A celebrated result by Zehnder in the ’70s states that a generic analytic area-preserving map of the disk, having the origin as elliptic fixed point, exhibits a transverse homoclinc orbit in every neighborhood of the origin. In an ongoing project with Inmaculada Baldomà, Martin Leguil and Tere Seara, we adapt the strategy of Zehnder and use Aubry-Mather theory for twist maps in order to show that a generic analytic strongly convex billiard has, for every rational rotation number, a periodic orbit with a transverse homoclinic intersection.