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.

动力学系统的理论研究了某个迭代程序描述的轨迹的长期行为，以及该阶段肖像取决于系统参数的方式。非常有趣的分形对象可能是此类系统的相位和参数图。主要研究者专注于该项目中简单二次方程描述的复杂的低维动力系统。尽管描述很简单，但已知这些系统表现出复杂的混沌行为，可作为出现在天体力学，流体动力学，生物学和自然科学的其他分支中的各种现象的良好模型。这项活动将导致对动态系统的小规模结构，在培训高素质的博士后和研究生的培训中，他们将在学术界和行业中运用技能，在真实和复杂动态的各个分支中的专家之间进行更广泛的互动，在出版一本书中，可以通过一本书出版一本书，可以帮助广泛的学生和研究社区在该领域中促进沟通，并在促进该领域，并促进一定的沟通，并促进一定的沟通，并促进一定的沟通，并促进一定的沟通，并促进一定的沟通，并促进促进界面的界限，并促进宣传的界限，并促进宣传界的界限，并促进宣传界面，并促进宣传界的界限，并促进宣传界面，并促进宣传的界限，并促进宣传的界限。动态网站：http // www.math.stonybrook/dynamics.in在这项研究中，研究了一项有关复杂低维动力学的几个相互缠绕的几何主题的广泛研究计划。首席研究人员将逐步从一维逐步过渡到二维世界。首席研究者将追求几个一维项目，该项目由重新归一化的想法统一，作为渗透到旨在完成其分类的动态物体的小规模结构中的强大工具。它们包括PACMAN重新归一化理论，Mandelbrot肢体的缩放和无限重新划分的二次多项式的先验界限。主要研究者将继续探索各种朱莉娅集合的准对象群的结构，并开发出一种新理论：施瓦茨在正交域中施瓦茨反射产生的动力学。在两个复杂的维度中，主要研究人员计划继续致力于耗散复杂的亨逊地图的动态。特定的主题将包括探索徘徊域的存在问题，搜索双曲线痛苦地图的新示例以及其结构的描述。首席研究人员还计划完成一本书的前两卷，“二次多项式的共形几何学和动态”。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估标准来评估的。