KAM theory: from the Herman conjecture to the quasi-analytic properties of divergent series
In 1994, M. Herman conjectured the existence of invariant tori carrying quasi-periodic motions in the vicinity of an elliptic critical point, assuming an arithmetic condition on the frequency. The conjecture was solved in 2012 by me and Duco van Straten using a new type of normal form which replaces the Birkhoff normal form. The Birkhoff normal form becomes the asymptotic expansion of a generating series. The solution of the Herman conjecture leads to the general notion of Birkhoff series associated to a generating series and belongs to the class of Borel monogenic functions. Unlike holomorphic functions, monogenic functions are not uniquely defined by their Taylor expansions. However, with Duco van Straten, we introduced a quasi-analytic class that we call meandromorphic functions because their zero locus sometimes have the form of a meander. This class is large enough to include q-analogues and classical perturbative expansions of classical and quantum mechanics. We then prove a general theorem of divergence: when a meandromorphic function have poles accumulating at a point then the asymptotic expansion at that point diverges. (Joint works with Duco Van Straten). Text: arxiv 2410.04583