Dynamics Seminar

Event Information E pluribus unum: natural boundaries and uniqueness of KAM curves
15:10 on Wednesday October 30, 2013
16:00 on Wednesday October 30, 2013
BA6180, Bahen Center, 40 St. George St.
Stefano Marmi

Scuola Normale Superiore di Pisa

Addressing a question raised by Kolmogorov and Herman, we show that KAM curves of area-preserving twist maps are uniquely determined by their knowledge on a set of positive 1-dimensional Hausdorff measure in frequency space. This result is obtained by complexifying the rotation number and by an extension of the classical theory of quasianalytic functions.

The parametrization of KAM curves is naturally defined in a complex domain containing the real Diophantine frequencies. Real frequencies constitute a natural boundary for the analytic continuation from the Weierstrass point of view because of the density of the resonances.