论文标题
在随机分叉上，有关$ z^{2} + c_ {n} $的多项式随机迭代
On the stochastic bifurcations regarding random iterations of polynomials of the form $z^{2} + c_{n}$
论文作者
论文摘要
在本文中，我们考虑了多项式映射的随机迭代$ z^2 +c_n $，其中$ c_n $是复杂值的独立随机变量，遵循了封闭磁盘上的均匀分布，带有中心$ c $ and radius $ r $。本文的目的是双重的。首先，我们研究随机朱莉娅集的（dis）连接性。在这里，我们揭示了分叉半径与随机朱莉娅集的连接性之间的关系。其次，我们研究了随机迭代的分叉，并给出了分叉参数的定量估计。特别是，我们证明，对于中央参数$ c = -1 $，几乎每个随机的朱莉娅集合都完全断开了径向参数$ r $比预期的要小得多。我们还介绍了值得讨论的几个公开问题。
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.
