Holomorphic Dynamics and Small Divisors

本课题的主要目的是给出一维全纯映射的局部动力学的完整分析，该全纯映射在一个由于小因子的存在而非线性化的不动点附近。特别地，我们想要了解映射（Siegel compacta）的不变集的拓扑结构和度量结构及其稳定性。这将需要解析圆微分同态的尖锐几何估计，这将用于研究西格尔盘的边界的工具。本研究的长期目标是获得适用于高维问题的小因子几何理论和KAM理论的改进。动力系统理论研究描述一个物理系统的微分方程组的解的演化。确定物理系统的未来演变在应用中具有根本的重要性。也许最重要的问题是稳定性问题：系统会永远稳定地进化，还是会在某个时刻崩溃？对于制造飞机或其他移动设备以及更复杂的情况的工程师来说，这是一个常见的问题。数学家研究最简单的问题，以开发处理更复杂问题的工具。小除数理论是研究保守情况下的稳定性的基本工具，在保守情况下，一些结构（体积，保形结构，…）是保留的。这项工作涉及研究平面上的保角映射，也就是保角映射。在这种情况下，有经典的小因子定理来保证稳定性。这项研究的很大一部分致力于小除数的工具无法证明稳定性和在这种情况下发展较弱形式的稳定性的情况。

The main goal of the project is to give a complete analysis of the local dynamics of a one-dimensional holomorphic map near an indifferent irrational fixed point which is nonlinearizable due to the existence of small divisors. In particular, we want to understand the topological and metric structure of the invariant sets of the mapping (Siegel compacta) and its stability properties. This will entail sharp geometric estimates for analytic circle diffeomorphisms, tools which will be used to study the boundary of Siegel disks. A long range objective of this research is to obtain a geometric theory of small divisors applicable to higher dimensional problems and a refinement of KAM theory. The theory of dynamical systems studies the evolution of solutions of a system of differential equations describing a physical system. It is of fundamental importance in applications to determine the future evolution of the physical system. Probably the most important problem is the question of stability: Will the system have a stable evolution forever or it is going to break down at some moment? This is a common concern for engineers building planes or other moving devices, as well as with more complex situations. Mathematicians study the simplest problems to develop the tools to treat the more complex. The theory of small divisors is a fundamental tool in the study of stability in conservative situations, where some structure (a volume, a conformal structure, ...) is preserved. This work involves the study of conformal maps in the plane, that is, maps that preserve angles. In this case, there are classical small divisors theorems which guarantee stability properties. A large part of this research is devoted to the situation where the tools of small divisors fail to prove stability and the development of weaker forms of stability in that situation.