Perfect retroreflectors and billiard dynamics

Abstract

We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in the limit when the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties.

A solution to Newton's least resistance problem is uniquely defined by its singular set

“…We assume that the particles move upward along the z-axis. Let the lower part of the body's surface be the graph of a convex radially symmetric function z = u(x 1 , x 2 ) = ϕ( x 2 1 + x 2 2 ), x 2 1 + x 2 2 ≤ L 2 ; then the resistance equals…”

“…The problem for nonconvex bodies is by now well understood [7,8,15,1,17,18]. Generalizations of the problem to the case of rotating bodies have been studied [12,13,22,23], and connections with the phenomena of invisibility, retro-reflection, and Magnus effect in geometric optics and mechanics have been established [21,19,24,2,14,22]. The methods of billiards, Kakeya needle problem, optimal mass transport have been used in these studies.…”

“…in the class C M , where f : R 2 → R is a continuous function. 2 Taking f (ξ) = 1/(1 + |ξ| 2 ), one obtains the functional (1).…”

Mathematical retroreflectors

“…They form a small collection of four objects; the first, the second and the fourth one are asymptotically perfect retroreflectors, and the third one is a retroreflector which is very close to perfect. The first three objects mushroom, tube and helmet have already been published or submitted for publication [12,1,6]. Note that the proof of retroreflectivity for the tube reduces to a quite nontrivial ergodic problem considered in [1].…”

Geometry of Refractions and Reflections Through a Biperiodic Medium

“…This is sometimes called open billiards. Typical questions asked in this context are how many orbits never fall in the hole, how does this change with the width of the hole, how long do orbits take to leave through a hole [5,7] (because many billiard systems are ergodic, almost all solutions do eventually leave), how does this change with the shape of the billiard table [3]?…”