Themes in Holomorphic Low-Dimensional Dynamics

全纯低维动力学主题

基本信息

批准号：
1901357
负责人：
Mikhail Lyubich
金额：
$ 31万
依托单位：
SUNY at Stony Brook
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-15 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901357&HistoricalAwards=false
关键词：
Themes Holomorphic Low Dimensional Dynamics

项目摘要

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.

动力系统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可能会以此类系统的相位图和参数图的形式出现。在该项目中，主要研究人员重点研究由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，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。该活动将导致对动力系统小规模结构的更深入了解，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，出版这本书将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、提供迷你课程和维护动态网站来促进该领域的交流：http//www.math .stonybrook/dynamics.在此研究针对复杂低维动力学的几个相互交织的几何主题进行了广泛的研究计划。首席研究员将逐渐从一维世界过渡到二维世界。首席研究员将开展几个一维项目，这些项目通过重正化的思想统一起来，作为渗透到动态对象的小规模结构的强大工具，旨在完成其分类。它们包括 Pacman 重整化理论、Mandelbrot 肢体的缩放以及无限重整二次多项式的先验界限。首席研究员将继续探索各类 Julia 集的拟对称群的结构，并发展一种新理论：正交域中施瓦茨反射产生的动力学。在两个复杂的维度中，首席研究员计划继续研究耗散复数 Henon 贴图的动力学。具体主题将包括探索徘徊域的存在问题、寻找双曲Henon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 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。