Semi - dispersing billiards with an infinite cusp Marco Lenci

Let f : [0,+∞) → (0,+∞) be a sufficiently smooth convex function, vanishing at infinity. Consider the planar domain Q delimited by the positive x-semiaxis, the positive y-semiaxis, and the graph of f . Under certain conditions on f , we prove that the billiard flow in Q has a hyperbolic structure and, with some extra requirements, that it is also ergodic. This is done using the cross section corresponding to collisions with the dispersing part of the boundary. The relevant invariant measure for this Poincaré section is infinite, whence the need to surpass the existing results, designed for finite-measure dynamical systems. Mathematics Subject Classification: 37D50, 37D25. 1 Historical introduction There is a long tradition in the study of hyperbolic billiards, especially billiards in the plane. This was initiated by Sinai as early as 1963 [S1], in connection with the Boltzmann Hypothesis in statistical mechanics. In his celebrated 1970 paper [S2], Sinai proved the first cornerstone theorem of the field: theK-property of a billiard in a 2-torus endowed with a finite number of convex scatterers (of positive curvature). The result was polished in a later joint work with Bunimovich [BS] and extended to a larger class of dispersing billiards. This terminology was introduced precisely in that paper and designates billiard tables whose boundaries are composed of finitely many convex pieces, when seen from the interior. In particular, the new theorem allowed for positive-angle corners at the boundary.