A Poincar\'e section for horocycle flow on the space of lattices

TL;DR

This paper constructs a Poincaré section for horocycle flow on the modular surface, analyzes the first return map (BCZ map), and derives results on measure classification, equidistribution, and applications to Farey sequences and lattice vectors.

Contribution

It introduces a Poincaré section for the horocycle flow and studies the BCZ map, providing new insights into measure classification and distribution properties.

Findings

Classified ergodic invariant measures for the BCZ map.

Proved equidistribution of periodic orbits.

Derived results on cusp excursions and Farey sequence gaps.

Abstract

We construct a Poincar\'e section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. 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.