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On the preimages of parabolic periodic points. (English) Zbl 0982.37038

Let \(R\) be a rational function of degree at least two and \(J(R)\) is the Julia set corresponding to dynamics generated by \(R\). The author gives a relation between a geometrical property of \(J(R)\) and dynamical properties of \(R\). The author characterizes the preimages of parabolic periodic points in the family of rational functions for which the Julia set does not lie on a smooth Jordan curve and does not contain any recurrent critical point.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37G10 Bifurcations of singular points in dynamical systems
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
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