Complex and Real Low Dimensional Dynamics

我们提出了一个研究计划的复杂和真实的低维动力学的几个相互交织的几何主题，主要是在黎曼球面上的多项式动力学和耗散的真实的和复杂的Henon家庭的动力学。这些主题中的大多数都是由重整化的思想统一起来的，重整化是一个强大的工具，可以渗透到动力学对象的小尺度结构中，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，正区域的Feigenbaum Julia集和Siegel重整化理论，强耗散真实的Henon映射吸引子有限性的Palis猜想，以及中等耗散复Henon映射的稳定性和分支，这些活动将使我们对动力系统的小尺度结构有更深入的了解，在高素质的研究生和博士后谁将在学术界和工业应用他们的技能的培训，在真实的和复杂的动力学的各个分支之间的专家更广泛的互动，在促进动力学领域和物理学和应用数学的相关领域之间的沟通。

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.