On the random Julia sets of the family of polynomials $z^n+c_m{z}^k$

Abstract

For a sequence of complex parameters $\omega =(c_m)$ we consider the non-autonomous composition of functions $P_{c_m}(z)=z^n+c_m{z}^k$ with $n>k\ge 2$. The definitions of Julia, ${\mathcal{J}}_{\omega }$, and Fatou, ${\mathcal{F}}_{\omega }$, sets are naturally generalized to this setting. In this paper, we study the connectedness and some dynamic properties of the Julia set provided that the sequence $\omega$ is randomly chosen from some bounded Borel set $W\subset \mathbb{C}$.