Themes in Holomorphic Low-Dimensional Dynamics

动力系统理论研究的是由一定的迭代过程描述的轨迹的长期行为，而这种阶段描述的方式取决于系统的参数。这种系统的相位图和参数图可能会出现非常有趣的分形对象。本项目主要研究由简单二次方程描述的复杂低维动力系统。尽管描述简单，但众所周知，这些系统显示出复杂的混沌行为，作为天体力学、流体动力学、生物学和其他自然科学分支中出现的各种现象的良好模型。这项活动将使人们对动力系统的小规模结构有更深入的了解，将培养高素质的博士后和研究生，使他们将把他们的技能应用于学术界和工业界，在真实和复杂动力学的各个分支的专家之间进行更广泛的互动，出版一本有助于广大学生和研究团体获得该领域背景的书。为了促进该领域的交流，通过组织会议和科学计划，提供迷你课程和维护一个动力学网站：http//www.math.stonybrook/dynamics.In这项研究是一个广泛的研究计划，研究了复杂低维动力学的几个相互交织的几何主题。首席研究员将从一维世界逐步过渡到二维世界。首席研究员将追求几个一维项目统一的重整化的想法，作为一个强大的工具，渗透到动态对象的小尺度结构，旨在完成他们的分类。它们包括吃豆人重整化理论，Mandelbrot分支的缩放，以及无限重整化二次多项式的先验界。首席研究员将继续探索各种Julia集合的准对称群结构，并发展一个新的理论：正交域中Schwarz反射产生的动力学。在两个复杂的维度上，首席研究员计划继续研究耗散复杂Henon图的动力学。具体的主题将包括探索漫游域的存在问题，寻找双曲Henon地图的新例子，以及对其结构的描述。首席研究员还计划完成《共形几何和二次多项式动力学》一书的前两卷。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Theory of Dynamical systems 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 may appear as phase and parameter diagrams for such systems. The principal investigator focuses on complex low-dimensional dynamical systems described by simple quadratic equations in this project. 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 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.In this research a broad research program on several intertwined geometric themes of complex low-dimensional dynamics is investigated. The principal investigator will make a gradual transition from the one-dimensional to the two-dimensional world. The principal investigator will pursue several one-dimensional projects unified by the idea of renormalization as a powerful tool of penetrating into small-scale structure of dynamical objects aimed towards complete their classification. They include the Pacman Renormalization Theory, scaling of Mandelbrot limbs, and a priori bounds for infinitely renormalizable quadratic polynomials. The principal investigator will keep exploring the structure of the group of quasisymmetris for various classes of Julia sets and develop a new theory: the dynamics generated by Schwarz reflections in quadrature domains. In two complex dimensions, the principal investigator plans to keep working on the dynamics of dissipative complex Henon maps. Specific themes will include exploring the problem of existence of wandering domains, search for new examples of hyperbolic Henon maps, and description of their structure. The principal investigator also plans to finish the first two volumes of a book "Conformal Geometry and Dynamics of Quadratic Polynomials".This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.