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Connectedness of Julia sets for a quadratic random dynamical system

Published online by Cambridge University Press:  02 December 2003

ZHIMIN GONG ,
WEIYUAN QIU  and
YING LI
ZHIMIN GONG
Affiliation:
Business School, Xiangtan University, Xiangtan 411105, People's Republic of China (e-mail: gggzm@vip.sina.com) and Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, People's Republic of China and Department of Mathematics, Xiangtan University, Xiangtan 411105, People's Republic of China
WEIYUAN QIU
Affiliation:
Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, People's Republic of China
YING LI
Affiliation:
Department of Mathematics, Xiangtan University, Xiangtan 411105, People's Republic of China
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Abstract

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For a sequence (cn) of complex numbers, the quadratic polynomials $f_{c_n}:=z^2+c_n$ and the sequence (Fn) of iterates $F_n:=f_{c_n}\circ\dotsb \circ f_{c_1}$ are considered. The Fatou set $\mathcal{F}(c_n)$ is defined as the set of all $z\in \hat{\mathbb{C}}: =\mathbb{C}\cup \{\infty\}$ such that (Fn) is normal in some neighbourhood of z, while the complement $\mathcal{J}(c_n)$ of $\mathcal{F}(c_n)$ (in $\hat{\mathbb{C}}$) is called the Julia set. In this paper we discuss the conditions for $\mathcal{J}(c_n)$ to be totally disconnected. A problem posed by Brück is solved.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 23 , Issue 6 , December 2003 , pp. 1807 - 1815
Copyright
2003 Cambridge University Press