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.

中文翻译：

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关于形式多项式随机迭代的随机分岔

在本文中，我们考虑多项式映射的随机迭代 $z^{2} + c_{n}$ ， 在哪里 $c_{n}$ 是复值独立随机变量，遵循以中心为中心的闭圆盘上的均匀分布C和半径r。本文的目的是双重的。首先，我们研究随机 Julia 集的（断开）连通性。在这里，我们揭示了随机 Julia 集的分叉半径和连通性之间的关系。其次，我们研究随机迭代的分叉并给出分叉参数的定量估计。特别地，我们证明对于中心参数 $c = -1$ ，几乎每个随机 Julia 集都与更小的径向参数完全断开r比预期的。我们还介绍了几个值得讨论的开放性问题。