Complex and Real Low Dimensional Dynamics

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方案，主要是关于Riemann球面上的多项式动力学和耗散实和复Henon族中的动力学。这些主题中的大多数都被重整化的思想统一起来，重整化是渗透到动力学对象的小尺度结构的强大工具，目的是为了实现它们的完全分类。我们特别强调了以下主题：Mandelbrot集的局部连通性问题，正区域的Feigbaum Julia集和Siegel重整化理论，关于强耗散实Henon映射吸引子的有限的Palis猜想，以及中等耗散复Henon映射的稳定性和分叉。所提出的活动将导致对动力系统小尺度结构的更深入的了解，在培养将在学术和工业中应用他们的技能的高素质的研究生和博士后，在真实的和复杂的动力学的各个分支的专家之间的更广泛的互动，促进动力学领域与相关的物理和应用数学领域之间的交流。

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.