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

### Submission summary

 As Contributors: Maxim Olshanii Preprint link: scipost_202108_00061v2 Date accepted: 2022-08-01 Date submitted: 2022-06-24 05:34 Submitted by: Olshanii, Maxim Submitted to: SciPost Physics Academic field: Physics Specialties: Mathematical Physics Approaches: Theoretical, Computational

### 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 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: this generalization allows to include rational billiards into consideration. We show that in the case of a $\ang{45} \!\! : \!\ang{45} \!\! : \! \ang{90}$ billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

### Author comments upon resubmission

We thank the Referee 2 for s/his help with improving the accessibility of our manuscript. According to s/his suggestion we
(a) completely changed the order of presentation; (b) expanded the introduction by a factor of two; (c) clearly presented the position of our system of interest on the integrable->rational->Liouvillian->generic axis for right-triangular billiards. We want to express our wholehearted thank you: our manuscript improved visibly. Hoping for an affirmative answer.