Journal article

Publication year: 1999

Pages: 1221-1231

Journal: Ergodic Theory and Dynamical Systems

Volume number: 19

ISSN: 0143-3857

DOI Link: https://doi.org/10.1017/S0143385799141658

Publisher: Cambridge University Press

Abstract: 
For a sequence (c(n)) of complex numbers we consider the quadratic polynomials f(cn) (z) := z(2) + c(n) and the sequence (F-n) of iterates F-n := f(cn) o ... o f(c1). The Fatou set F-(cn) is by definition the set of all z is an element of (C) over cap such that (F-n) is normal in some neighbourhood of z, while the complement of F-(cn) is called the Julia set J((cn)). The aim of this paper is to study the connectedness of the Julia set J((cn)) provided that the sequence (c(n)) is bounded and randomly chosen. For example, we prove a necessary and sufficient condition for the connectedness of J((cn)) which implies that J((cn)) is connected if \c(n)\ less than or equal to 1/4 while it is almost surely disconnected if \c(n)\ less than or equal to delta for some delta > 1/4.

Harvard Citation style: Brück, R., Büger, M. and Reitz, S. (1999) Random iterations of polynomials of the form z²+c_n:: connectedness of Julia sets, Ergodic Theory and Dynamical Systems, 19, pp. 1221-1231. https://doi.org/10.1017/S0143385799141658

APA Citation style: Brück, R., Büger, M., & Reitz, S. (1999). Random iterations of polynomials of the form z²+c_n:: connectedness of Julia sets. Ergodic Theory and Dynamical Systems. 19, 1221-1231. https://doi.org/10.1017/S0143385799141658