Analytic Low Dimensional Dynamics: From Dimension One to Two

动态系统理论（离散时间）研究由特定迭代过程描述的轨迹的长期行为，并且这种相图取决于系统的参数。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将专注于由简单的二次方程描述的复杂和真实的低维动力系统。 尽管描述简单，但已知这些系统显示复杂的混沌行为，作为天体力学、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的良好模型。 拟议的活动将导致更深入地了解小规模结构的动力系统，在培训高素质的博士后和研究生谁将适用于他们的技能在学术界和工业界，在更广泛的互动专家在各分支的真实的和复杂的动力学，在出版一本书，这将有助于广泛的学生和研究社区获得背景，在该地区，通过组织会议和科学计划、开设小型课程和维护动态网站，促进该领域的交流（http//www.math.stonybrook/dynamics）。PI将对复杂和真实的低维动力学的几个交织的几何主题进行广泛的研究计划，从一维世界逐渐过渡到二维世界。PI将研究耗散复Henon映射的动力学和典型耗散真实的Henon映射的吸引子。具体的主题包括探索问题的存在游荡域，建立拼图技术，并研究局部动力学附近的半克雷默不动点。PI将继续追求由重整化思想统一的几个一维项目，这是一个强大的工具，可以渗透到动力学对象的小尺度结构中，旨在完成它们的分类。它们包括西格尔重整化理论，标度的曼德尔布罗特肢体，和先验界的constitutively无限renormalizable二次多项式。PI将完成一本书的第一卷“二次多项式的共形几何和动力学”。

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".