Spherical billiards with almost complete escape

Abstract

A dynamical billiard consists of a point particle moving uniformly except for mirror-like collisions with the boundary. Recent work has described the escape of the particle through a hole in the boundary of a circular or spherical billiard, making connections with the Riemann Hypothesis. Unlike the circular case, the sphere with a single hole leads to a non-zero probability of never escaping. Here, we study variants in which almost all initial conditions escape, with multiple small holes or a thin strip. We show that equal spacing of holes around the equator is an efficient means of ensuring almost complete escape and study the long time survival probability for small holes analytically and numerically. We find that it approaches a universal function of a single parameter, hole area multiplied by time.

Chaos Detection by Fast Dynamic Indicators in Reflecting Billiards

“…Deformations of the boundary cause the loss of integrability and the emergence of chaotic behavior in the particles or rays trajectories [ 22 ]. Even if the dynamics of integrable billiards are still an exciting research topic that continuously unveils novel phenomena, a growing interest has been developing within the scientific community for the properties of chaotic billiards [ 23 , 24 , 25 , 26 , 27 ].…”