Arnold diffusion in multidimensional convex billiards (Q6177376)

Planar strictly convex curves generate billiard maps that correspond to a subclass of area-preserving twist maps of the (bounded) cylinder. \textit{V. F. Lazutkin} [Math. USSR, Izv. 7, 185--214 (1974; Zbl 0277.52002)] showed that these maps have the property that there are always invariant curves near the boundaries of the cylinder. This means that orbits remain away from those boundaries, that is, angles of the billiard trajectories cannot tend to zero. The authors show that is not the case in higher dimensions. For billiards in convex bodies with sufficiently smooth boundaries, it is possible to have orbits that get arbitrarily closer to the phase space boundaries. This phenomenon is known in more general settings as Arnold diffusion. It is not generic in two-dimensional maps, as KAM-stable invariant curves prevent it from happening. Mechanisms of diffusion are known in higher-dimensional cases. This paper show that in the considered restricted class of billiards, Arnold diffusion is also a generic property. The main idea of the proof is based on approximating the billiard trajectory with small reflection angles by a geodesic flow on the boundary of the convex body. This relation is used to construct the diffusive orbits.