For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

For a bounded sequence ω = (λn) ∞ n=1 of positive real numbers we consider the exponential functions f λn (z) = λne z and the compositionsThe definitions of Julia and Fatou sets are naturally generalized to this setting. We study how the Julia set depends on the sequence ω. Among other results, we prove that for the sequence λn = 1/e + 1/n p with p < 1/2, the Julia set is the whole plane.