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In view of classical results of Masur and Veech almost every element in themoduli space of compact translation surfaces is recurrent. In this paper wefocus on the problem of recurrence for elements of smooth curves in the modulispace. We give an effective criterion for the recurrence of almost everyelement of a smooth curve. The criterion relies on results developed byMinsky-Weiss in \cite{Mi-We}. Next we apply the criterion to the billiard flowon planar tables confined by arcs of confocal conics. The phase space of suchbilliard flow splits into invariant subsets determined by caustics. We provethat for almost every caustic the billiard flow restricted to the correspondinginvariant set is uniquely ergodic. This answers affirmatively to a questionraised by Zorich.

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