Growth of the Wang-Casati-Prosen counter in an integrable billiard
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, morally 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 degrees:45 degrees:90 degrees billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.
Hwang, Zaijong, Christoph A. Marx, Joseph J. Seaward, et al. 2023. "Growth of the Wang-Casati-Prosen counter in an integrable billiard." SciPost Physics 14(2): Article 017.