Analytic Low Dimensional Dynamics: From Dimension One to Two

动力系统（离散时间）理论研究由特定迭代过程描述的轨迹的长期行为，以及该相图取决于系统参数的方式。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位图和参数图。在该项目中，PI 将重点研究由简单二次方程描述的复杂且真实的低维动力系统。 尽管描述很简单，但这些系统显示出复杂的混沌行为，可以作为天体力学、流体动力学、生物学和自然科学其他分支中出现的各种现象的良好模型。 拟议的活动将导致对动力系统的小规模结构有更深入的了解，培训高素质的博士后和研究生，他们将在学术界和工业界应用他们的技能，真实和复杂动力学各个分支的专家之间更广泛的互动，出版一本帮助广大学生和研究界获得该领域背景的书，通过组织会议和科学项目、提供迷你课程和维护动态网站来促进该领域的交流(http//www.math.stonybrook/dynamics)。PI将针对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，逐步从一维世界过渡到二维世界。 PI 将致力于耗散复 Henon 贴图的动力学和典型耗散真实 Henon 贴图的吸引子。具体主题包括探索徘徊域的存在问题、建立谜题技术以及研究半 Cremer 不动点附近的局部动力学。 PI 将继续追求几个由重整化思想统一的一维项目，重整化是渗透到动态对象小规模结构的强大工具，旨在完成其分类。它们包括西格尔重整化理论、曼德尔布罗特肢的缩放以及原始无限可重整化二次多项式的先验界限。 PI将完成《共形几何和二次多项式动力学》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".