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. 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.
Keywords
billiard flows, unique ergodicity, the moduli space of
translation surfaces, recurrent points of the Teichmüller
flow
