Random iterations of polynomials of the form z(2)+c(n): connectedness of Julia sets

For a sequence (c(n)) of complex numbers we consider the quadratic polynomi als 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 a im of this paper is to study the connectedness of the Julia set J((cn)) pro vided 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 t o 1/4 while it is almost surely disconnected if \c(n)\ less than or equal t o delta for some delta > 1/4.