# 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