Small divisors and large multipliers

2006/07/30 by B. L. J. Braaksma, Braaksma, B., Laurent Stolovitch +1
Arts and Humanities · Health Professions · #Aging, Elder Care, and Social Issues #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Health, Medicine and Society #Hermeneutics and Narrative Identity

paper · doi:10.48550/arxiv.math/0607787

openalex publication_date 2006/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study germs of singular holomorphic vector fields at the origin of \Bbb Cⁿ of which the linear part is 1-resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some "sectorial domains" which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms. We show that the Gevrey ordrer of the latter is linked to the diophantine type of the small divisors.

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