Holomorphic Dynamics

动力系统在所有科学中都起着重要的作用。它们是捕食者和猎物数量、行星轨道和天气条件的模型。一些最容易描述的动力系统是通过单一映射的迭代得到的，可以看作是从实线到实线的函数，或者是从复平面到复平面的函数。最近，物理科学领域的研究人员对高维映射给出的模型很感兴趣。在复杂的环境中，人们可以使用复杂的解析和几何工具，而这些工具在实际环境中是不可用的。这个项目的目的是描述一个复杂的解析映射的动力学，从几个变量的空间到它本身，在一个不动的点是抛物线的存在。在一个复杂动力学中，对接近不动点的全纯映射的局部研究已经得到了很好的理解（尽管不完整）。然而，在几个维度上，情况变得复杂得多。PI还将关注这种情况下的全局动力学，并研究轨道的可能行为及其与固定点乘数的关系。最后，PI将探讨可和性理论在多维抛物线映射中的应用。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。

Dynamical systems play an important role in all the sciences. They occur as models for predator and prey populations, orbits of planets and weather conditions. Some of the dynamical systems that are easiest to state are obtained by the iteration of a single map, viewed as a function from the real line to the real line, or from the complex plane to the complex plane. More recently researchers in physical sciences have been interested in models given by higher dimensional mappings. In the complex setting one can make use of complex analytic and geometric tools that are not available in the real one.The aim of this project is to describe the dynamics of a complex analytic map from a space of several variables to itself in the presence of a fixed point that is parabolic. The local study of holomorphic maps close to a fixed point is well understood (although not complete) in one complex dynamics. In several dimensions however, the situation becomes much more complicated. The PI will also focus on global dynamics in this setting and investigate the possible behaviors of orbits and their relation with multipliers at fixed points. Finally, the PI will explore applications of summability theory in the context of parabolic maps in several dimensions.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.