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.
