Complex and Real Low Dimensional Dynamics

我们提出了一个关于复杂和真实低维动力学的几个相互交织的几何主题的研究计划，主要是关于黎曼球上的多项式动力学和耗散实数和复杂Henon族中的动力学。这些主题中的大多数都统一于重整化的思想，重整化是一种深入动态对象的小尺度结构的强大工具，旨在实现它们的完整分类。我们特别强调以下主题：Mandelbrot集合的局部连通性问题，Feigenbaum Julia正面积集和Siegel重整化理论，关于强耗散实Henon映射吸引子有限的Palis猜想，中等耗散复Henon映射的稳定性和分岔。所提议的活动将导致对动力系统的小尺度结构的更深入的了解，培养高素质的研究生和博士后，他们将把他们的技能应用于学术界和工业界，在真实和复杂动力学的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。

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.