Holomorphic Dynamics in one and several variables

动力系统是允许研究人员研究随时间演变的物理现象的数学模型：从行星的运动到股票市场。PI研究全纯动力学，它指的是允许丰富几何图形的某些复值映射的迭代。这门学科与数学的其他领域有许多互动，特别是与泰希穆勒理论和多维复分析等重要学科的互动。PI将继续她关于全纯动力学几何方面的工作，特别是她将致力于将理论的某些部分从一个复变量的动力学扩展到多个复变量。PI将继续为本科生和研究生提供专题讲座，希望激发学生对动力系统领域以及更广泛的数学领域的兴趣。全纯动力学和泰希穆勒理论之间的联系在很大程度上可以用全纯运动来解释。PI与合作者一起提出了一种几何方法来证明Slodkowski的Lambda引理。我们将全纯运动与用全纯圆盘填充全实流形的技巧联系起来。我们打算用这个方法来研究黎曼曲面上全纯运动的h-原理。关于后临界有限有理映射的瑟斯顿定理是一维全纯动力学的中心结果之一。在瑟斯顿工作的基础上，A.爱泼斯坦引入了形变空间，它给出了全纯动力学中横截性结果的统一方法。PI将与合作者一起研究这些变形空间的某些拓扑性质。在一维全纯动力学中，临界点的轨道在很大程度上决定了映射的动力学。对于复平面的自同构，不存在临界点，但在不同的动态定义区域，临界点有不同的类比。这项研究将研究这些关键轨迹及其在构建拓扑模型中的应用。我们还研究了复平面解析微分同胚的不可线性化芽。

Dynamical systems are mathematical models that allow researchers to study physical phenomenon that evolve with time: from motion of the planets to stock markets. The PI studies holomorphic dynamics, which refers to iterations of certain complex-valued maps that admit a rich geometry. The subject has numerous interactions with other areas of mathematics, in particular with the important subjects of Teichmuller Theory and Multidimensional Complex Analysis. The PI will continue her work concerning geometric aspects of holomorphic dynamics and in particular she will work to extend certain parts of the the theory from dynamics of one complex variable to several complex variables. The PI will continue to provide special topical lectures to undergraduate and graduate students in the hope of exciting the interest of students in the field of dynamical systems and, more broadly, in mathematics.The link between holomorphic dynamics and Teichmuller Theory can be to a large extent explained by holomorphic motions. The PI, together with collaborators, suggest a geometric approach to the proof of Slodkowski's lambda lemma. We relate holomorphic motions to the technique of filling totally real manifolds by holomorphic discs. We plan to use this approach to investigate the h-principle for holomorphic motions over Riemann surfaces. Thurston's theorem on postcritically finite rational maps is one of the central results in one-dimensional holomorphic dynamics. Building on Thurston's work, A. Epstein introduced deformation spaces, which give a unified approach to transversality results in holomorphic dynamics. The PI, together with collaborators, will investigate certain topological properties of these deformation spaces. In one-dimensional holomorphic dynamics to a large extent the orbits of critical points determine the dynamics of the map. For an automorphism of complex plane there are no critical points however there are various analogs of the critical points in different dynamically defined regions. The research will investigate these critical loci and their application to the construction of topological models. We also study the non-linearizable germs of analytic diffeomorphisms of complex plane.