Growth of the Wang-Casati-Prosen counter in an integrable billiard

TL;DR

This paper investigates the behavior of a generalized integer counter in irrational billiards, demonstrating indefinite growth in a specific case, which supports the idea of non-ergodicity and extends prior conjectures about billiard dynamics.

Contribution

It introduces a generalized counter for billiards, including rational cases, and shows its indefinite growth in a specific billiard, supporting non-ergodic behavior.

Findings

Counter grows indefinitely in a 45°:45°:90° billiard

Supports non-ergodic behavior in certain billiards

Extends analysis to rational billiards

Abstract

This work is motivated by an article by Wang, Casati, and Prosen [Phys. Rev. E vol. 89, 042918 (2014)] devoted to a study of ergodicity in two-dimensional irrational right-triangular billiards. Numerical results presented there suggest that these billiards are generally not ergodic. However, they become ergodic when the billiard angle is equal to $\pi/2$ times a Liouvillian irrational, a Liouvillian irrational, a class of irrational numbers which are well approximated by rationals. In particular, Wang et al. study a special integer counter that reflects the irrational contribution to the velocity orientation; they conjecture that this counter is localized in the generic case, but grows in the Liouvillian case. We propose a generalization of the Wang-Casati-Prosen counter: this generalization allows to include rational billiards into consideration. We show that in the case of a $45^{\circ} \!\! : \! 45^{\circ} \!\! : \! 90^{\circ}$ billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.