Total disconnectedness of Julia sets of random quadratic polynomials
Abstract
For a sequence of complex parameters \cn\ we consider the compositions of functions fcn (z) = z2 + cn, which is 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 Br\"uck, B\"uger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values cn are chosen randomly from a large disk. 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 disks, in particular if one picks cn randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.