Abstract
In the absence of friction and other external forces, a billiard ball on a rectangular billiard table follows a predictable path. As it reflects off the sides of the table, the ball will either (1) return to its original position and direction, or (2) travel all over the table, spending equal time in regions with equal area. Because of this dichotomy, we say the rectangular table is optimal: billiard trajectories define an easily understood dynamical system on the phase space of positions and directions. In this article, we ask:
Highlights
This article provides an introduction to some recent results in billiard dynamics
We concentrate on the billiard dynamics in polygons T ⊂ R2 with all interior angles equal to rational multiples of π
The path of a billiard ball, which bounces around the table by reflecting off the sides, unfolds into a straight line on the surface X, with respect to the Euclidean metric determined by ω
Summary
A billiard table means a connected polygon in R2 with all angles equal to rational multiples of π. The simplest example is the square table of side length 1, with sides parallel to the coordinate axes In this table, it is easy to see that any billiard trajectory of rational slope will either hit a vertex or eventually return to its original configuration (position and angle). We require that the trajectory be dense on the finite union of tables corresponding to different directions under reflection. The billiard dynamics on the isosceles triangle with angles (2π/5, 3π/10, 3π/10) are topologically optimal but not ergodically optimal: for each direction, either all infinite trajectories are closed or all are dense, but there exist trajectories which are dense but not uniformly distributed.
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