Holomorphic Dynamics and Small Divisors

该项目的主要目标是给出一个完整的分析的局部动力学的一维全纯映射附近的一个中立的无理不动点，这是非线性化由于存在小因子。 特别是，我们要了解的拓扑和度量结构的不变集的映射（Siegel Alfreta）和它的稳定性。这将需要尖锐的几何估计解析圆同态，工具，将被用来研究边界的西格尔磁盘。 本研究的一个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物理系统的微分方程组的解的演化。 在应用中，确定物理系统的未来演化具有根本的重要性。 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？ 这是建造飞机或其他移动设备的工程师以及更复杂情况下的常见问题。 数学家研究最简单的问题，以开发处理更复杂问题的工具。小因子理论是研究保守情况下稳定性的基本工具，其中一些结构（体积，共形结构，.）被保存了下来 这项工作涉及研究的共形地图在平面上，即地图，保持角度。 在这种情况下，有经典的小因子定理，保证稳定性。 这项研究的很大一部分是致力于小因子的工具无法证明稳定性的情况下，在这种情况下，较弱形式的稳定性的发展。

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.