TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.14.2.017
TI  - Growth of the Wang-Casati-Prosen counter in an integrable billiard
PY  - 2023/02/10
UR  - https://www.scipost.org/SciPostPhys.14.2.017
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 14
IS  - 2
SP  - 017
A1  - Hwang, Zaijong
AU  - Marx, Christoph A.
AU  - Seaward, Joseph J.
AU  - Jitomirskaya, Svetlana
AU  - Olshanii, Maxim
AB  - 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.
ER  -

@Article{10.21468/SciPostPhys.14.2.017,
	title={{Growth of the Wang-Casati-Prosen counter in an integrable billiard}},
	author={Zaijong Hwang and Christoph A. Marx and Joseph J. Seaward and Svetlana Jitomirskaya and Maxim Olshanii},
	journal={SciPost Phys.},
	volume={14},
	pages={017},
	year={2023},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.14.2.017},
	url={https://scipost.org/10.21468/SciPostPhys.14.2.017},
}