Complex and Real Low Dimensional Dynamics

复杂而真实的低维动力学

• 批准号: 1301602
• 负责人: Mikhail Lyubich
• 金额: $ 43.5万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301602&HistoricalAwards=false
• 关键词: Complex; Real; Low; Dimensional; Dynamics

We propose a research program on several intertwined geometric themes of complex and real low-dimensional dynamics, primarily on the polynomial dynamics on the Riemann sphere and the dynamics in the dissipative real and complex Henon family. Most of these themes are unified by the idea of renormalization as a powerful tool of penetrating into small-scale structure of dynamical objects, aimed towards their complete classification. We particularly emphasize the following themes: the problem of local connectivity of the Mandelbrot set, Feigenbaum Julia sets of positive area and Siegel Renormalization Theory, the Palis Conjecture on finiteness of attractors for strongly dissipative real Henon maps, and stability and bifurcations for moderately dissipative complex Henon maps.The proposed activity will result in deeper insights into small scale structure of dynamical systems, in training of highly qualified graduate students and postdocs who will apply their skills in academia and industry, in broader interactions between experts in various branches of real and complex dynamics, in promotion of communication between the field of dynamics and related areas of physics and applied mathematics.

我们提出了一个关于复数和实数低维动力学的几个相互交织的几何主题的研究计划，主要是黎曼球上的多项式动力学以及耗散实数和复数 Henon 族中的动力学。这些主题中的大多数都通过重整化的思想统一起来，作为渗透到动态对象的小规模结构的强大工具，旨在实现它们的完整分类。我们特别强调以下主题：Mandelbrot 集的局部连通性问题、Feigenbaum Julia 正面积集和 Siegel 重整化理论、强耗散实 Henon 映射吸引子有限性的 Palis 猜想以及中耗散复形的稳定性和分岔Henon 地图。拟议的活动将更深入地了解动力系统的小规模结构，培训高素质的研究生和博士后，他们将应用他们的技能学术界和工业界，在真实和复杂动力学各个分支的专家之间进行更广泛的互动，促进动力学领域与物理和应用数学相关领域之间的交流。