# Talk

Modul:   MAT760  Ergodic Theory and Dynamical Systems Seminar

## On the divergence of Birkhoff Normal Forms

Prof. Dr. Raphael Krikorian talk

Speaker invited by: Prof. Dr. Artur Avila

Date: 29.10.18  Time: 15.30 - 16.30  Room: Y27H25

An analytic hamiltonian system (or a symplectic diffeomorphism) admitting an elliptic fixed point is always formally conjugated to a formal integrable normal form, the Birkhoff Normal Form. It is known since Siegel (1954) that the formal conjugacy cannot in general converge and H. Eliasson asked whether the Birkhoff Normal Form itself could be divergent. Perez-Marco (2001) proved that for any given frequency vector at the origin, one has the following dichotomy: either the BNF always converges or it generically diverges and Gong (2012) exhibited a divergent example with Liouville frequency vector. I will explain in this talk the proof of the following theorem: given any diophantine frequency vector at the origin, the BNF is generically divergent.