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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
##### Keywords:
random iterates; Fatou set
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##### References:
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