Siegel's linearization theorem

Abstract

[en] The purpose of this study is to give an insight to Siegel’s linearization theorem, a result in discrete dynamics of one-dimensional holomorphic maps that claims the existence of a change of coordinates in a neighbourhood of a map’s fixed point to its linear part, whenever the multiplier for such point satisfies the Diophantine condition. This overall approach aims to provide an understanding of the theorem and all it encompasses. It firstly puts forward necessary knowledge in Diophantine approximations as well as complex and functional analysis and introduces some background to Schröder’s equation, the conjugacy problem in which the theorem originates. Once set, the theorem is proved in great detail and the dissertation concludes with a numerical exploration performed to visualize and ponder about the most relevant results forementioned.