Title: Diamond structures in a family of KAM solution for billiards

Abstract: Strictly convex billiards give nice and concrete examples of symplectic maps of the cylinder with a twist. When the angle of reflection is small, Lazutkin showed that the corresponding billiard map is close to be integrable. He used this property to show that 1) there is a Cantor set of invariant curves (so called KAM curves) accumulating to the boundary; 2) on each of these curves, the billiard map is conjugated to a rotation of diophantine angle; 3) these curves depend smoothly - in a Whitney sense - on the rotation number. In this talk, I will present a joint result with Vadim Kaloshin extending this work to convex analytic billiards. More precisely, we showed that the dependence in the rotation number can be extended to a true analytic map on a domain in the complex space. This result follows a similar work from Carminati, Marmi and Sauzin. It has a few nice corollaries such as the unicity of KAM curves, or length spectral rigidity of generic billiards by deformation under trigonometric polynomials