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Stability of Julia sets for a quadratic random dynamical system. (English) Zbl 1099.37038
Summary: For a sequence \((c_n)\) of complex numbers, the quadratic polynomials \(f_{c_n}: =z^2+c_n\) and the sequence \((F_n)\) of iterates \(F_n:=f_{c_n}\circ\cdots \circ f_{c_1}\) are considered. The Fatou set \({\mathcal F}(c_n)\) is defined as the set of all \(z\in\check\mathbb{C}:=\mathbb{C} \cup\{\infty\}\) such that \((F_n)\) is normal in some neighbourhood of \(z\), while the complement \({\mathcal J}(c_n)\) of \({\mathcal F}(c_n)\) (in \(\check \mathbb{C})\) is called the Julia set. The aim of this paper is to study the stability of the Julia set \({\mathcal J}(c_n)\) in the case where \((c_n)\) is bounded. A problem put forward by R. Brück [Ergodic Theory Dyn. Syst. 19, 1221–1231 (1999; Zbl 0942.37041) and J. Lond. Math. Soc., II. Ser. 61, 462–470 (2000; Zbl 1033.37026)] is solved.
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
39B12 Iteration theory, iterative and composite equations
37H10 Generation, random and stochastic difference and differential equations
Full Text: DOI
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