Themes in Holomorphic Low-Dimensional Dynamics

动力系统理论研究由一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可能会出现这样的系统的相图和参数图。主要研究者在这个项目中专注于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。该活动将导致更深入地了解动力系统的小尺度结构，培养高素质的博士后和研究生，他们将在学术界和工业界应用自己的技能，在真实的和复杂动力学的各个分支专家之间更广泛的互动，出版一本书，这将有助于广泛的学生和研究社区获得该领域的背景，通过组织会议和科学计划，提供迷你课程，并维护一个动力学网站：http//www.math.stonybrook/dynamics，促进该领域的交流。首席研究员将从一维世界逐渐过渡到二维世界。 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小尺度结构的动力学对象，旨在完成他们的分类。它们包括Pacman重整化理论，Mandelbrot肢体的缩放，以及无限可重整化二次多项式的先验界。 主要研究者将继续探索各种类别的Julia集的quasiqueries群的结构，并开发一个新的理论：由正交域中的施瓦茨反射产生的动力学。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映射的动力学。具体的主题将包括探索流浪域的存在问题，寻找双曲Henon映射的新例子，并描述其结构。主要研究者还计划完成一本书的前两卷“二次多项式的共形几何和动力学”。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.