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Convergence of rays with rational argument in hyperbolic components – an illustration in transcendental dynamics. (English) Zbl 1351.37188
This paper is suggested by the work of A. Douady and J. H. Hubbard [C. R. Acad. Sci., Paris, Sér. I 294, 123–125 (1982; Zbl 0483.30014)] for the ray landing for quadratic family of complex dynamical systems. Here the author takes the following transcendental dynamical system $f_a(z) = a(e^z(z-1)+1), \qquad a\in {\mathbb C}^*.$ It is proved that every rational internal ray of the family will land at a boundary point of the main hyperbolic component. Moreover, the landing point is either a parabolic, a repelling, or a Misiurewicz parameter.
Compared with the quadratic family, the following two facts are remarkable. The first is that, due to the existence of the finite preimage of the the asymptotic value of $$f_a$$, the second case for landing points is new. The second is that the proof is based on the Caratheodory topology concerning with the convergence of sequences of marked domains. Because of these, this is an interesting and important paper.

MSC:
 37F15 Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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