On convergence of normal form transformations
Valery G. Romanovski, Sebastian Walcher
TL;DR
The paper analyzes convergence of normal form transformations for analytic and formal vector fields near a stationary point, surveying Bruno’s diophantine framework and Condition $\omega$, and extending the approach to Stolovitch’s simultaneous normalization for abelian Lie algebras. It introduces a unified formalism to reprove Bruno’s normalization, derives a convergence theorem under a generalized Condition AS (and its Stolovitch variant AL) and the diophantine hull, and examines how integrability and formally meromorphic first integrals influence convergence. The work clarifies when a convergent normalization exists, provides elementary, accessible proofs in simplified settings, and discusses limitations of the diophantine hull in higher dimensions. Together, these results enhance understanding of when local dynamical systems admit convergent normalizations and how symmetry and integrability interact with normalization.
Abstract
We discuss various aspects concerning transformations of local analytic, or formal, vector fields to Poincare-Dulac normal form, and the convergence of such transformations. We first review A.D. Bruno's approach to formal normalization, as well as convergence results in presence of certain (simplified) versions of Bruno's ``Condition A'', and along the way we also identify a large class of systems that satisfy Bruno's diophantine ``Condition omega''. We retrace the proof steps in Bruno's work, using a different formalism and variants in the line of arguments. We then proceed to show how Bruno's approach naturally extends to an elementary proof of L. Stolovitch's formal and analytic simultaneous normalization theorems for abelian Lie algebras of vector fields. Finally we investigate the role of integrability for convergence, sharpening some existing and adding new results. In particular we give a characterization of formally meromorphic first integrals, and their relevance for convergence.