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

### Submission summary

 As Contributors: Maxim Olshanii Preprint link: scipost_202108_00061v1 Date submitted: 2021-08-25 07:59 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 $\pi$ 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.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission scipost_202108_00061v2 on 24 June 2022

Submission scipost_202108_00061v1 on 25 August 2021

## Reports on this Submission

rigour

### Weaknesses

presentation, accessibility

### Report

The paper addresses a seemingly important problem of the properties of billiards in polygons (triangles), and whether ergodicity is or not attained depending on the arithmetic properties of the angles of the billiard. While the contribution is mathematically rigorous, and since the journal covers a wide spectrum of subjects, the paper deserves to be self-contained and accessible to non-specialists. I would suggest therefore that the authors start with a more thorough introduction, and maybe move the chapter named “Motivation” to the beginning. A more intuitive presentation of the methods used (with e.g. some extra figures) may help a quicker access to the results.

• validity: good
• significance: good
• originality: good
• clarity: low
• formatting: good
• grammar: good

### Strengths

Understanding the dynamcal and statistical properties of linearly unstable, parabolic maps and of billiards in polygons is a very important task. So far there are several numerical works on this subject and few rigourous mathematical results. On the other hand, numerical analysis is very delicate and mahematical analysis seems to be very hard. The present manuscript gives a rigourous contribution to this problem.

### Report

I recommend acceptance of this paper

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Maxim Olshanii  on 2022-04-21  [id 2404]

(in reply to Report 1 on 2022-03-29)

We are grateful to the Referee for s/his assessment of our manuscript and s/his warm words.