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集和Mandelbrot集）可能会显示为此类系统的相图和参数图。本计画主要研究由简单二次方程式所描述的复杂且真实的低维动力系统。尽管模型简单，但已知这样的系统显示出复杂的混沌行为，指示出现在天体力学、流体动力学、统计力学、生物学和自然科学的其他分支中的各种现象。拟议的活动将导致更深入地了解小规模结构的动力系统，在高素质的博士后研究员和研究生的培训，在更广泛的互动高级和初级专家之间的各个分支的真实的和复杂的动力学，并在编写一本书，以协助研究界在获取背景在该地区。此外，首席研究员将通过组织国际会议和科学方案以及维持一个与动力学有关的网站，促进该领域内的交流，该项目涉及复杂低维动力学中的几个几何主题，使一维世界逐渐过渡到二维世界。在一维动力学方面，重整化将被研究为一个统一的和强大的工具，用于阐明动力学对象的小尺度结构。具体的议题正在审议中包括一个半本地理论的中立映射和先验界的无限renormalizable二次多项式的应用问题的局部连通性的Mandelbrot集。其他研究课题包括施瓦茨反射在正交域和耗散复杂的Henon映射的动力学产生的动力学。与后者，具体的主题包括建设的二维例子的真实的和复杂的野生吸引和发展的一般理论单峰Henon地图。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，首席研究员将继续致力于一本关于二次多项式的共形几何和动力学的多卷书。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。

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.