# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  [B] A. Beurling.The collected works of Arne Beurling. Birkhäuser I (1989). · Zbl 0732.01042  [C,G] L. Carleson and T. W. Gamelin.Complex dynamics. Springer-Verlag, Universitext: Tracts in Mathematics (1993). · Zbl 0782.30022  [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  [D,H] A. Douady and J. Hubbard.Études dynamique des polynômes complexes. Publ. Math. Orsay (1984-02, 1985-04). · Zbl 0552.30018  [H] M. Herman, personal communication.  [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).  [Ma] R. Mañé.On a lemma of Fatou. Bol. Soc. Bras. Mat. 24 (1993), 1-12. · Zbl 0781.30023 · doi:10.1007/BF01231694  [Mi] M. Misiurewicz.Absolutely continuous measures for certain maps of an interval. Publ. I.H.E.S. 53 (1981), 17-52. · Zbl 0477.58020  [N,V] R. Näkki and J. Väisälä.John disks. Expositiones Math. (1991), 3-43.  [P] Ch. Pommerenke.Uniformly perfect sets and the Poincaré metric. Arch. Math. 32 (1979), 192-199. · Zbl 0393.30005 · doi:10.1007/BF01238490
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.