A new quasi-analytic class

TL;DR

This paper introduces a new quasi-analytic class relevant to perturbation theory, explaining divergence in perturbative expansions through geometric analysis of pole accumulation points.

Contribution

It defines a novel quasi-analytic class applicable to dynamical systems and clarifies divergence phenomena in perturbation expansions.

Findings

Introduces a quasi-analytic class relevant to KAM theory.

Explains divergence of perturbative expansions via pole accumulation.

Provides geometric interpretation of divergence in perturbation series.

Abstract

Spaces of quasi-analytic classes are defined by the existence and uniqueness of Taylor expansions, which are not necessarily convergent. First examples were given by Borel in his theory of monogenic functions, a generalisation of holomorphic functions defined on locally closed sets. Denjoy and Carleman then gave simpler examples of quasi-analytic classes which are now widely known. Unfortunately, in most examples coming from mathematical physics and number theory, the power series are neither of Borel nor Denjoy-Carleman's classes. In this paper we introduce a quasi-analytic class which is relevant to perturbation theory and especially to KAM theory and dynamical systems. Our theorems also explain geometrically the divergence of most perturbative expansions by the presence of accumulation points of poles.