发表机构
Technische Universität Braunschweig; Liberality Research; Max Planck Institute for Solar System Research; Tokyo Metropolitan University（布伦瑞克工业大学; Liberality研究; 马克斯·普朗克太阳系统研究所; 东京都立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过引入实变形参数，证明可积极限下Julia集与反Julia集发生临界碰撞并湮灭，类似宇宙Big Crunch，并解析刻画了前反Julia集的渐近行为。
AI 中文摘要
Julia集定义为混沌系统中排斥周期点的闭包。为什么这种结构不出现在可积系统中？在本文中，我们通过证明周期方程发散点的闭包存在性来回答这个问题，并将其命名为“反Julia集”。我们还分别称取闭包之前的集合为前Julia集和前反Julia集。我们通过引入实变形参数$a$，考虑一个在可积与非可积动力学之间插值的复映射，来说明这一转变机制。对于$0<a\le 1/2$，我们证明在复平面上Julia集与反Julia集重合，尽管前Julia集与前反Julia集保持完全不相交。在可积极限$a\to 0$处，这两个对偶结构经历一次临界碰撞并随后湮灭，类似于宇宙学中的Big Crunch现象。另一方面，当$1/2<a<1$时，前反Julia集的边界包含Julia集。我们从解析上刻画了这些现象，特别关注前反Julia集在接近可积极限时的渐近行为，并提供数值可视化以阐明这一Big Crunch现象背后的机制。
英文摘要
The Julia set is defined by the closure of repelling periodic points in chaotic systems. Why does this structure not appear in integrable systems? In this paper, we address this question by demonstrating the existence of the closure of divergences of the periodic equations, which we designate as the "anti-Julia set." We also call the sets before taking the closures the pre-Julia and pre-anti-Julia sets, respectively. We illustrate the transition mechanism by considering a complex map that interpolates between integrable and non-integrable dynamics, by introducing a real deformation parameter $a$. For $0<a\le 1/2$, we show that the Julia set and the anti-Julia set coincide in the complex plane, although the pre-Julia and pre-anti-Julia sets remain completely disjoint. At the integrable limit $a\to 0$, these two dual structures undergo a critical collision and subsequent annihilation, reminiscent of a cosmological Big Crunch. When $1/2<a<1$, on the other hand, the boundary of the pre-anti-Julia set includes the Julia set. We analytically characterize these phenomena, focusing in particular on the asymptotic behavior of the pre-anti-Julia set as it approaches the integrable limit, and provide numerical visualizations that elucidate the underlying mechanisms of this Big Crunch phenomenon.
Comments16 pages, 5 figures
