可积台球中Wang-Casati-Prosen计数器的增长
Growth of the Wang-Casati-Prosen counter in an integrable billiard
AI总结：
针对二维无理直角台球的遍历性相关Wang-Casati-Prosen计数器猜想，本文推广该计数器以纳入有理台球，证明45°:45°:90°可积台球中计数器无限增长，验证了刘维尔情形下的相关结论。
摘要：
本工作的研究动机来自Wang、Casati与Prosen的一篇论文[Phys. Rev. E vol. 89, 042918 (2014)]，该论文聚焦二维无理直角台球的遍历性研究。文中给出的数值结果表明，这类台球通常不具备遍历性；但当台球角度等于π/2乘以一个刘维尔无理数（一类可被有理数良好逼近的无理数）时，系统会呈现遍历性。\nWang等人特别研究了一种特殊的整数计数器，该计数器反映了速度取向的无理数贡献；他们推测，在一般情形下该计数器是局域化的，但在刘维尔情形下会发生增长。我们提出了Wang-Casati-Prosen计数器的一种推广形式：该推广可将有理台球纳入研究范畴。我们证明，在45°:45°:90°台球的情形下，该计数器会无限增长，这与Wang等人提出的刘维尔情形的结论一致。
英文摘要：
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 $π/2$ times a Liouvillian irrational, a Liouvillian irrational, 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 $45^{\circ} \!\! : \! 45^{\circ} \!\! : \! 90^{\circ}$ billiard, the counter grows indefinitely, consistent with the Liouvillian scenario suggested by Wang et al.
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