Winners
On the local Birkhoff conjecture for convex billiards
Vadim Kaloshin and Alfonso Sorrentino
Winners
On the local Birkhoff conjecture for convex billiards
Vadim Kaloshin and Alfonso Sorrentino
Period
works until January 31, 2019
Award ceremony
Date
will be announced
Place
Institute of Catalan Studies
Place
Institute of Catalan Studies
Resolution of the Barcelona Dynamical Systems Prize 2019 under the patronage of Professor Carles Simó i Torres
In this article, the authors prove that the boundary of a strictly convex integrable billiard table sufficiently close to an ellipse or a circle is necessarily an ellipse or a circle. This is a perturbative version of the Birkhoff conjecture, which is commonly considered one of the oldest (and most impenetrable) problems in dynamical systems. To prove this result, the authors introduce a notion of integrability called »rational integrability» (the existence of caustics of all periods 1/q near the boundary of the domain), and show that for these caustics to be preserved under a perturbation of an elliptic domain, a family of subharmonic Melnikov integrals must be identically zero. After a technically demanding analysis of this class of perturbations, they show that the only integrable deformations of an elliptic motion (i.e., those preserving the class of ellipses), thus completing the proof of their main result.
Committee
Alain Chenciner
Observatoire de Paris
Freddy Dumortier
University of Hasselt
Daniel Peralta
ICMAT
Amadeu Delshams
UPC, Secretary of the jury without vote