Recurrence for smooth curves in the moduli space and an application to the billiard flow on nibbled ellipses

Abstract

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. (4) 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.