On the stochastic bifurcations regarding random iterations of polynomials of the form $z^{2} + c_{n}$

TL;DR

This paper investigates how randomness in quadratic polynomial iterations affects the connectedness of Julia sets and identifies bifurcation parameters, revealing that at certain points, most Julia sets become totally disconnected even with small randomness.

Contribution

It provides new insights into the relationship between bifurcation radius and Julia set connectedness in random polynomial iterations, including quantitative bifurcation estimates.

Findings

Almost all Julia sets are totally disconnected at c=-1 with small radius r.

Connectedness of Julia sets depends on the bifurcation radius.

Quantitative estimates of bifurcation parameters are established.

Abstract

In this paper, we consider random iterations of polynomial maps $z^2 +c_n$ where $c_n$ are complex-valued independent random variables following the uniform distribution on the closed disk with center $c$ and radius $r$. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. Second, we investigate the bifurcation of our random iterations and give quantitative estimates of bifurcation parameters. In particular, we prove that for the central parameter $c = -1$, almost every random Julia set is totally disconnected with much smaller radial parameters $r$ than expected. We also introduce several open questions worth discussing.