Analytic Low Dimensional Dynamics: From Dimension One to Two

动力系统理论(离散时间)研究由某种迭代过程描述的轨迹的长期行为，以及这种相图的方式取决于系统的参数。非常有趣的分形对象(如Julia集和Mandelbrot集)可能会显示为此类系统的相图和参数图。在该项目中，PI将专注于用简单的二次方程描述的复杂和真实的低维动力系统。尽管描述很简单，但众所周知，这些系统表现出复杂的混沌行为，为天体力学、流体动力学、生物学和自然科学的其他分支中出现的各种现象提供了良好的模型。拟议的活动将有助于更深入地了解动力系统的小尺度结构，培养将在学术界和工业中应用技能的高素质博士后和研究生，促进真实和复杂动力学各分支专家之间的更广泛互动，出版一本书，帮助广大学生和研究团体获得该领域的背景，通过组织会议和科学计划，提供迷你课程，以及维护动力学网站(http//www.math.stonibrook/Dynamics)，促进该领域的交流。国际动力学会将就复杂和真实的低维动力学的几个相互交织的几何主题开展广泛的研究计划，从一维世界逐步过渡到二维世界。PI将研究耗散复Henon映射的动力学和典型耗散实Henon映射的吸引子。具体的主题包括探索游荡域的存在问题，建立拼图技术，以及研究半Cremer不动点附近的局部动力学。PI将继续追求由重整化思想统一的几个一维项目，重整化是穿透动力学对象的小尺度结构的强大工具，旨在完成它们的分类。它们包括Siegel重整化理论，Mandelbrot分支的尺度，以及本原无限可重整化的二次多项式的先验界。PI将完成《二次多项式的共形几何和动力学》一书的第一卷。

The theory of Dynamical systems (with discrete time) studies the long-term behavior of trajectories described by a certain iteration procedure, and the way this phase portrait depends on the parameters of the system. Very interesting fractal objects (like Julia sets and the Mandelbrot set) may appear as phase and parameter diagrams for such systems. In the project, the PI will focus on complex and real low-dimensional dynamical systems described by simple quadratic equations. Despite simplicity of the description, these systems are known to display complicated chaotic behavior serving as a good model for various phenomena that appear in celestial mechanics, fluid dynamics, biology, and other branches of natural science. The proposed activity will result in deeper insights into small scale structure of dynamical systems, in training of highly qualified postdocs and graduate students who will apply their skills in academia and industry, in broader interactions between experts in various branches of real and complex dynamics, in publishing a book that would help a broad student and research community to acquire background in the area, in promotion of communication in the field by organizing conferences and scientific programs, giving mini-courses, and maintaining a dynamics web site (http//www.math.stonybrook/dynamics).The PI will conduct a broad research program on several intertwined geometric themes of complex and real low-dimensional dynamics, making a gradual transition from the one-dimensional to the two-dimensional world. The PI will work on the Dynamics of dissipative complex Henon maps and attractors for typical dissipative real Henon maps. Specific themes include exploring the problem of existence of wandering domains, building up puzzle techniques, and the study of local dynamics near semi-Cremer fixed points. The PI will keep pursuing several one-dimensional projects unified by the idea of renormalization, a powerful tool of penetrating into small-scale structure of dynamical objects aimed towards completing their classification. They include the Siegel Renormalization Theory, scaling of Mandelbrot limbs, and a priori bounds for primitively infinitely renormalizable quadratic polynomials. The PI will finish the first volume of a book "Conformal Geometry and Dynamics of Quadratic Polynomials".