Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons

Abstract

This paper proves the existence of non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons. For any n ^ 3 there exists a corresponding convex rc-agon (for n = 3 this will be a right triangle with a small acute angle), while in three-dimensional space it will be a prism, the n-agon serving as the base.The results are applied for investigating a mechanical system of two absolutely elastic balls on a segment, and also for proving the existence of an infinite number of periodic trajectories in the given polygons.The exchange transformation of two intervals is used for proving the theorems. An arbitrary exchange transformation of any number of intervals can also be modeled by a billiard trajectory in some convex polygon with many sides.

Negative refraction and tiling billiards

“…It is also possible to construct a table so that some trajectory on that table is dense in one part of the table, but never visits another part of the table at all. McMullen constructed an L-shaped example of this phenomenon [11], Galperin constructed examples on polygons with 4 or more sides [4], and recently Tokarsky asserted that Galperin's triangle construction is flawed, so the problem of constructing a not-everywhere-dense non-periodic billiard orbit on a triangle is still open [20].…”

On the mathematical transport theory in microporous media: The billiard approach

“…G.A. Galperin has recently proved that the conjecture is wrong [76]. He constructed a trajectory which is everywhere dense in some subdomain of a polygon P. Apparently this situation often occurs.…”

Trapping light rays aperiodically with mirrors

“…We would also like to mention that the approach used to prove Theorem 1 can be applied to polygons other than the square; our original proof was based on a quadrilateral with a nonperiodic billiard path constructed in [Galperin 1983].…”