Billiards

The discrete system I consider here is analogous to continuous so-called billiard systems consisting of circular balls in the plane. The simplest case involves one ball bouncing around in a region of a definite shape. In a rectangular region, the position is given by Mod[a t, {w, h}] and every point will be visited if the parameters have irrational ratios. In a region that contains fixed circular obstructions, the motion can become sensitively dependent on initial conditions. (This setup is similar to a so-called Lorentz gas.) For a system of balls in a region with cyclic boundaries, a complicated proof due to Yakov Sinai from the 1960s purports to show that every ball eventually visits every point in the region, and that certain simple statistical properties of trajectories are consistent with randomness. (See also page 971.)