A poincaré section for the horocycle flow on the space of lattices

Jayadev S. Athreya; Yitwah Cheung

doi:10.1093/imrn/rnt003

A poincaré section for the horocycle flow on the space of lattices

Jayadev S. Athreya, Yitwah Cheung

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We construct a Poincaré section for the horocycle flow on the modular surface, and study the associated first return map, which coincides with a transformation [the Boca-Cobeli-Zaharescu (BCZ) map] defined by Boca et al. [7]. We classify ergodic invariant measures for this map and prove equidistribution of periodic orbits. As corollaries, we obtain results on the average depth of cusp excursions and on the distribution of gaps for Farey sequences and slopes of lattice vectors.

Original language	English (US)
Pages (from-to)	2643-2690
Number of pages	48
Journal	International Mathematics Research Notices
Volume	2014
Issue number	10
DOIs	https://doi.org/10.1093/imrn/rnt003
State	Published - 2014

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Mathematics(all)

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abstract = "We construct a Poincar{\'e} section for the horocycle flow on the modular surface, and study the associated first return map, which coincides with a transformation [the Boca-Cobeli-Zaharescu (BCZ) map] defined by Boca et al. [7]. We classify ergodic invariant measures for this map and prove equidistribution of periodic orbits. As corollaries, we obtain results on the average depth of cusp excursions and on the distribution of gaps for Farey sequences and slopes of lattice vectors.",

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AB - We construct a Poincaré section for the horocycle flow on the modular surface, and study the associated first return map, which coincides with a transformation [the Boca-Cobeli-Zaharescu (BCZ) map] defined by Boca et al. [7]. We classify ergodic invariant measures for this map and prove equidistribution of periodic orbits. As corollaries, we obtain results on the average depth of cusp excursions and on the distribution of gaps for Farey sequences and slopes of lattice vectors.

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A poincaré section for the horocycle flow on the space of lattices

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