A Complete Classification for Pairs of Real Analytic Curves in the Complex Plane with Tangential Intersection

Abstract

We study the group structure of centralizers of a holomorphic mapping that is tangent to the identity. We give a complete classification for pairs of real analytic curves intersecting tangentially via the Ecalle--Voronin theory. We also classify pairs of one-dimensional holomorphic involutions that have a high-order tangency, real analytic mappings on the real line, and exceptional non-Abelian subgroups of Cerveau and Moussu. Consequently, we prove the existence of holomorphic mappings z ? z + O(2) that are reversible by a formal antiholomorphic mapping, but not by any (convergent) antiholomorphic one. We also prove the existence of local real analytic mappings on the real line that are not equivalent with respect to any local real analytic diffeomorphism and yet whose complexifications are holomorphically equivalent.