Family of intersecting totally real manifolds of $(C^n ,0)$ and germs of holomorphic diffeomorphisms

Laurent Stolovitch (1)

1 LJAD - Laboratoire Jean Alexandre Dieudonné

We prove the existence (and give a characterization) of a germ of complex analytic set left invariant by an abelian group of germs of holomorphic diffeomorphisms at a common fixed point.We also give condition that ensure that such a group can be linearized holomorphically near the fixed point. It rests on a " small divisors condition " of the family of linear parts. The second part of this article is devoted to the study families of totally real intersecting n-submanifolds of $(C^n , 0)$. We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it formally, we construct a germ of complex analytic set at the origin which interesection with the family can be holomorphically straightened. The second part is an application of the first.

Mots clés: Linearization; Analytic diffeomorphisms; Fixed Point; Small divisors; Invariant set; CR geometry

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Laurent Stolovitch. Family of intersecting totally real manifolds of $(C^n ,0)$ and germs of holomorphic diffeomorphisms. Bulletin de la société mathématique de France, 2015, 143 (2), pp.247-263. ⟨hal-01285115⟩