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Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps: Geometric coding trees technique. (English) Zbl 0817.58033
Let \(f\) be a rational self-map of the Riemann sphere and let \(J(f)\) denote its Julia set. A set \(A\) is called an immediate basin of attraction to a sink or parabolic periodic point \(p\) of period \(m\) if \(A\) is a component of \(\overline{\mathbb{C}} \setminus J(f)\) such that \(f^{nm}_{| A} \rightarrow p\) as \(n \to \infty\) and \(p \in A\) for \(p\) attracting, and \(p \in \partial A\) for \(p\) parabolic.
The main result of the paper states that if \(A\) is a basin of immediate attraction for a periodic attracting or parabolic point for a rational self-map \(f\) of the Riemann sphere then the periodic points contained in the boundary of \(A\) are dense in the boundary of \(A\).
Reviewer: M.Mrozek (Krakow)

MSC:
37F99 Dynamical systems over complex numbers
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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