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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.
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
39B12 Iteration theory, iterative and composite equations
37H10 Generation, random and stochastic difference and differential equations
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