SciPost logo

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

Zaijong Hwang, Christoph A. Marx, Joseph J. Seaward, Svetlana Jitomirskaya, Maxim Olshanii

SciPost Phys. 14, 017 (2023) · published 10 February 2023


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 $45°\!\!:\!45°\!\!:\!90°$ billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.

Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.
Funders for the research work leading to this publication