Sectorial normalization of tangent to the identity diffeomorphisms in dimension one.

Abstract

The aim of this work is to review the basics on local diffeomorphisms, both formaland holomorphic, in one complex dimension, focusing on the ones which are tangent to the identity. We include the study of the exponential map and ofnormal forms of holomorphic vector fields. In particular, we offer a new proof of the analyticity of the flow of such objects. On the other hand, we give a detailed proof of Leau-Fatou theorem, and two constructions of the solutionsof Abel’s equation related to the problem of conjugating such diffeomorphisms to translations.