Poincarè-Dulac Normal Forms: Formal and Analytic Aspects

Abstract

We discuss various aspects concerning transformations of local analytic, or formal, vector fields to Poincarè-Dulac normal form, and the convergence of such transformations. After introducing a new transparent approach to transformations, we review Bruno's approach to formal normalization, as well as convergence results in presence of certain versions of Bruno's Condition A. Retracing the proof steps in Bruno's work, we use a different formalism and variants in the line of arguments, which are easily transferrable to more general scenarios. In particular we show how Bruno's approach naturally extends to an elementary proof of Stolovitch's formal and analytic simultaneous normalization theorems for abelian Lie algebras of vector fields. Furthermore we investigate the role of (meromorphic and formally meromorphic) integrability in normal forms and convergence properties. In particular we establish a bridge to Zung's convergence theorems and show that these are can be complemented by Stolovitch's results.

57 pages. Extensive rewrite of first version, with new material added. Title has been adjusted