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

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, 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.

Publisher

SciPost Foundation

Publication Date

2-10-2023

Publication Title

SciPost Physics

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.21468/SciPostPhys.14.2.017

Keywords

Dimension

Language

English

Format

text

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