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

by Z. Hwang, C. A. Marx, J. Seaward, S. Jitomirskaya, M. Olshanii

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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 $\pi$ times a Liouvillian irrational, a class of numbers with properties lying in between irrational and rational. 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 for rational billiards. We show that in the case of a $\ang{45} \!\! : \!\ang{45} \!\! : \! \ang{90}$ billiards, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.