Random Dynamics of a Family of Cubic Polynomials

TL;DR

This paper investigates the complex dynamics of randomly iterated cubic polynomials, revealing conditions for Julia set disconnectedness and showing that such properties are prevalent under certain probabilistic assumptions.

Contribution

It establishes the density of parameter sequences with totally disconnected Julia sets and constructs examples where disconnectedness occurs without hyperbolicity.

Findings

The set of sequences with totally disconnected Julia sets is dense.

Almost every sequence under certain distributions yields a totally disconnected Julia set.

Examples show disconnected Julia sets can occur without hyperbolicity.

Abstract

In this work, we study the non-autonomous dynamics generated by random iterations of the cubic family of the form $z^3 + cz$. The parameter sequence is chosen randomly from a bounded Borel subset of $\mathbb{C}$. We investigate topological properties of the corresponding Julia sets, with particular emphasis on conditions leading to total disconnectedness. We prove that the set of parameter sequences for which the Julia set is totally disconnected is dense in the parameter space. We also construct examples where the Julia set is totally disconnected but the associated non-autonomous system is not hyperbolic. Finally, under suitable probabilistic assumptions on the parameter distribution, we show that almost every sequence produces a totally disconnected Julia set.