HOLOMORPHIC DYNAMICS AND RELATED THEMES

全态动力学及相关主题

• 批准号: 2247613
• 负责人: Mikhail Lyubich
• 金额: $ 36.94万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247613&HistoricalAwards=false
• 关键词: HOLOMORPHIC; DYNAMICS; RELATED; THEMES

The theory of dynamical systems studies the long-term behavior of trajectories described by iteration procedures, and how such behavior depends on the parameters of the system. Intricate fractal objects (like Julia sets and the Mandelbrot set) may appear as phase and parameter diagrams for such systems. This project focuses on complex and real low-dimensional dynamical systems described by simple quadratic equations. Despite the simplicity of the model, such systems are known to display complicated chaotic behavior indicative of various phenomena appearing in celestial mechanics, fluid dynamics, statistical mechanics, biology, and other branches of natural science. The proposed activity will result in deeper insights into the small scale structure of dynamical systems, in the training of highly qualified postdoctoral fellows and graduate students, in broader interactions between senior and junior experts in various branches of real and complex dynamics, and in the preparation of a book to assist the research community in acquiring background in the area. In addition, the Principal Investigator will facilitate communication within the field through the organization of international conferences and scientific programs and by maintaining a dynamics-related web site.The project addresses several geometric themes within complex low-dimensional dynamics, making a gradual transition from the one-dimensional to the two-dimensional world. In connection with dynamics in one dimension, renormalization will be investigated as a unifying and powerful tool for elucidating the small-scale structure of dynamical objects. Specific topics under consideration include a semi-local theory of neutral maps and a priori bounds for infinitely renormalizable quadratic polynomials with applications to the problem of local connectivity of the Mandelbrot set. Other topics of study include the dynamics generated by Schwarz reflections in quadrature domains and the dynamics of dissipative complex Henon maps. In connection with the latter, specific themes include the construction of two-dimensional examples of real and complex wild attractors and the development of a general theory of unimodal Henon maps. The project will also explore applications of higher dimensional holomorphic dynamics to the spectral theory of self-similar groups. Finally, the principal investigator will continue to work on a multi-volume book on the conformal geometry and dynamics of quadratic polynomials.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

动力系统理论研究迭代过程描述的轨迹的长期行为，以及这种行为如何取决于系统的参数。复杂的分形对象（如 Julia 集和 Mandelbrot 集）可能会显示为此类系统的相位图和参数图。该项目重点研究由简单二次方程描述的复杂且真实的低维动力系统。尽管模型很简单，但众所周知，此类系统会表现出复杂的混沌行为，表明天体力学、流体动力学、统计力学、生物学和自然科学的其他分支中出现的各种现象。拟议的活动将更深入地了解动力系统的小规模结构，培养高素质的博士后研究员和研究生，在真实和复杂动力学的各个分支的高级和初级专家之间进行更广泛的互动，并在准备过程中帮助研究界获取该领域背景的一本书。此外，首席研究员将通过组织国际会议和科学计划以及维护与动力学相关的网站来促进该领域内的交流。该项目解决了复杂的低维动力学中的几个几何主题，逐步从一维世界到二维世界。与一维动力学相关，重整化将被研究作为阐明动态对象的小尺度结构的统一且强大的工具。正在考虑的具体主题包括中性映射的半局部理论和无限可重整二次多项式的先验界限以及曼德尔布罗特集局部连通性问题的应用。其他研究主题包括正交域中施瓦茨反射产生的动力学和耗散复数 Henon 映射的动力学。与后者相关的具体主题包括真实和复杂的野生吸引子的二维示例的构建以及单峰Henon图的一般理论的发展。该项目还将探索高维全纯动力学在自相似群谱理论中的应用。最后，首席研究员将继续撰写一本关于共形几何和二次多项式动力学的多卷书。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。