Mathematical Sciences: Holomorphic Dynamical Systems and Small Divisions

摘要佩雷斯-马尔科 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冷漠的无理不动点时，它是非线性化的，由于存在小因子。 特别是，我想尽可能彻底地理解，映射不变量的拓扑和度量结构 集合（Siegel紧集）和稳定性。 实现这些目标将需要尖锐的几何估计解析圆同态。 使用这些工具，我建议研究边界的所谓西格尔磁盘的第一种类型。 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。 动力系统理论研究描述物理系统演化的微分方程系统解的演化。 通常这些方程的解决方案出现在物理科学，气象学，工程等.不能通过一个封闭的公式获得（这与多项式的根的情况相同：我们知道总是有复数根，但我们不能通过简单的代数运算从系数中表达根）。 确定物理系统的未来演化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： 它会永远保持稳定的进化，还是会在某个时刻崩溃？ 当然，对于建造飞机或其他移动设备的工程师来说，这是一个日常问题，但我们可以找到更复杂的情况，在这些情况下，解决这个问题至关重要。 例如，在托卡马克机器的聚变实践中，最困难的问题之一是在极高温度下找到等离子体稳定结构的磁流体动力学问题。 控制聚变将从水中提供清洁能源！ 这是一项具有根本重要性的探索。 作为数学家，我们研究最简单的情况，以开发处理更复杂情况的工具。 小因子理论是研究保守情形下稳定性问题的基本工具。 保守的情况意味着一些结构（体积，共形结构，.）被保存了下来这在实际应用中很常见，例如无摩擦力的力学方程是保守的。 在我的研究中，我研究了共形映射在平面上的情况，即保角映射。在这种情况下，有经典的小因子定理，保证稳定性 特性. 我研究的很大一部分是专门研究小除数工具无法证明稳定性的情况。 新的弱稳定性概念已被发现在此设置。 我的研究项目的主要部分包括对这些新的稳定性特征的详细研究。

Abstract Perez-Marco 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 when it is nonlinearizable due to the existence of Small Divisors. In particular, I want to understand as thoroughly as possible, the topological and metrical structure of the mapping invariant sets (Siegel compacts), and stability properties. Achieving these aims will entail sharp geometric estimates for analytic circle diffeomorphisms. Using these tools I propose to study the boundary of what are called Siegel disks of the first type. A long range objective of this research is to obtain a geometric theory of Small Divisors applicable to higher dimensional problems. The theory of Dynamical Systems studies the evolution of solutions of a system of differential equations, that describe the evolution of a physical system. Typically the solutions of these equations arising in Physical Sciences, meteorology, engineering, etc... cannot be obtained by a close formula (it is the same situation than for the roots of a polynomial: we know that there are always complex roots, but we cannot express the roots from the coefficients by simple algebraic operations). It is of fundamental importance in the applications to determine the future evolution of the physical system. Probably the most important problem is the question of stability: Is the system going to have a stable evolution forever or it is going to break at some moment ? Certainly, this is an every day problem for engineers building planes or other moving devices, but we can find much more complex situations where the solution of this problem is of capital importance. For example, one of the hardest problems in the practice of fusion in the Tokamak machines, is the magnetohydrodynamical problem of finding stable configurations of a plasma at extremely high temperatures. The control of fusion will provide clean energy from the water! It is a quest ion of fundamental importance. As mathematicians, we study the simplest situations to develop the tools to treat the more complex ones. The theory of Small Divisors is a fundamental tool to study problems of stability in conservative situations. Conservative situation meaning that some structure (a volume, a conformal structure,...) is preserved. This is very common in the applications, the equations of mechanics without friction are conservative for example. In my research, I study the situation of a conformal map in the plane, that is a map that preserves angles. In this situation, there are classical Small Divisors theorems which guarantee stability properties. A large part of my research is devoted to study the situation where the tools of Small Divisors fail to prove stability. New weak stability notions have been found in this setting. The major part of my research project consists in the detailed study of these new stability features.