Vol. 14, No. 3, 2021

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

Krzysztof Frączek

Vol. 14 (2021), No. 3, 793–821

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.

billiard flows, unique ergodicity, the moduli space of translation surfaces, recurrent points of the Teichmüller flow
Mathematical Subject Classification 2010
Primary: 37A10, 37D40, 37E35, 37J35, 37D50
Received: 18 December 2018
Revised: 27 August 2019
Accepted: 21 November 2019
Published: 18 May 2021
Krzysztof Frączek
Faculty of Mathematics and Computer Science
Nicolaus Copernicus University