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External rays to periodic points. (English) Zbl 0854.30020
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.

MSC:
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37E99 Low-dimensional dynamical systems
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