Prevalent divergence of Birkhoff Normal Forms

Raphael Krikorian

A real analytic symplectic diffeomorphism of $\R^{2d}$ admitting a non resonant elliptic fixed point is always formally conjugated to a formal integrable system, its Birkhoff Normal Form (BNF). Siegel proved in 1954 that the involved formal conjugation does not in general define a converging series. I will give a proof of the fact that, in any dimension, the same phenomenon holds for the Birkhoff Normal Form itself (the formal integrable model). The key result is that the convergence of the BNF of a real analytic symplectic diffeomorphism of the plane has strong dynamical consequences on the diffeomorphism: the measure of the set of its invariant curves is abnormally large. In other words, an information on the formal dynamics has consequences on the "real" dynamics.

http://scgp.stonybrook.edu/video_portal/video.php?id=4522

Simons- Program: Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory