Impact of symmetry on ergodic properties of triangular billiards

Abstract

Polygonal billiards constitute some of the simplest yet counterintuitive dynamical systems in physics. Even basic features of the dynamics, such as ergodicity of the microcanonical distribution or the decay of correlations have not been settled in general. In this numerical study, we will highlight the importance of symmetries of the billiard table for the resulting dynamics. While typical triangular billiards appear to show correlation decay, symmetric billiards may not even be ergodic with respect to the uniform distribution in phase space.

Quantum chaos in triangular billiards

“…Based on numerical evidence, generic triangles (all angles have irrational ratios with π) are believed to be strongly-mixing [15]. On the other hand, irrational triangles with one rational angle, like for instance right-triangles, have weaker ergodic properties [16][17][18] and the most recent numerical evidence suggests they are not ergodic in the Lebesgue measure [19][20][21]. The quantum triangular billiards of these classes have been explored in [22].…”