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Discretely shrinking targets in moduli space

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Abstract

We consider the discrete shrinking target problem for Teichmüller geodesic flow on the moduli space of abelian or quadratic differentials and prove that the discrete geodesic trajectory of almost every differential will hit a shrinking family of targets infinitely often provided the measures of the targets are not summable. This result applies to any ergodic \(\mathrm {SL}(2,\mathbb {R})\)–invariant measure and any nested family of spherical targets. Under stronger conditions on the targets, we moreover prove that almost every differential will eventually always hit the targets. As an application, we obtain a logarithm law describing the rate at which generic discrete trajectories accumulate on a given point in moduli space. These results build on work of Kelmer (Geom Funct Anal 27:1257–1287, 2017) and generalize theorems of Aimino, Nicol, and Todd (Ann Inst Henri Poincaré Probab Stat 53:1371–1401, 2017).

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Acknowledgements

The authors would like to thank Jayadev Athreya and Vaibhav Gadre for several helpful conversations and their willingness to answer numerous questions concerning the topics of this paper. We also thank the anonymous referee for their careful reading of the paper and helpful suggestions. The first named author was partially supported by NSF grant DMS-1711089.

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    Spencer Dowdall

  2. Department of Mathematics, University of Wisconsin, 480 Lincoln Drive 213 Van Vleck Hall, Madison, WI, 53706, USA

    Grace Work

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Correspondence to Spencer Dowdall.

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Dowdall, S., Work, G. Discretely shrinking targets in moduli space. Geom Dedicata 216, 53 (2022). https://doi.org/10.1007/s10711-022-00716-4

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