Levin, G.; Przytycki, F. External rays to periodic points. (English) Zbl 0854.30020 Isr. J. Math. 94, 29-57 (1996). Let \(P\) be a polynomial, \(\deg P \geq 2\). The basin of infinity is defined as \[ D_P = \bigl\{ z \in \mathbb{C} \cup \{\infty\} : P^n (z) = P \circ \cdots \circ P(z) \to \infty,\;n \to \infty \bigr\}. \] It is known that the Julia set \(J(P)\) is the closure of the repelling periodic points of \(P\) and coinsides with the boundary of \(D_P\). Let \(u(z)\) be the Green function of \(D_P\) with the pole at infinity. An external ray \(R\) of \(P\) is a trajectory of the vector field grad \(u\). The \(\lim_{{w \to \infty \atop w \in R}} {\arg w \over 2 \pi} \pmod 1\) exists and is called the external argument of \(R\). Let \(z \in J(P)\). The set of the arguments of all external rays having \(z\) as the end is denoted by \(\Lambda(z)\). The authors describe the set \(\Lambda (a)\) for repelling or parabolic periodic points for \(P\). The case of a simply connected \(D_P\) was investigated earlier by Douady and Hubbard (1985) considers the polynomial like mapping. This more general cases are investigated by the authors as well. Reviewer: A.F.Grishin (Khar’kov) Cited in 1 ReviewCited in 19 Documents MSC: 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 37E99 Low-dimensional dynamical systems Keywords:Julia set; accesiblee points; periodic points PDFBibTeX XMLCite \textit{G. Levin} and \textit{F. Przytycki}, Isr. J. Math. 94, 29--57 (1996; Zbl 0854.30020) Full Text: DOI References: [1] Ahlfors, L.; Beurling, A., Conformal invariants and function-theoretic nullsets, Acta Mathematica, 83, 101-129 (1950) · Zbl 0041.20301 [2] [CL] E. F. Collingwood and A. J. Lohwater,The Theory of Cluster Sets, Cambridge University Press, 1966. · Zbl 0149.03003 [3] [D] A. Douady,Informal talk at Durham Symposium, 1988. [4] Douady, A.; Hubbard, J. H., On the dynamics of polynomial-like mappings, Annales Scientifiques de l’École Normale Supérieure, 18, 287-343 (1985) · Zbl 0587.30028 [5] Douady, A.; Hubbard, J. H., Iteration des polynomes quadratiques complexes, Comptes Rendus de l’Académie des Sciences, Paris, Ser. I, 294, 123-126 (1982) · Zbl 0483.30014 [6] [E] A. Eremenko,Accessible fixed points in the Julia set, Manuscript (1990). [7] Eremenko, A.; Levin, G., On periodic points of polynomials, Ukrainskii Matematicheskii Zhurnal, 41, no. 11, 1467-1471 (1989) · Zbl 0686.30019 [8] Goluzin, G. M., Geometric theory of functions of a complex variable (1969), Providence, R.I.: American Mathematical Society, Providence, R.I. · Zbl 0183.07502 [9] Goldberg, L. R.; Milnor, J., Fixed points of polynomial maps. Part II. Fixed point portraits, Annales Scientifiques de l’École Normale Supérieure, 4e, 26, 51-98 (1993) · Zbl 0771.30028 [10] Goldberg, L., On the multiplier of a replling fixed point, Inventiones Mathematicae, 118, 85-108 (1994) · Zbl 0837.30020 [11] [K] J. Kiwi,Non-accessible critical points of Cremer polynomial, Preprint IMS 1995/2, SUNY at Stony Brook, 1995. [12] [LS] G. Levin and M. Sodin,Polynomials with disconnected Julia sets and Green maps, The Hebrew University of Jerusalem. Preprint No. 23, 1990/91. [13] [McM] C. McMullen,Complex Dynamics and Renormalizations, Princeton University Press, to appear. [14] [Mi] J. Milnor,Dynamics in one complex variable: Introductory lectures, Preprint IMS 1990/5, SUNY at Stony Brook. [15] [P-M] R. Perez-Marko,Topology of Julia sets and hedgehogs, Preprint 94-48, Université de Paris-Sud, 1994. [16] Przytycki, F., Riemann map and holomorphic dynamics, Inventiones Mathematicae, 85, 439-455 (1986) · Zbl 0616.58029 [17] Przytycki, F., Accessibility of typical points for invariant measures of positive exponents for iterations of holomorphic maps, Fundamenta Mathematicae, 144, 259-278 (1994) · Zbl 0812.58058 [18] Sario, L.; Nakai, M., Classification Theory of Riemann Surfaces (1970), Berlin: Springer-Verlag, Berlin · Zbl 0199.40603 [19] [Y] J.-Ch. Yoccoz,Sur la taille des membres de l’ensemble de Mandelbrot, Manuscript (1986). [20] [Th] W. Thurston,The combinatorics of iterated rational maps, Preprint 1984. 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.