In this talk we study chaotic dynamics generated by analytic convex billiards. We consider the set S of analytic billiards with negative curvature satisfying the following property: for any rational rotation number, there exists a hyperbolic periodic orbit whose stable and unstable manifolds intersect tansversally along a homolinic orbit. And we prove that the set S is residual among analytic billiards with negative curvature with the ususal analytic topology. This result is a consequence of the Baire property and the main result of this work, which reads: Fixing a rational rotation number p/q, we can prove that the set of analytic billiards with negative curvature having a hyperbolic periodic orbit of rotation umber p/q whose stable and unstable manifolds intersect tansversally along a homolinic orbit, is open and dense. As a consequence of our results, we have that chaotic billiards are dense among analytic biliards. Our proof combines Aubrey-Mather theory to study periodic orbits of any rotation number as well as their heteroclinic trajectories, with the work by Zehnder on planar twist maps with elliptic points in the 1970's, which provides a methodology for constructing analytic perturbations of maps in order to obtain transversality between the invariant manifolds of hyperbolic periodic orbits. This is a joint work with Imma Baldomá, Anna Florio and Martin Leguil. Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).
Per informazioni, rivolgersi a: sorrentino@mat.uniroma2.it