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

2021/04/01 by Jon Chaika, Chaika, Jon, Osama Khalil +3 · 1 citation
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2104.00554

openalex publication_date 2021/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the horocycle flow on the stratum of translation surfaces 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 SL₂(ℝ). As a consequence, we show that there are points in H(2) whose horocycle flow orbits do not equidistribute towards any invariant measure.

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• Veech Surfaces and Expanding Twist Tori on Moduli Spaces of Abelian Differentials

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