Ergodicity of hyperbolic billiards.
Gianluigi Del Magno (UFBA)
A billiard is a mechanical system consisting of a point-particle moving freely inside a domain, the billiard table. When the particle reaches the boundary of the table it bounces off so that the angle of reflection equals the angle of incidence. A billiard is hyperbolic if its trajectories behave like the orbits of a geodesic flow on a surface of negative curvature: typically two nearby trajectories diverge exponentially fast. The study of hyperbolic billiards was started by Sinai. In his seminal work, he proved that if the billiard table is a planar domain bounded by convex arcs, then the billiard is hyperbolic. Later on, Bunimovich discovered that also billiards in domains bounded by concave arcs and possibly by straight lines may be hyperbolic. All the concave arcs in Bunimovich billiards are arcs of circles though. This limitation was eventually overcome by Wojtkowski, Markarian, Donnay and Bunimovich himself who independently constructed new examples of hyperbolic billiards. The first part of the talk is a rather informal introduction to hyperbolic billiards. In the second part, I will describe some general principles for constructing hyperbolic billiards, and address the problem of their ergodicity. The talk is based on a joint work with R. Markarian.