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Stability of Julia sets for a quadratic random dynamical system

Authors:	Gong Zhimin Qiu Weiyuan and Wang Jian

Institution:	1. Lab of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, China;Department of Mathematics, Xiangtan University, Xiangtan 411105, China 2. Lab of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, China 3. Department of Mathematics, Xiangtan University, Xiangtan 411105, China;Business School, Xiangtan University, Xiangtan 411105, China

Abstract:	For a sequence (cⁿ) of complex numbers, the quadratic polynomials f_cn:= z² + C_n and the sequence (F_n) of iterates F_n: = f_cn ο ⋯ ο f_c1 are considered. The Fatou set F(C_n) is defined as the set of all $z \in \hat {\mathbb{C}}: = {\mathbb{C}} \cup \left\{ \infty \right\}$ such that (Fⁿ) is normal in some neighbourhood of z, while the complement J(C_n) of F(c_n) (in $\hat {\mathbb{C}}$ ) is called the Julia set. The aim of this paper is to study the stability of the Julia set J(C_n) in the case where (c_n) is bounded. A problem put forward by Brück is solved.

Keywords:	Fatou set Julia set random iterates stability
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