In view of classical results of Masur and Veech almost every element in the moduli space of compact translation surfaces is recurrent; i.e., its Teichmüller positive semiorbit returns to a compact subset infinitely many times. We focus on the problem of recurrence for elements of smooth curves in the moduli space. We give an effective criterion for the recurrence of almost every element of a smooth curve. The criterion relies on results developed by Minsky and Weiss (Ann. Sci. École Norm. Sup. $\left(4\right)$ 47:2 (2014), 245–284). Next we apply the criterion to the billiard flow on planar tables confined by arcs of confocal conics. The phase space of such a billiard flow splits into invariant subsets determined by caustics. We prove that for almost every caustic the billiard flow restricted to the corresponding invariant set is uniquely ergodic. This answers affirmatively a question raised by Zorich.

鉴于Masur和Veech的经典结果，紧致平移曲面的模空间中的几乎每个元素都是递归的。即，它的Teichmüller正半轨道无限次返回紧凑的子集。我们关注模空间中光滑曲线元素的递归问题。我们为平滑曲线的几乎每个元素的重复提供了有效的标准。该标准依赖于明斯基和Weiss（开发结果安，科学。É油菜规范。 燮。$（4）$ 47：2（2014），245-284）。接下来，我们将该标准应用于由共焦圆锥曲线的弧约束的平面桌上的台球流动。这样的台球流的相空间分裂成由焦散确定的不变子集。我们证明，对于几乎每个苛性碱，限制在相应不变集上的台球流量都是唯一遍历遍历的。这肯定回答了佐里奇提出的一个问题。