Princeton University Press (1993).
[7] [Ma] R. Mañé.On a lemma of Fatou. Bol. Soc. Bras. Mat. 24 (1993), 1-12. · Zbl 0781.30023 · doi:10.1007/BF01231694
[8] [Mi] M. Misiurewicz.Absolutely continuous measures for certain maps of an interval. Publ. I.H.E.S. 53 (1981), 17-52. · Zbl 0477.58020
[9] [N,V] R. Näkki and J. Väisälä.John disks. Expositiones Math. (1991), 3-43.
[10] [P] Ch. Pommerenke.Uniformly perfect sets and the Poincaré metric. Arch. Math. 32 (1979), 192-199. · Zbl 0393.30005 · doi:10.1007/BF01238490
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