2019/06/03 by Raphaël Krikorian, Krikorian, Raphaël
Chemistry · Mathematics · #37J #Axial and Atropisomeric Chirality Synthesis #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Molecular spectroscopy and chirality
paper · doi:10.48550/arxiv.1906.01096
openalex publication_date 2019/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well known that a real analytic symplectic diffeomorphism of the 2d-dimensional disk (d≥ 1) admitting the origin as a non-resonant elliptic fixed can be \it formally conjugated to its Birkhoff Normal Form, a formal power series defining a \it formal integrable symplectic diffeomorphism at the origin. We prove in this paper that this Birkhoff Normal Form is in general divergent. This solves, in any dimension, the question of determining which of the two alternatives of Perez-Marco's theorem \citePM is true and answers a question by H. Eliasson. Our result is a consequence of the fact that when d=1 the convergence of the formal object that is the BNF has strong dynamical consequences on the Lebesgue measure of the set of invariant circles in arbitrarily small neighborhoods of the origin. Our proof, as well as our results, extend to the case of real-analytic diffeomorphisms of the annulus admitting a Diophantine invariant torus.