Linearització conforme de punts fixos el·líptics

Abstract

[en] The study of dynamics of holomorphic functions near a fixed point has led to numerous works since the end of the nineteenth century, and the Siegel linearization problem plays an important role in this branch of the theory of dynamical systems in one complex variable. The natural way of studying the dynamics of a system near a fixed point is finding a local change of cordinates to represent this system in a simpler way. If $f+$ is a holomorphic function with a fixed point $z_{0} = f (z_{0})$, and multiplier $\lambda = f'(z_{0}) \in S^{1}, \lambda = e^{2\pi i \alpha}$ for an irrational number $\alpha$, we say that f is linearizable if it’s locally conjugated to the linear system $g(z) =\lambda z$. Then, Siegel’s problem consists in describing completely the family of numbers $\alpha$ for which every local system $f$ with multiplier $\lambda$ is linearizable. The contributions of H. Cremer and, specially, of C.L. Siegel to the problem, represent a big step in understanding it's trickyness, as well as the importance of the role that the arithmetical nature of $\alpha$ plays in it. The techniques introduced by J.C. Yoccoz in his resolution of Siegel’s problem, at the end of the past century, have inspired other results to help understanding the dynamics of $f$ in the non-linearizable case, yet not fully understood nowadays.