Total disconnectedness of Julia sets of random quadratic polynomials

Abstract

For a sequence of complex parameters $\{c_n\}$ we consider the compositions of functions $f_{c_n}(z)=z^2+c_n$, 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 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 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 $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

Cited by in corpus (1)

• On the stochastic bifurcations regarding random iterations of polynomials of the form $z^2+c_n$