A Poincaré Section for the Horocycle Flow on the Space of Lattices

Abstract

Abstract. We construct a Poincaré 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 BCZ map) defined by Boca-Cobeli-Zaharescu [8]. 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.

Entry and return times for semi-flows

“…A further possibility is to fix D and consider a sequence of invariant measures µ n ∈ A . The latter setting is for instance relevant for the gap statistics of Farey fractions via the horocycle flow: Take ν n to be the invariant measure supported on a closed horocycle of length ℓ n → ∞, and µ n the measure supported on periodic points of the return map T for a certain section (the Boca-Cobeli-Zaharescu map) [5]. The equidistribution of long closed horocycles implies the weak convergence ν n w −→ ν and µ n w −→ µ, which in turn implies ξ n d −→ ξ in this setting; see [4] for more examples of this type and [34] for the higher-dimensional analogue.…”

Equidistribution of Farey sequences on horospheres in covers of and applications

“…In dimension n = 1, Athreya and Cheung [1] provided a unified explanation for various statistical properties of Farey fractions, which were originally proven by analytic methods, by realizing the horocycle flow in SL(2, R)/SL(2, Z) as a suspension flow over the BCZ map introduced by Boca, Cobeli, and Zaharescu [5] in their study of Farey fractions. The author's recent work [10] used a process of Fisher and Schmidt [9] to lift the Poincaré section of Athreya and Cheung to obtain sections of the horocycle flow in covers SL(2, R)/∆ of SL(2, R)/SL (2, Z), which in turn yielded results on the spacing statistics of the various subsets of Farey fractions related to those lifted sections.…”

Poincaré sections for the horocycle flow in covers of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ and applications to Farey fraction statistics

“…The significant work of Elkies and McMullen [13] and of Marklof and Strömbergsson [25] has demonstrated that ergodic properties of homogeneous flows can provide a very powerful device in the study of the limiting gap distributions of certain sequences of arithmetic origin. Recently, Athreya and Cheung [3] realized the horocycle flow on SL(2, R)/SL(2, Z) as a suspension flow over the BCZ map introduced by Boca, Cobeli, and Zaharescu [9] in their study of statistical properties of Farey fractions. Athreya and Cheung used this connection to rederive the limiting gap distribution and other properties of Farey fractions.…”

“…In this paper, we use the process in [16] to explicitly lift, for every finite index subgroup H ⊆ SL(2, Z), the section of the horocycle flow discovered in [3] to the cover SL(2, R)/H of SL(2, R)/SL (2, Z). As one application, we prove in Theorem 1 the existence of the limiting gap measure of certain subsets of Farey fractions following the ideas in [3] and the more general framework of [1,Theorem 2.5].…”