Random iterations of polynomials of the form z2+cn:: connectedness of Julia sets

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.