HOLOMORPHIC DYNAMICS AND RELATED THEMES

动力系统理论研究由迭代过程描述的轨迹的长期行为，以及这种行为如何依赖于系统的参数。复杂的分形对象（如Julia集合和Mandelbrot集合）可能会以这样的系统的相位图和参数图的形式出现。这个项目的重点是用简单的二次方程描述的复杂和真实的低维动力系统。尽管模型很简单，但已知这样的系统显示出复杂的混沌行为，表明在天体力学、流体力学、统计力学、生物学和其他自然科学分支中出现的各种现象。拟议的活动将导致对动力系统的小规模结构的更深入的见解，培养高素质的博士后研究员和研究生，在真实和复杂动力学的各个分支的高级和初级专家之间进行更广泛的互动，并准备一本书，以帮助研究界获得该领域的背景。此外，首席研究员将通过组织国际会议和科学项目以及维护一个与动力学相关的网站来促进该领域内的交流。该项目在复杂的低维动态中解决了几个几何主题，从一维世界逐渐过渡到二维世界。在一维动力学中，重整化将作为一种统一而有力的工具来研究动态对象的小尺度结构。正在考虑的具体主题包括中性映射的半局部理论和无限可重整二次多项式的先验界，并应用于Mandelbrot集的局部连通性问题。其他的研究课题包括Schwarz反射在正交域中产生的动力学和耗散复Henon映射的动力学。与后者相关的具体主题包括真实和复杂野生吸引子的二维示例的构建以及单峰Henon映射的一般理论的发展。该项目还将探索高维全纯动力学在自相似群谱理论中的应用。最后，首席研究员将继续撰写一本关于二次多项式保形几何和动力学的多卷书。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。

The theory of dynamical systems studies the long-term behavior of trajectories described by iteration procedures, and how such behavior depends on the parameters of the system. Intricate fractal objects (like Julia sets and the Mandelbrot set) may appear as phase and parameter diagrams for such systems. This project focuses on complex and real low-dimensional dynamical systems described by simple quadratic equations. Despite the simplicity of the model, such systems are known to display complicated chaotic behavior indicative of various phenomena appearing in celestial mechanics, fluid dynamics, statistical mechanics, biology, and other branches of natural science. The proposed activity will result in deeper insights into the small scale structure of dynamical systems, in the training of highly qualified postdoctoral fellows and graduate students, in broader interactions between senior and junior experts in various branches of real and complex dynamics, and in the preparation of a book to assist the research community in acquiring background in the area. In addition, the Principal Investigator will facilitate communication within the field through the organization of international conferences and scientific programs and by maintaining a dynamics-related web site.The project addresses several geometric themes within complex low-dimensional dynamics, making a gradual transition from the one-dimensional to the two-dimensional world. In connection with dynamics in one dimension, renormalization will be investigated as a unifying and powerful tool for elucidating the small-scale structure of dynamical objects. Specific topics under consideration include a semi-local theory of neutral maps and a priori bounds for infinitely renormalizable quadratic polynomials with applications to the problem of local connectivity of the Mandelbrot set. Other topics of study include the dynamics generated by Schwarz reflections in quadrature domains and the dynamics of dissipative complex Henon maps. In connection with the latter, specific themes include the construction of two-dimensional examples of real and complex wild attractors and the development of a general theory of unimodal Henon maps. The project will also explore applications of higher dimensional holomorphic dynamics to the spectral theory of self-similar groups. Finally, the principal investigator will continue to work on a multi-volume book on the 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.