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

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arxiv 2104.00554 v3 pith:OTUO6ZLA submitted 2021-04-01 math.DS math.GT

Jon Chaika , Osama Khalil , John Smillie This is my paper · ORCID
classification math.DSmath.GT
keywords horocycleflowergodicinvariantmathcalmeasuremeasuresthere
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.DS 2025-07 conditional novelty 7.0 of 10

    For certain Veech surfaces, the paper shows that the Teichmüller geodesic flow pushes twist tori to dense sets, and under an algebraic monodromy condition every weak-* limit is fully supported.

  2. Effective Exponential Drifts on Strata of Abelian Differentials

    math.DS 2025-01 conditional novelty 7.0 of 10

    For the stratum H(2) of translation surfaces, long horocycle orbits have discrete transverse dimension arbitrarily close to 1, with effective exponential error rates.

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