e 0 e t/2 : t R 1 0 N = h s = s 1

Transcription

1 Athreya, JS, and YCheung 009 A Poincaré section for the horocycle flow on the space of lattices, International Mathematics Research Notices, Vol 009, Article ID rnn999, 3 pages doi:0093/imrn/rnn999 A Poincaré section for the horocycle flow on the space of lattices Jayadev S Athreya and Yitwah Cheung Department of Mathematics, University of Illinois Urbana-Champaign, 409 W Green Street, Urbana, IL 680 Department of Mathematics, San Francisco State University, Thornton Hall 937, 600 Holloway Ave, San Francisco, CA 943 Correspondence to be sent to: jathreya@illinoisedu We construct a Poincaré section for the horocycle flow on the modular surface SL, R/SL, 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 Introduction Let X SL, R/SL, Z be the space of unimodular lattices in R X is a non-compact finite-volume with respect to Haar measure on SL, R homogeneous space X can also be viewed as the unit-tangent bundle to the hyperbolic orbifold H /SL, Z The action of various one-parameter subgroups of SL, R on X via left multiplication give several important examples of dynamics on homogeneous spaces, and have close links to geometry and number theory For example, the action of the subgroup A : { g t e t/ 0 0 e t/ } : t R yields the geodesic flow on X, whose orbits are hyperbolic geodesics when projected to H /SL, Z This flow can be realized as an suspension flow over the natural extension of the Gauss map Gx { x}, and this connection can be exploited to give many connections between the theory of continued fractions and hyperbolic geometry see the beautiful articles by Series [33] or Arnoux [] for very elegant expositions In this paper, we study the horocycle flow, that is, the action of the subgroup { } 0 N h s : s R s The main result Theorem of this paper displays the horocycle flow as a suspension flow over the BCZ map, which was constructed by Boca-Cobeli-Zaharescu [8] in their study of Farey fractions Equivalently, we construct a transversal to the horocycle flow so that the first return map is the BCZ map This enables us to use well-known ergodic and equidistribution properties of the horocycle flow to derive equivalent properties for the BCZ map in particular that it is ergodic, zero entropy, and that periodic orbits equidistribute, and give a unified explanation of several number-theoretic results on the statistical properties of Farey gaps We also give a piecewise-linear description of the cusp excursions of the horocycle flow, and derive several new results on the geometry of numbers relating to gaps of slopes of lattice vectors Plan of paper We describe the organization of the paper, and also give a guide for readers Received January 009; Revised January 009; Accepted January 009 Communicated by A Editor c The Author 009 Published by Oxford University Press All rights reserved For permissions,

2 JS Athreya and YCheung Organization This paper is organized as follows: in the remainder of the introduction, we state our results: in 3, we describe the transversal to the horocycle flow, and state Theorem In 4, we discuss the ergodic properties of the BCZ map 4 and the structure and equidistribution of periodic orbits 43 A piecewise-linear description of horocycle cusp excursions is given in 5; a unified approach to Farey statistics is described in 6; and a similar approach to statistics of gaps in slopes of lattice vectors is the subject of 7 In, we prove Theorem, and show how to describe it using Euclidean and hyperbolic geometry The structure of periodic orbits is the subject of 4; and in 5, these structure results are used to obtain equidistribution properties and the corollaries on Farey statistics in 5 3 contains the proof of ergodicity and the calculation of entropy of the BCZ map We prove our results on cusp excursions in 6, and in 7 we prove our results on geometry of numbers Finally, in 8, we outline some questions and directions for future research Readers guide Since this paper touches on several different topics, it can be read in several different ways We suggest different approaches for the ergodic-theoretic and number-theoretic minded readers We recommend that the ergodictheoretic reader start with sections 3 and 4 and perhaps 5, and follow it with, 3 and 6 before exploring the more number theoretic parts of the paper The number-theoretic reader should also start with 3 and 4, but then may be more intrigued by the results of 43, 6, and 7 and their proofs in 4, 5, and 7 respectively Acknowledgments and Funding The original motivation for this project was the study of cusp excursions for horocycle flow on X as described in 5 We developed the first return map T as a technical tool in order to study these excursions Later, the first-named author was attending a talk by F Boca in which the map T was written down in the context of studying gaps in angles between hyperbolic lattice points After many invaluable and illuminating conversations with F Boca and A Zaharescu, we understood the deep connections they had developed between orbits of this map and the study of Farey sequences It is a pleasure to thank them for their insights and assistance We would also like to thank Jens Marklof for numerous helpful discussions and insightful comments on an early version of this draft We thank the organizers of the Summer School and Conference on Teichmüller dynamics in Marseilles in June 009, when this project was originally developed In addition, JSA would like to thank San Francisco State University for its hospitality JSA supported by National Science Foundation grant number DMS YC supported by National Science Foundation CAREER grant number DMS Description of transversal Recall that X can be explicitly identified with the space of unimodular lattices via the identification gsl, Z gz Let P { } a b p a,b 0 a : a R, b R denote the group of upper-triangular matrices in SL, R Let We will use Ω to denote the set Ω : {p a,b SL, Z : a, b 0, ], a + b > } X {a, b R : a, b 0, ], a + b > } R In, we will show that Ω can be viewed as the set of lattices with a horizontal vector of length at most, can also be identified with the subset { z x + iy H : < x, y >, } H of the upper-half plane Our main theorem is that the Ω is a Poincaré section for the horocycle flow, and that the first return map is the BCZ map We see the space Ω together with the roof in Figure

3 A Poincaré section for horocycle flow 3 Fig A picture of the suspension space over Ω Trajectories of the flow are vertical lines Starting from a, b Ω, h s trajectories move vertically until they hit the roof at time Ra, b upon which they return to the floor at position T a, b Theorem Ω X is a Poincaré section for the action of N on X That is, every N-orbit {h s Λ} s R with the exception of the codimension set of lattices Λ with a length vertical vector, Λ X, intersects Ω in a non-empty, countable, discrete set of times {s n } n Z Given Λ a,b p a,b SL, Z, the first return time Ra, b min{s > 0 : h s Λ a,b Ω } is given by The first return map T : Ω Ω defined implicitly by Ra, b ab 3 Λ T a,b h Ra,b x a,b is given explicitly by the BCZ map T a, b b, a + + a b b 4 Remarks: Short periodic orbits We will see below that lattices with a length vertical vector consist of those whose horocycle orbits are periodic of period at most, and are embedded closed horocycles in H /SL, Z, foliating the cusp Thus, from a dynamical point of view, there is no loss in missing them Integrability of R A direct calculation shows that π Rdadb Ω 3, which is the volume of X when viewed as the unit tangent bundle of H /SL, Z with respect to the measure dxdydθ y It is also immediate that R L p dm for p <, where dm dadb 4 Return map and applications The BCZ map has proved to be a powerful technical tool in studying various statistical properties of Farey fractions [5, 8, 0], and the distribution of angles of families of hyperbolic geodesics []

4 4 JS Athreya and YCheung Fig The Farey triangle Ω k Ω k 0, / Ω Ω /3 Ω 3 Ω 4 /4 /5 /6 /7 Ω 7, 0, 0 4 Tiles and images We briefly describe the basic structure of the piecewise linear decomposition of T 4 tells us that the map T acts via T a, b a, ba T k, where the superscript T denotes transpose, and 0 A k k see Figure below Ω is a triangle with 3, 3 ; and for k, Ωk is a quadrilateral with vertices at, k,, k+, Note that areaω 6, andfor k, areaω 4 k kk+k+ Ok 3 so that on the region Ω k : {a, b Ω : κa, b k}, where κa, b +a b vertices at 0,,,, and k k+, k+, and k k+, k+ κ L p m for p < The image T Ω k is the reflection of Ω k about the line a b see Figure 3 However, T is orientationpreserving, and thus does not act by the reflection For k, T acts by the elliptic matrix A, for k the parabolic matrix A, and for k 3, by the hyperbolic matrices, since tracea k k 4 Ergodic properties of the BCZ map In [7, 3], Boca-Zaharescu posed a series of questions on the ergodic properties of T : uestion Is T ergodic with respect to Lebesgue measure? Weak mixing? What is the entropy of T? A corollary of Theorem and the ergodicity and entropy properties of the horocycle flow is: Theorem T is an ergodic, zero-entropy map with respect to the Lebesgue probability measure dm dadb Moreover, m is the unique absolutely continuous invariant probability measure, and in fact is the unique ergodic invariant measure not supported on a periodic orbit 43 Periodic orbits The map T has a rich and intricate structure of periodic orbits, closely related to the Farey sequences A direct calculation shows that for N the point Λ, p, Z is h s -periodic with minimal period Given N, the Farey sequence F is the collection in increasing order of fractions 0 < p q < with q Let N N denote the cardinality of F We write F { 0 γ < γ < < γ N} For notational convenience, we write γ N+ Writing γ i pi q i, with q i, we have q i + q i+ >, and p i+ q i p i q i+

5 A Poincaré section for horocycle flow 5 Fig 3 The images T Ω k 0, T Ω 7 T Ω 4 T Ω 3 T Ω T Ω, 0 0, The following fundamental observation is due to Boca-Cobeli-Zaharescu [8]: qi T, q i+ qi+, q i+ 5 The indices are viewed cyclically in Z/NZ We give a geometric explanation for 5 in 4 below Thus,, is a periodic point of order N for T To relate this to the periodic orbit of Λ, for h s, we first record two simple observations First, q, q,, and second, i q i q i+ Multiplying both sides by, and applying 5, we obtain i q i q i+ i γ i+ γ i i R T i, Thus, as we sum the roof function R over the periodic orbit for T, we obtain the length of the associated periodic orbit for the flow We recall Sarnak [3] showed that long periodic orbits for h s ie, long closed horocycles become equidstributed with respect to µ, the Haar measure on X Our main result on the distribution of periodic orbits is the following discrete corollary of Sarnak s result: let ρ,i denote the probability measure supported on a long piece of the orbit of, : given I [α, β] [0, ], let N I : F I, and define ρ,i N I δ T i, i:γ i I

6 6 JS Athreya and YCheung If I [0, ], we write ρ,i ρ Theorem 3 For any non-empty interval I [α, β] [0, ], the measures ρ,i become equidistributed with respect to m as That is, ρ,i m, where convergence is in the weak- topology Remark The case I [0, ] of Theorem 3 was proven in [7] using different methods Also, Theorem 3 can be deduced from Theorem 6 in [8], which is proven in the more general context of horospherical flows using a section that reduces to the same one in Theorem Theorem 4 A point a, b Ω is T -periodic if and only if it has rational slope In particular, the set of periodic points is dense While periodic points are abundant, there are strong restrictions on the lengths of periodic orbits, and the associated matrices, governed by the relationship between the discrete period P a, b under T of a point a, b and the continuous period sa, b of the associated lattice p a,b Z under h s We fix notation: for n, we write so A n a, b A T n a, b A T n a, b Aa, b, T n a, b a, ba n a, b T As a starting point for our observations, note that the diagonal a, a Ω consists of fixed ie, period P a, a points for T, the associated horocycle period is the value roof function Ra, a a, and the associated matrix 0 A P a,a a, a Aa, a A is parabolic Our main result on periodic points is that appropriate versions of these observations hold for all segments of periodic points Theorem 5 For any periodic point a, b Ω, we have P a, b N sa, b Thus the set of possible periods is given by the cardinalities {N : N} of Farey sequences Moreover, the matrix A P a,b a, b A T P a,b a, b A T P a,b a, b Aa, b associated { to the periodic orbit is always parabolic, and in fact constant along the segment ta, tb : t a+b, ]} Precisely, for k, l N, k l relatively prime, and a l l+r, l l+r ], r k, we have P a, a k Nl + r, l and for a l l+k, ], A Pa,a k l a, a k kl l l k + kl 5 Piecewise linear description of horocycles In this section, we describe the results that originally motivated this project, on cusp excursions and returns to compact sets for the horocycle flow Let denote the supremum norm on R For Λ gz X, define and let α : X R + be given by lλ : inf v, 0 v gz αλ lλ By Mahler s compactness criterion, a subset A of X is precompact if and only if there is an ɛ > 0 so that for all Λ A, lλ > ɛ,

7 A Poincaré section for horocycle flow 7 or, equivalently if α A is bounded Dani [5] showed that for any lattice Λ X, the orbit {h s Λ} s 0 is either closed or uniformly distributed with respect to the Haar probability measure µ on X Thus for an Λ so that {h s Λ} s 0 is not closed, we have and lim sup α h s Λ s lim sup lh s Λ 0 s In fact, this does not require equidistribution but simply density due to Hedlund [4] {h s Λ} is closed if and only if Λ has vertical vectors A natural question is to understand the rate at which these excursions to the non-compact part cusp of X occur The first-named author, in joint work with G Margulis [4], showed that for any Λ X without vertical vectors, lim sup s log α h s Λ log s and related the precise limit to Diophantine properties of the lattice Λ see also [] Our results concern the average behavior of all visits to the cusp, as defined by local minima of the function l Λ s lh s Λ or, equivalently, local maxima of α Λ s αh s Λ The function l Λ s is a piecewiselinear function of s, and helps give a picture of the height of the horocycle at time s In this way it is similar to work of the second author [4], where a piecewise linear description of diagonal flows on SL3, R/SL3, Z was studied We also consider returns to the compact part of X, given by local minima of l Λ s Given Λ X so that {h s Λ} s 0 is not closed, let {s n } and {S n } denote the sequences of minima and maxima of l Λ s respectively To be completely precise, the local minima for the supremum norm occur in intervals, and we take s n to be the midpoint of the n th interval Define the averages a N Λ : N l N Λ : N, α Λ s n ; A N Λ : N n l Λ s n ; L N Λ : N n Theorem 6 For any Λ without vertical vectors, we have α Λ S n, n l Λ S n n lim a NΛ 6 N lim N l NΛ 3 7 Let M : Ω [0, ] be given by { } Ma, b max a, b, a + b Theorem 7 For any Λ without vertical vectors, we have lim A N Λ N lim L NΛ N Ω 8 M dm Ω Mdm Theorem 6 and Theorem 7 follow from applying the ergodic theorem to the BCZ transformation T : Ω Ω The assumption that Λ does not have vertical vectors is to ensure it does not have a periodic orbit under T equivalently, h s There is a version of this result for periodic orbits Corollary given in 65 The function M gives the maximum of the function l on a sojourn from the transversal Ω The limits 6 and 7 in Theorem 6 are the integrals of the functions ga, b Λ and ha, b Λ over Ω respectively The times s n correspond to the return times of {h s Λ} to Ω

8 8 JS Athreya and YCheung 6 Farey Statistics Theorem 3 has several number theoretic corollaries We record results on spacing and indices of Farey fractions, originally proved using analytic methods in [5, 8, 0] Our results give a unified explanation for these equidistribution phenomena For similar applications in the context of higher dimensional generalizations of Farey sequences we refer the reader to [30] and [9] 6 Spacings and h-spacings We now fix an interval I [0, ] It is well known that the Farey sequences F I become equidistributed in [0, as A natural statistical question is to understand the distribution of the spacings γ i+ γ i Since there are N I points, and N I I 3 π, the natural normalization yields the following question: given 0 c d, what is the limiting behavior of 3 {γ i F I : π I γ i+ γ i [c, d]}? N I Since γ i+ γ i R T i,, the above quantity reduces to [ ] Since R π 3 I c, π 3II d R R Hall [0]: ρ,i R [ π 3 I ] π c, 3 I d is compact, we can directly apply Theorem 3 to obtain a result of Corollary 8 As, 3 {γ i F I : π I γ i+ γ i c, d} π m R π c, N I 3 I 3 I d We call this distribution Hall s distribution A generalization of this result to higher-order spacings was considered by Augustin-Boca-Cobeli-Zaharescu [5] They considered h-spacings: the vector of h-tuples h of spacings v i,h v i,h γ i+j γ i+j h j Rh Given B k i [c i, d i ], we have, as above, { γ i F I : 3 π I v i,h B } N I ρ,i R B h, where R h : Ω R h is given by and R h a, b R T j a, b h j, k [ ] B π π 3 I B 3 I c i, π 3 I d i i Applying our equidistribution result to, we recover Theorem of [5]: Corollary 9 As, {γ i F I : 3 π v i,h B} N I m R B h

9 } A Poincaré section for horocycle flow 9 Fig 4 The distribution function for the gaps [7, Figure ] ˆ ˆ ˆ and density functions of ν 6 Density of the limiting distribution Corollary 8 allows one to explicitly calculate the density of the limiting distribution of consecutive gaps, that is, of Hall s distribution For d > 0, we define the distribution function G c d to be the asymptotic proportion of normalized gaps of size at most b, that is, By Corollary 8, we can write 3 {γ i F I : π I γ G c d lim i+ γ i 0, d} N I G c d m R 0, π 3 I d Since for d < 3 I π, the curve Ra, b π 3 I d does not intersect Ω, we have G cd 0 In particular, the distribution does not have any support at 0 For d 3 I π, G c d > 0 The distribution has another point of non-differentiability at d 4 3 π, when the curve Ra, b 3 I π d intersects the bottom line a + b The picture of the distribution for I [0, is given in Figure 4, reproduced from [7, Figure ] In Figure 5, we display a picture of the region R [c, d] Fig 5 The region R [c, d] Ω Indices of Farey fractions Following [], we define the index of the Farey fraction γ i ai q i νγ i : q i + q i+ q i F by + qi q i

10 0 JS Athreya and YCheung +x In [8], this was reinterpreted in terms of the map T and the function κ : Ω N Recall that κx, y y Precisely, we have qi νγ i κ, q i κ T i, Fix a non-trivial interval I [0,, and an exponent α 0, We consider the average ρ,i κ α N I γ i F I νγ i α Applying a strengthening oftheorem 3 to κ α for which we need to estimate the integrals of κ α, see 5, we obtain a result originally due to Boca-Gologan-Zaharescu [0]: Corollary 0 As, Here Ω k κ k N I γ i F I νγ i α Ω κ α dm k α areaω k k 64 Powers of denominators We can also obtain general results on sums associated to Farey fractions by applying Theorem 3 to various functions in L Ω, m For example, if we define f s,t : Ω C by for s, t C with Rs, Rt >, and define B s,t : f s,t Ω f s,t a, b a s b t, f s,t x, ydm we obtain results originally due to Hall-Tanenbaum [] Γs + Γt + s + t + Γs + t + 3 Theorem Fix a non-trivial interval I [0, Let s, t C Then, if Rs, Rt >, lim N I s+t qi s qi+ t x s y t dm B s,t 3 In particular, for s and t 0, we have lim N I γ i F I γ i F I Ω, q i xdm Ω 3 4 In addition, for s and t 0, lim N I γ i F I dm, 5 q i Ω x and for s t, we have lim N I γ i F I π dm q i q i+ Ω xy 3, 6 which yields the classical result N I I 3 π

11 A Poincaré section for horocycle flow 65 Excursions Finally, we remark that applying the equisitribution result Theorem 3 to the functions M and M defined in 8 above, we obtain an amusing statistical result on Farey fractions: Corollary Let I [0, be a non-trivial interval Then lim lim N I N I γ i F I γ i F I min,, q i + q i+ 3 8 q i q i+ 3 qi max, q i+, q i + q i Geometry of numbers Let Λ X be a unimodular lattice, and fix t > 0 Let S t Λ {s < s < < s n < } denote the slopes of the lattice vectors in the vertical strip V t R given by V t : {x, y : x 0, t], y > 0}, written in increasing order Let G N,t Λ {s n+ s n : 0 n N} denote the sequence of gaps in this sequence, viewed as a set, so G N,t Λ N Our main geometry of numbers result states that for lattices Λ without vertical vectors, this sequences has the same limiting distribution as the gaps for Farey fractions, namely, Hall s distribution That is: Theorem 3 Suppose Λ does not have vertical vectors, and let 0 c d Then lim N N G N,tΛ c, d mr c, d This result will follow from the application of the Birkhoff ergodic theorem to the orbit of Λ under the BCZ map, with observable given by the indicator function of the set R c, d If Λ does have a vertical vector, and thus a periodic orbit under the BCZ map, we have that for any t > 0, there is an N 0 N 0 t > 0 so that G N,t Λ G N0,tΛ as sets of numbers for all N > N 0 We can use Theorem 3 to obtain the following: Corollary 4 Let 0 c d Then lim t N 0 t G N 0t,tΛ c, d mr c, d Construction of Transversal In this section, we prove Theorem We first give an interpretation of the transversal Ω in terms of geometry of numbers in ; and prove Theorem in We also record some observations on slopes in We show how Ω can be interpreted in terms of hyperbolic geometry in In 3, we show a certain self-similarity property of the BCZ map T

12 JS Athreya and YCheung Short horizontal vectors Fix t > 0 We say a lattice Λ X is t-horizontally short if it contains a non-zero horizontal vector v a, 0 T so that a t Note that since Λ is a group, we can assume a > 0 We will call -horizontally short lattices simply horizontally short We also have the analgous notions of t-vertically and vertically shortto prove Theorem, we break it up into several lemmas Our first lemma is: Lemma Ω {Λ a,b : a, b 0, ], a + b > } is the set of horizontally short lattices Proof Since the horizontal vector a, 0 T is in Λ a,b, and a, clearly every lattice in Ω is short To show the reverse containment, suppose Λ is short Then we can write Λ Λ a,b p a,b Z, with 0 < a, and b 0 Let m Z be such that a < ma + b, ie, m a b Set b ma + b Then, since p,m SL, Z, Λ Λ a,b p a,b Z p a,b p,m Z Λ a,b Λ a,b Ω Next, we show that for any a, b Ω, that there is a s 0, so that h s Λ a,b Ω, and give a formula for the minimum s Lemma Let a, b Ω, so Λ a,b Ω Let s 0 Ra, b ab Then h s 0 Λ Ω, and for every 0 < s < s 0, h s Λ a,b / Ω Furthermore, h s0 Λ a,b Λ T a,b Proof Since Ω consists of horizontally short lattices, we first note that the first time h s Λ a,b will have a horizontal vector is given by the equation Thus we set s 0 ab, and a direct calculation shows Let κa, b +a b Applying the matrix we obtain Thus, h s0 Λ a,b Λ T a,b, as desired h s0 p a,b 0 κa, b sb + a 0 a b b 0 Aa, b T h s0 p a,b Aa, b T b a + κa, bb 0 b p T a,b Finally, we show that for any lattice Λ X which is not vertically short, the orbit under {h s } s 0 will intersect Ω Lemma 3 Let Λ X so that Λ is not vertically short Then there is a s R so that h s Λ Ω Proof Let S [, ] R S is a square centered at 0, so is convex, centrally symmetric, and has area 4 Thus by the Minkowski convex body theorem, any lattice Λ must have a nonzero vector v S Since Λ is not vertically short, we can assume that v a, b T is not vertical, that is, a 0 Further, we can assume that a 0, otherwise we multiply by Let s b a Then a, 0T h s Λ, so h s Λ Ω

13 A Poincaré section for horocycle flow 3 Proof of Theorem To prove Theorem, we combine the above lemmas Lemma 3 guarantees that all non-vertically short lattice horocycle orbits {h s Λ} s R intersect Ω, and Lemma shows that if an h s -orbit intersects Ω once, it must intersect Ω infinitely often both forward and backward in time To see that the set of intersection times is discrete, observe that the roof function Ra, b ab is bounded below by on Ω, so visits must be spread out at least time apart The calculation of the roof function R and the return map T are also given by Lemma Finally, we address the remark following Theorem A direct calculation shows that if a lattice Λ has a vertical vector, then it is periodic under h s with period t, where t is the length of the shortest vertical vector If Λ is vertically short, this means that it is periodic under h s of period, so the h s -orbit of Λ cannot intersect Ω We will see in below that these orbits correspond to embedded closed horocycles, the family of which foliates the cusp of H /SL, Z Observations on slopes Given a unimodular lattice Λ, let {0 s < s < < s N } denote the sequence of slopes of vectors in Λ V, where V {0 < x, y > 0} R is the vertical strip as in 7 Then we see that s is in fact the first hitting time of Ω of the positive orbit h s Λ, and the s n are the subsequent hitting times, since Ω is the set of horizontally short lattices The slopes of vectors decrease by s under the horocycle flow h s in particular, while h s does not preserve slopes, it preserves differences of slopes, and the BCZ map records them when they become horizontal, while also keeping track of their horizontal component the coordinate a and the next vector in the strip to become horizontal the vector b, a T The differences in slopes are the return times R, that is, for n, s n+ s n RT n h s Λ This observation will be crucial for the proofs our results on the geometry of numbers, see 7 Hyperbolic geometry We recall that the there is a natural identification of the the upper-half plane H {z x + iy : y > 0} with the space of unimodular lattices with a horizontal vector, via z x Z Λ y 0 y y, x y That is, we identify z with the unimodular lattice homothetic to the lattice generated by and z This extends to an identification of all unimodular lattices with the unit-tangent bundle T H, where the point z, i where i denotes the upward pointing tangent vector is identified to the lattice with a horizontal vector, and the point z, e iθ i is identified to the lattice rotated by angle θ This identification is well-defined up to the action of SL, Z by isometries on H In the standard fundamental domain for SL, Z, { z x + iy H : z >, x < the set of horizontally short lattices can be identified with the strip { C : z x + iy H : y >, x < }, together with their upward pointing tangent vectors, see Figure 6 The explicit map between this region and Ω is given by x, y Λ Ix,y, where where }, Ix, y y /, xy / + my /, y / m xy / xy / x The choice of m is as in the proof of Lemma 3, which guarantees that the image IC Ω, since x, y is already naturally identified with Λ y /,xy /

14 4 JS Athreya and YCheung Fig 6 A hyperbolic picture of Ω C The green circle is an h s -orbit Vertical lines are g t -orbits and horizontal lines are u s -orbits Ω - Geodesics and horocycles The orbits of the one-parameter subgroups and A : U : { g t e t/ 0 0 e t/ { s u s 0 } : t R } : t R, the geodesic and opposite horocycle flows respectively, also have nice interpretations in terms of hyperbolic geometry Under our choices, the action of {g t : t 0} moves the upward pointing tangent vectors vertically upward in H, and the flow {u s } moves them horizontally In particular, Ω is preserved by the action of {g t : t 0} and {u s } In dynamical language, orbits of h s are leaves of the stable foliation for {g t : t 0}, and {u s } are leaves of the strong unstable foliation We are constructing a cross section of h s by taking a piece of the unstable foliation AU That is, Ω AU In the Euclidean picture, direct calculations show that g t Λ a,b Λ e t/ a,e t/ b and u s Λ a,b Λ a,b+sa, so geodesic orbits are radial lines and opposite horocyclic orbits are vertical lines, see Figure 7 We will use these observations in crucial ways in 3 below Identifications Note that in the fundamental domain {z x + iy : z, x } for SL, Z acting on H, there is a natural identification of the vertical sides x and x transformation z z +, which is a fractional linear transformation associated to the unipotent matrix 0 via the In the Euclidean picture, we can also identify two boundaries of the region Ω via the above unipotent matrix, namely the lines {x + y : 0 < x } and {y : 0 < x } The resulting loop formed by the vertical segment {x : 0 < y }

15 A Poincaré section for horocycle flow 5 Fig 7 Geodesic and opposite horocyclic orbits in the Euclidean picture of Ω u s Ω g t corresponds to the closed horocycle {z + iy : 0 y } in the hyperbolic picture Thus, the topology of Ω is that of a sphere with one puncture corresponding to the point at in the hyperbolic picture and the point 0, in the Euclidean picture and one boundary component corresponding to the loop described above 3 Self-similarity The BCZ map has an extraordinary self-similarity property which can be seen naturally in both the hyperbolic and Euclidean pictures Let t > 0, and let Ω t denote the set of t-horizontally short lattices Arguing as in Lemma, we have the identification with the set Ω t : {a, b 0, t] : a + b > t} 3 Appropriately modifying Lemma 3, we can define a return map the t-bcz map T t : Ω t Ω t, which captures the h s -orbits of all lattices except those of t -vertically short lattices Modifying the argument in Lemma, we have t + a T t a, b b, a + b b 4 A direct calculation shows that the t-bcz map is conjugate to the original - BCZ map via the linear transformation M t : Ω Ω t given by M t a, b ta, tb, that is T t M t M t T 5 Now assume t < if t >, the roles of t and below should be reversed Then the set of t-horizontally short lattices can also be identified with the subset Ω t Ω given by Ω t : {a, b Ω : a < t} Let T t : Ω t Ω t be the first return map of the BCZ map T to Ω t Thus, T t also represents the first return map of h s to the set of t-horizontally short lattices, and so T t and T t are conjugate This conjugacy can be made explicit via the map L t : Ω t Ω t given by L t a, b a, b + t b a b,

16 6 JS Athreya and YCheung Fig 8 Ω t and Ω t Ω can be identified with Ω t via the scaling M t, and Ω t and Ω t via the map L t y Ω t Ω y t Ω t x t x Fig 9 A hyperbolic picture of Ω t Ω, that is, C t C Ω t Ω y t - that is, T t L t L t T t In the hyperbolic picture, the set of t-horizontally short lattices can be identified with the subset C t : { z x + iy H : y > t, x < } Figures 8 and 9 show the Euclidean and hyperbolic pictures respectively We can summarize this discussion with the following: Observation T is self-similar in the following sense: For any 0 < t <, the BCZ map T is conjugate to its own first-return map T t Essentially, this self-similarity is a consequence of the following conjugation relation for g t and h s g t h s g t h se t, 6 which in particular implies that the time- map h and time-s map h s are conjugate for any s > 0 via the map g log s

17 A Poincaré section for horocycle flow 7 3 Ergodic properties of the BCZ map In this section, we prove Theorem, using Theorem and well-known properties of suspension flows We first recall these properties, which can be found, eg, in [3] 3 Suspension Flows Let X, µ be a measure space, and T : X X a measure-preserving bijection Let R : X R + be in L X, µ The suspension flow over T with roof function R is defined on the space where x, Rx T x, 0 The suspension flow φ R,T is given by X R : {x, t : x X, t [0, Rx}/ 3 φ R,T s x, t x, s + t The measure dν R,T R dµdt is a φ R,T -invariant probability measure on X R It is ergodic if and only if µ is T -ergodicthe natural dual construction to suspension flow is the construction of a first return map for a flow Given a measure-preserving flow φ : Y, ν Y, ν, and a subset X Y so that for almost all y Y, {t R : φ t y X} is discrete, we define the first return map T : X X by T x φ Rx x, where Rx inf{t > 0 : φ t x X} is the first return time There is a natural possibly infinite T -invariant measure µ on X so that R L X, µ and the suspension flow φ R,T : X R, ν R,T X R, ν R,T is naturally isomorphic to the original flow φ There is also a map between the sets of invariant measures for the flow φ R,T and the map T, whose properties are summarized in the following: Lemma 3 Let X, µ be a measure space, and let T : X, µ X, µ be a µ-preserving bijection Let R be a positive function in L X, µ The map η η R,T, with dη R,T R,η dηdt is a bijection between the set of T -invariant measures η on X so that R L X, η, and the set of φ R,T invariant probability measures on X R and thus also between the ergodic invariant measures Moreover, we have Abramov s formula relating the entropy of the flow φ R,T that is, of its time -map to the entropy of the map T : h η R,T φ R,T h ηt R,η 3 3 Ergodicity In this section, we prove the ergodicity and measure-classification parts of Theorem The ergodicity of the BCZ map with respect to dm dadb follows from Ergodicity of the horocycle flow with respect to Haar measure µ on X Suspension The measure dadbds on X viewing it as the suspension space over Ω is an absolutely continuous invariant measure for {h s } Measure classification By Dani s measure classification [5], this must be a constant multiple of µ To show uniqueness, suppose ν was another ergodic T invariant measure on Ω Then it must be supported on a periodic orbit for T, since the measure dνds on X is a non-haar ergodic invariant measure for {h s } and thus, by Dani s measure classification, must be supported on a periodic orbit for {h s }

18 8 JS Athreya and YCheung 33 Entropy We now prove the fact that the entropy h m T of T with respect to the Lebesgue probability measure m is 0, completing the proof of Theorem We use Lemma 3, which states that the entropy h m T of the BCZ map with respect to m is proportional to the entropy of the horocycle flow h µ h with respect to Haar measure To show that h µ h 0, we record a fact which seems to be well-known but does not have a complete proof in the literature as far as we are aware: Theorem 3 Let Γ SL, R be a lattice Let µ denote the finite measure on SL, R/Γ induced by Haar measure Then h µ h 0 In fact, the topological entropy h top h 0 Proof When Γ is a uniform lattice, that is, the quotient H /Γ is compact, the second assertion is a theorem of Gurěvic [9], and together with the standard variational principle see, eg, Walters [35], which states that the topological entropy is the supremum of the measure-theoretic entropies When Γ is non-uniform, we can use a version of the variational principle for locally compact spaces due to Handel-Kitchens [3], to again reduce the measure-theoretic statement to a topological statement Since the horocycle flow is C, we can apply a theorem of Bowen [] to conclude that h top h < On the other hand, since h s is conjugate to h for any s > 0, we have which implies that h top h 0 h top h h top h s sh top h, 4 Structure of Periodic Orbits In this section we prove Theorems 4 and 5 and describe in detail the relationship between periodic orbits for T and h s We first record in 4 our geometric proof of the key observation 5 that certain periodic orbits of the BCZ map parameterize Farey fractions 4 Farey fractions and lattices In this section, we give a geometric proof the following Lemma 4 Let be a positive integer, and let F { 0 γ < γ < < γ i p i q i < < γ N }, N N q ϕq denote the Farey sequence Then 5 holds, that is, T i, qi, q i+ Remark: Here, i is interepreted cyclically ie, in Z/NZ In particular, this is a periodic orbit for T and for h s Note also that the return time gives normalized gaps between the Farey fractions, that is, R T i, q i q i+ γ i+ γ i

19 A Poincaré section for horocycle flow 9 Fig 0 The triangles T 4 and g t4 T 4, with lattice points Primitive lattice points in blue, others are in red, 6 4, 4, 0 4, 0 Proof Geometrically, the sequence F correspond to the slopes of primitive integer vectors in the p i closed triangle T with vertices at 0, 0,, 0, and, Now note that since Z contains a vertical vector, it is periodic under h s, with period Using the conjugation relation 6, we have that the lattice g t Z has period e t under h s, since h e tg t Z g t h Z g t Z Note that g t scales slopes and their differences by e t For N, setting t ln, consider the lattice g t Z 0 0 Z qi Note that we can also write this as 0 0 Z 0 0 Z 0 Z This lattice corresponds to the point, in the Farey triangle Ω By our earlier observation, this has period under h s We are interested in the behavior of the orbit of this point under T Recall that the BCZ map only sees primitive vectors with horizontal component less than that is, in the strip V V, since h s does not change the horizontal component of vectors Originally we were interested in Farey fractions of level, that is, the slopes of primitive integer vectors in the region T Applying the matrix g t, the region T is transformed qi into a subset V Thus, the vectors T p correspond to vectors q i g i p t T V Therefore, i they are seen by the BCZ map Precisely, g t T is the triangle with vertices at 0, 0,, 0, and, See Figure 0

20 0 JS Athreya and YCheung As noted above, the primitive vectors q i g p t Z will be seen by the BCZ map in the order of their i slopes that is, at time γ i, and the time in between the i th and i + st vectors will precisely be the difference in their slopes γ i+ γ i At time 0 we are seeing the horizontal vector corresponding to γ 0 0, and the next vector to get short is corresponding to γ, which will become horizontal in time R, 0 γ γ 0 More generally, we have that T i, corresponds to the lattice h γ i g t Z, and noting we obtain, as desired and qi q h γ i g i+ t q i q i+ p i p i+ 0 q i T i, qi, q i+, RT i, γ i+ γ i In particular, the point, is T -periodic with period N Also note that N i0 RT i,, and thus, the sum of the return times over the discrete periodic orbit correspond, as they must, to the continuous period 4 Proof of Theorem 4 We first note that any periodic point a, b Ω for T corresponds to a periodic point p a,b Z Ỹ for h s on X and vice-versa Precisely, suppose a, b Ω is periodic under T, and n P a, b > 0 is the minimal period, so and T j a, b a, b for j < n Then where T n a, b a, b h sa,b p a,b Z p a,b Z, n sa, b RT i a, b i0 Note that the points a, a Ω have period P a, a, and that a direct calculation shows that sa, a Ra, a a Periodic orbits for h s occur in natural families, due the conjugation relation 6 In particular, if Λ is periodic with period s 0, h s0e tg tλ g t h s0 Λ g t Λ, 4 so the flow period of g t gz is s 0 e t We have g t Λ a,b Λ e t/ a,e t/ b, so we see that if a, b is periodic for T, so is the entire line segment { ta, tb, t > } a + b

21 A Poincaré section for horocycle flow Thus, to analyze which points have periodic orbits, it suffices to consider points of the form, b or a, Since T a,, a, we consider points of the first kind, or, equivalently, lattices of the form Λ,b This is a periodic point under h s if and only if there exists an s 0 so that h s0 Λ,b Λ,b, or, equivalently, p,b h s 0 p,b SL, Z A direct calculation shows that this is impossible if y is irrational, and if y k l the minimal such s 0 l, and we have p + kl k,b h s 0 p,b l 4 kl This proves Theorem 4, since we have shown that any point with rational slope is periodic 43 Proof of Theorem 5 As above, we consider the point, b If b k l, k l N relatively prime, then the period under h s is l The line segment associated to this point is Applying 6 to p t,t k Z, we can calculate the period l { t, t k ]} l : t l l + k, s t, t k l l t To calculate the discrete period, we analyze the scalings of the Farey periodic orbits {T i, } N i0 using the following proposition, a special case of Theorem 5 which we will use to prove the general result: Proposition 4 For t +, ], P t, t N { This will follow from showing that T in fact is linear along the orbit of the segment t, t : t + ]},, that is, the segment does not break up into pieces Precisely, we have: Lemma 4 For t +, ], i N, κ T i t, t κ T i, Proof Since T i, ] qi, qi+, we need to show, for t +, qi κ, q i+ tqi κ, tq i+

22 JS Athreya and YCheung We have tqi κ, tq i+ t + q i q i+ 43 t + q i q i+ < + + q i q i+ q i+ + q i+ + q i q i+ + qi + q i+ where in the last line we are using the identity On the other hand, Thus, so as desired Now Proposition 4 follows from P + qi q i+ q i + q i+ q i+ a + q i + q i q i+ q i+ tqi κ, tq i+ + qi q i+ + qi q i+ tqi κ, tq i+ < +, N + qi q i+ + qi q i+, qi κ, q i+ To prove Theorem 5, we observe that the point, k l is contained in the periodic orbit corresponding to the point l,, since we can find consecutive Farey fractions γ i ai l and γ i+ ai+ k in Fl we are assuming that gcdk, l By Proposition 4, for a l l+, ], P a, a k Nl l If we put a l l+, we have the point l l+, k l+, which is contained in the periodic orbit of the point l+,, by similar reasoning as above Applying Proposition 4 to this point, we have that for b l+, ], bl P l +, bk Nl + l + Rewriting, we have that for a l l+, l l+], P t, t k Nl + l Continuing to reason in this fashion, we have that for r k, t l P t, t k Nl + r l l+r, l l+r ], Finally, we need to calculate A n t, t k l calculation to 4 that, where n P t, t k l Letting s0 s a, a k l l + kl k p h t,t k s0 p t,t k l l l kl a, we have, by a similar

23 A Poincaré section for horocycle flow 3 On the other hand, can be re-written as the conjugation Iterating, we obtain, for any m > 0, a, b Ω, Applying this to a, b t, t k l and m n, we have p T a,b h Ra,bp a,b Aa, b T p T m a,b h Ra,bp a,b A m a, b T An a, b T + kl k l kl since T n t, t k l t, t k l Inverting and taking transposes, we obtain, as desired A Pt,t k l t, t k kl l l k, + kl completing the proof, 44 Segments and the shearing matrix Note that the matrix A Pa,a k l a, a k kl l l k + kl is parabolic, and fixes the segment { a, a k l : a l l+k, ]} In fact, it shears along this segment in the following fashion Note that k + l l k kl l l k l k l k + k + kl k l 0 Thus for points a, b sufficiently near the segment, they will be sheared along the segment under the appropriate power of T which may vary depending on the period of the subsegment a, b is close to As it moves along { the segment, the period will change This is illustrated in Figure, which shows the periodic segment a, a 3 : a 3 5, ]}, which breaks up into two pieces, one of period 5 and one of period 6 The shearing matrix for this line is given by which is conjugate to 3 0, 45 Hierarchies of periodic orbits Theorem 5 has a nice geometric explanation, which we describe informally: If we start with the points a, a Ω of period under T, and move down the line, we hit,, which is not in Ω However, we can identify Λ, with Λ,, since, in general, we have Λ a, a Λ a, The orbit of, has period N, and continuing to move down the line, Proposition 4 shows that for a /3, ], a P, a N At a 3, we identify 3, 3 with the point 3,, which has period N3 5 Continuing this process, we obtain all the Farey periodic orbits, and the scaling segments { ]} t, t : t +, The set of T -periodic points in Ω is the union t {T i, t : t N i N ]} +,

24 4 JS Athreya and YCheung Fig The periodic segment { a, a 3 : a 3 5, ]} The segment a 3 4, is in blue and the segment a 3 5, 4] 3 is in red The labels next to the segments correspond to the power of T The red segment has period N4 6, and the blue segment has period N3 4 0,, , 0, 0 46 Hyperbolic Geometry of Periodic Orbits What we have done in X is to push the cuspidal horocycle correpsonding to Λ a,a, a > down using g t For a >, the lattice Λ a,a is vertically short, and the h s trajectory is an embedded loop in X which does not intersect Ω For > a >, the h s-orbit it intersects our transversal once, on the point Λ a,a, which is a fixed point for the BCZ map T As we push it down, the h s trajectory intersects the transversal more and more times However, it does not simply pick up one intersection at a time, but rather N + N ϕ + intersections when we transition from +, + to +, 5 Equidistribution of periodic orbits 5 Proof of Theorem 3 In this section, we prove Theorem 3, which is the key result for our number theoretic applications This is essentially a direct consequence of the well-known equidistribution principle for closed horocycles on X, originally due to Sarnak [3] and reproved by Eskin-McMullen [6] using ergodic theoretic techniques In our notation, and in terms of the map T, this result can be reformulated as follows Recall that given a non-empty interval I [α, β] [0, ], and >, we defined ρ,i as the probability measure supported on a long piece of the orbit of, That is, setting N I : F I, we had ρ,i N I i:γ i I δ T i, Define the measure ρ R,I on X by setting dρ R,I dρ,i dt, where we are identifying X with the suspension space over Ω Theorem 5 [[3],[6],[5]] As, ρ R,I µ, where the convergence is taken in the weak-* topology

25 A Poincaré section for horocycle flow 5 Remark In [3] and [6] the result is only established for the case α 0, β, although it is clear that the same arguments in [6] work for the case of fixed 0 < α < β < A proof of this was outlined in [5] Stronger results where the difference β α is permitted to tend to zero with have been obtained by Hejhal [6] and Strömbergsson [34] Let π be the projection map from the suspension onto Ω Note that π is continuous except for a set of measure zero with respect to the measures ρ R,I and µ Thus, we have as, which completes the proof of Theorem 3 ρ,i R π ρ R,I R π µ m, 5 Farey Statistics Our results on the statistics of Farey fractions are all immediate corollaries of Theorem 3, being statements of the form Gdρ,I Gdm 5 Ω for appropriate choices of functions G on Ω In the applications to spacings and h-spacings, the function G is in fact an indicator function, and so the convergence is immediate In the setting of indices and moments, the functions G are L m functions, we need the following lemma Lemma 5 For N, set m ρ,i and let m m Let W be the set of measurable functions G : Ω R satisfying G W : sup G dm < 5 N { } Ω where two functions in W are identified if their difference has zero norm Let C 0 Ω be the space of continuous functions vanishing at infinity equipped with the uniform norm and let W 0 be the closure of its image in W Then 5 holds for any G W 0 Proof Let η : C 0 Ω W be the natural inclusion map, which we note is a continuous injection that fails to be an embedding It induces a map η : W MΩ between the dual W with the weak- topology and the space MΩ of Radon measures on Ω Each m determines a linear map ηc 0 Ω R that is readily seen to be continuous even with the subspace topology on ηc 0 Ω coming from W This linear map extends uniquely to a linear map W 0 R of norm at most one Given ε > 0 and G W 0, there is sequence G k C 0 Ω converging to G Choose k so that G k G < ε/3, then choose 0 such that for all > 0 we have m G k mg k < ε/3 The triangle inequality now gives This shows that 5 holds m G mg G G k + m G k mg k < ε Below, we indicate which functions go with which corollaries, and show how to verify the condition 5 in these settings 5 Spacings and h-spacings [ Corollary 8 follows from applying 5 to the indicator function of the set R 3 π I c, 3 π I ] d Similarly, Corollary 9 follows from applying 5 to the indicator function of the set R h B SInce indicator functions are bounded, 5 holds immediately 5 Indices To verify 5 for κ α, we need: Lemma 5 [9], Lemma 34 Let r N, n 4r + Suppose κa, b n Then For i ±, κt i a, b For < i r, κt i a, b Ω

26 6 JS Athreya and YCheung That is, any large value n of κ must be followed by roughly n/4 small values Furthermore, by the definition of κ, we can see that the maximum of κ along the support of m is at most, which is on the order of the square root of the length of the orbit N I, since κ qi, q i+ + q i q i+ + qi Combining these two facts, we have that there is a constant c c β, depending on the interval I and the power β so that for any 0 < β <, and any N, Let 0 < α < β < Then Since mκ k 8 kk+k+ m κ α q i+ m κ > N c β N β m κ > N α < c β N β α < cβ ζ β/α N for k, we have N mκ α < 8ζ3 α, so we have verified 5 for κ α, proving Corollary 0 53 Moments Finally, to obtain Theorem, we apply 5 to the function G s,t a, b a s b t, t, s C, Rs, Rt To verify 5, we use that for a, b Ω, G s,t a, b < G, a, b, so it suffices to verify the condition in the case s, t, We need the following: Lemma 53 For any interval I [0, there is a constant c such that for any with N I > 0, we have N I c Proof N I counts the number of primitive lattice points in the triangle: {x, y R + : y x I, x }, which has area I There are two cases If N / I, then Theorem 3 of [3] with the sup norm implies 4 N I 3 7π 8 I I 8π On the other hand, if N / I 0, the length of I is bounded above by the largest gap in F /, which is at most /, so that N I I 4 Thus, the lemma follows with the choice I c min 8π, I 4 We have m G, N I γ i F I q i q i+ I N I Applying Lemma 53, we have that m G, is unfiormly bounded in, and we have that mg, π 3, completing the proof

27 A Poincaré section for horocycle flow 7 54 Excursions To prove Corollary, we apply 5 to the functions and G max a, b max { } a, b,, a + b { G min a, b min a, } b, a + b respectively Equation 5 is satisfied since G max is bounded by and G min is bounded by 6 Piecewise linear description of horocycles Before proving Theorems 6 and 7, we record an elementary lemma on the behavior of the length of vectors under h s, whose proof we leave to the reader Lemma 6 Let v x, y T R be a vector with x > 0 Let σ ± σ ± x where σ y x is the slope of v Then fs h s v sup is the continuous piecewise linear function given by i f s x for s < σ, ii fs x for σ s σ +, and ii f s + x for s > σ + 6 Minima and maxima Recall that Lemma states that Ω consists of lattices with a short length horizontal vector This observation shows that the visits of the trajectory {h s Λ} s 0 to the transversal Ω are precisely the times s n Λ of local minima of l Λ s, since a lattice cannot contain more than vector of supremum norm at most, and under h s, vectors are shorter when they are horizontal On Ω, we have By abuse of notation we write l : Ω 0, ] with l Λ a,b a la, b a Similarly we write αa, b a Thus when h s n Λ Λ an,b n, the length of the shortest vector in h sn Λ is given by an the horizontal vector Setting 0 and writing Λ a,b h s Λ, we have that and l N Λ N a N Λ : N s Λ min{s 0 : h s Λ Ω }, l Λ s n N n α Λ s n N n l T n a, b, n α T n a, b Since both α and l are in L Ω, dm, and we have assumed that Λ is not periodic under h s and so, a, b is not under T, we can apply the Birkhoff Ergodic Theorem and the fact that all non-periodic orbits are generic for m, by the measure classification result to conclude and proving Theorem 6 lim N lim N N N n n l Λ s n lim N N α Λ s n lim N N n l T n a, b n α T n a, b n Ω Ω ldm 3 αdm,