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Horocycle Dynamics: New Invariants and Eigenform Loci in the Stratum $\mathcal {H}$(1,1)
About this Title
Matt Bainbridge, John Smillie and Barak Weiss
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 280, Number 1384
ISBNs: 978-1-4704-5539-2 (print); 978-1-4704-7284-9 (online)
DOI: https://doi.org/10.1090/memo/1384
Published electronically: October 7, 2022
Keywords: Flat surfaces,
strata,
horocycle flow,
eigenform loci,
orbit closures,
invariant measures
Table of Contents
Chapters
- 1. Introduction
- 2. Strata
- 3. Blowups of Translation Surfaces
- 4. The Rel Foliation and Rel Vectorfields
- 5. Horizontal Equivalence of Surfaces
- 6. An Explicit Surgery for Real Rel
- 7. The Eigenform Locus
- 8. Construction of $U$-invariant Ergodic Measures in $\mathcal {E}_D$
- 9. Classification of Ergodic Measures
- 10. Injectivity and Nondivergence
- 11. All Horocycle Orbits are Generic
- 12. Equidistribution Results for Sequences of Measures
Abstract
We study dynamics of the horocycle flow on strata of translation surfaces, introduce new invariants for ergodic measures, and analyze the interaction of the horocycle flow and real Rel surgeries. We use this analysis to complete and extend results of Calta and Wortman classifying horocycle-invariant measures in the eigenform loci. In addition we classify the horocycle orbit-closures and prove that every orbit is equidistributed in its orbit-closure. We also prove equidistribution results describing limits of sequences of measures. Our results have applications to the problem of counting closed trajectories on translation surfaces of genus 2.- Artur Avila, Alex Eskin, and Martin Möller, Symplectic and isometric $\textrm {SL}(2,\Bbb R)$-invariant subbundles of the Hodge bundle, J. Reine Angew. Math. 732 (2017), 1–20. MR 3717086, DOI 10.1515/crelle-2014-0142
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