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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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