On the Space of Ergodic Measures for the Horocycle Flow on Strata of Abelian Differentials

Abstract

We study the horocycle flow on the stratum of translation surfaces $\mathcal{H}(2)$. We show that there is a sequence of horocycle ergodic measures, each supported on a periodic horocycle orbit, which weakly converges to an invariant, but non-ergodic, measure by ${\mathrm{SL}}_2(\mathbb{R})$. As a consequence, we show that there are points in $\mathcal{H}(2)$ whose horocycle flow orbits do not equidistribute towards any invariant measure.

arXiv commentCorrections based on referee reports, clarifications of statements of main results and of the arguments in Section 7, expanded and clarified some of the discussions in the introduction. Main results are not affected

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