For the stratum H(2) of translation surfaces, long horocycle orbits have discrete transverse dimension arbitrarily close to 1, with effective exponential error rates.
Quantitative recurrence and large deviations for teichmuller geodesic flow
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Effective Exponential Drifts on Strata of Abelian Differentials
For the stratum H(2) of translation surfaces, long horocycle orbits have discrete transverse dimension arbitrarily close to 1, with effective exponential error rates.