Themes in Holomorphic Low-Dimensional Dynamics

动力系统理论研究由某种迭代过程描述的轨迹的长期行为，而这种相图的方式取决于系统的参数。对于这样的系统，非常有趣的分形体可能会以相图和参数图的形式出现。本项目主要研究用简单二次方程描述的复杂低维动力系统。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。这项活动将导致对动力系统小尺度结构的更深入的了解，对将在学术界和工业中应用他们的技能的高素质博士后和研究生的培训，在真实和复杂动力学的不同分支的专家之间更广泛的互动，在出版一本书将帮助广大的学生和研究社区获得该领域的背景，在通过组织会议和科学计划促进该领域的交流，提供迷你课程，以及维护动力学网站：http//www.math.stonibrook/Dynamics。在这项研究中，关于复杂低维动力学的几个相互交织的几何主题的广泛的研究计划被调查。首席研究员将从一维世界逐步过渡到二维世界。首席研究人员将追求几个由重整化思想统一的一维项目，作为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Pacman重整化理论，Mandelbrot分支的比例，以及无限可重整化的二次多项式的先验界。主要的研究人员将继续探索各类Julia集的拟对称群的结构，并发展一种新的理论：由求积域中的Schwarz反射产生的动力学。在两个复数维中，首席研究员计划继续研究耗散复Henon映射的动力学。具体的主题将包括探索游荡域的存在问题，寻找双曲Henon映射的新例子，以及对其结构的描述。首席研究员还计划完成《二次多项式的保角几何和动力学》一书的前两卷。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.