On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces (Q2908143)

In this paper, the authors study the conservativity, ergodicity and related asymptotic properties of billiard dynamics on a non-compact polygonal surface \(\widetilde{P}\) admitting a free action of \(\mathbb{Z}\) or \(\mathbb{Z}^2\), such that the quotient surface \(P\) is compact.NEWLINENEWLINEThey prove that the billiard flow on a \(\mathbb{Z}\)-periodic surface \(\widetilde{P}\) is conservative if the base billiard on the quotient surface \(P\) is ergodic. Moreover, for some rational polygons, they prove that, along almost every direction, the directional billiard flow is conservative, with zero frequency and unbounded oscillations. They also strengthen the results if the table is arithmetic.NEWLINENEWLINEIn the \(\mathbb{Z}^2\)-periodic case, they focus on the rectangular Lorentz gas with small obstacles \(R(a,b)\) (in the sense that \(qa+pb\leq1\) with respect to some direction \((p,q)\)), and give explicitly a finite ergodic decomposition of the directional billiard system int the direction \((p,q)\).NEWLINENEWLINETheir approach is to identify the billiard systems with the skew products over interval-exchange transformations and reduce to the study of cocycles over irrational rotations.