Clay Mathematics Proceedings Volume 10 Homogeneous Flows, Moduli Spaces and Arithmetic Proceedings of the Clay Mathematics Institute Summer School Centro di Recerca Matematica Ennio De Giorgi, Pisa, Italy June 11– July 6, 2007 Manfred Leopold Einsiedler David Alexandre Ellwood Alex Eskin Dmitry Kleinbock Elon Lindenstrauss Gregory Margulis Stefano Marmi American Mathematical Society Jean-Christophe Yoccoz Clay Mathematics Institute Editors Homogeneous Flows, Moduli Spaces and Arithmetic Clay Mathematics Proceedings Volume 10 Homogeneous Flows, Moduli Spaces and Arithmetic Proceedings of the Clay Mathematics Institute Summer School Centro di Recerca Matematica Ennio De Giorgi, Pisa, Italy June 11– July 6, 2007 Manfred Leopold Einsiedler David Alexandre Ellwood Alex Eskin Dmitry Kleinbock Elon Lindenstrauss Gregory Margulis Stefano Marmi Jean-Christophe Yoccoz Editors American Mathematical Society Clay Mathematics Institute Cover photographs courtesy of the Scuola Normale Superiore, Pisa,Italy. 2000 Mathematics Subject Classification. Primary 37A17, 37A45, 37A35, 37C85, 37D40, 37E05, 11J13, 11J83,58J51, 81Q50. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2007 : Centro di recerca matematica Ennio de Giorgi) Homogeneous flows, moduli spaces and arithmetic : Clay Mathematics Institute Summer School,June11–July6,2007,CentrodirecercamatematicaEnniodeGiorgi,Pisa,Italy/Manfred LeopoldEinsiedler...[etal.],editors. p.cm. —(Claymathematicsproceedings;10) Includesbibliographicalreferences. ISBN978-0-8218-4742-8(alk.paper) 1.Ergodictheory—Congresses. 2.Analyticspaces—Congresses. 3.Differentiabledynamical systems—Congresses. I.Einsiedler,ManfredLeopold,1973– II.Title. QA313.C53 2007 515(cid:2).48—dc22 2010021098 Copying and reprinting. 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VisittheAMShomepageathttp://www.ams.org/ VisittheClayMathematicsInstitutehomepageathttp://www.claymath.org/ 10987654321 151413121110 Contents Introduction vii Interval Exchange Maps and Translation Surfaces 1 Jean-Christophe Yoccoz Unipotent Flows and Applications 71 Alex Eskin Quantitative Nondivergence and its Diophantine Applications 131 Dmitry Kleinbock Diagonal Actions on Locally Homogeneous Spaces 155 Manfred Einsiedler and Elon Lindenstrauss Fuchsian Groups, Geodesic Flows on Surfaces of Constant Negative Curvature and Symbolic Coding of Geodesics 243 Svetlana Katok Chaoticity of the Teichmu¨ller Flow 321 Artur Avila Orbital Counting via Mixing and Unipotent Flows 339 Hee Oh Equidistribution on the Modular Surface and L-Functions 377 Gergely Harcos Eigenfunctions of the Laplacian on Negatively Curved Manifolds : A Semiclassical Approach 389 Nalini Anantharaman v Introduction These are the proceedings of the 2007 Clay Summer School on Homogeneous Flows, Moduli Spaces and Arithmetic, which took place at the Centro di Ricerca Matematica Ennio De Giorgi in Pisa between June 11th and July 6th, 2007. More than100youngresearchersandgraduatestudentsattendedthisintensivefourweek school, as well as 18 lecturers and other established researchers. As suggested by the name, the topic of this summer school consisted of two connectedbutdistinctareasofactivecurrentresearch: flowsonhomogeneousspaces of algebraic groups (or Lie groups), and dynamics on moduli spaces of abelian or quadratic differentials on surfaces. These two subjects have common roots and have several important features in common; most importantly, they give concrete examples of dynamical systems with highly interesting behavior and a rich and powerful theory. Moreover, both have applications whose scope lies well outside that of the theory of dynamical systems. The first three weeks of the summer school were devoted to the basic theory, and consisted mostly of three long lecture series. Based on these lecture series, the following four sets of notes were written: [1] Interval exchange maps and translation surfaces by J. C. Yoccoz [2] Unipotent flows and applications by A. Eskin [3] QuantitativenondivergenceanditsDiophantineapplications byD.Klein- bock [4] Diagonal actions on locally homogeneous spaces by M. Einsiedler and E. Lindenstrauss Furthermore, there was a shorter lecture series [5] Fuchsian groups, geodesic flows on surfaces of constant negative curva- ture and symbolic coding of geodesics by S. Katok. Extensive notes for all the lecture series given in the first three weeks of the school are included in this proceedings volume (the content of the course by Eskin and Kleinbock has been separated into two different sets of notes). These papers were written to be read independently, and any of the five papers [1]–[5] could serve as agood startingpoint for the interested reader. More advancedtopics were covered by several lecture series and individual lectures mostly given in the last week of the summer school; it was left to the discretion of the lecturers in these shorter courses whether to provide notes for these proceedings (though they were strongly encouragedtocontribute). Alistoftheselecturenoteswithsomeadditionaldetails is given below. vii viii INTRODUCTION The common root of both main topics of the summer school mentioned above lie (at least in part) in the theory of flows on surfaces of constant negative cur- vature, particularly the modular surface SL(2,Z)\H, where pioneering work was done in the early 20th century by mathematicians such as Artin, Hedlund, Morse and others, and this theory has been developed much further in the times since. One highlight was the discovery that the geodesic flow on the modular surface is intimately connected to the continued fraction expansion of real numbers; indeed, when things are properly set up, one can view the continued fraction expansion as a symbolic coding of trajectories of the geodesic flow. These flows and their symbolic codings are carefully explained in Katok’s notes; in later sections of that work, recent extensions of this classical result are also discussed. One can view the modular surface SL(2,Z)\H in two ways: firstly, it can be viewedasthelocallyhomogeneousspaceSL(2,Z)\SL(2,R)/SO(2,R),inwhichcase thegeodesicflow aswellasanotherimportantgeometricflow —thehorocycleflow — can be viewed as in the projection of trajectories of the one parameter groups (cid:2) (cid:3) (cid:2) (cid:3) et/2 0 1 t (1) g = and u = t 0 e−t/2 t 0 1 on the quotient space SL(2,Z)\SL(2,R). Another way to view SL(2,Z)\H is as a moduli space of flat structure (up to rotations) on a two-dimensional torus. These twodifferentpointsofviewgeneralizetothetwomainthemesofthisClaySummer School: flows on homogeneous spaces, and flows on moduli spaces of abelian or quadratic differentials (which are essentially fancy names for flat structures in two related but slightly different senses). Flows on moduli spaces of flat structures. The torus is the only surface admitting a flat structure with no singularities. Whenone considers flat structures for surfaces of higher genus, one is forced to admit singularities: points where the totalanglesadduptomorethan2π. Itturnsoutthatinterval exchange maps play an important role in studying the analogue of the geodesic flow (sometimes called the Teichmu¨ller geodesic flow) on these moduli spaces of flat structures. We recall that interval exchange maps are the following simple yet intriguing dynamical sys- tem: dividetheunitinterval[0,1]intofinitelymanyintervalsI ,I ,...,I andthen 1 2 d permute these intervals according to a permutation π ∈ S . Yoccoz’ contribution d tothisproceedingsprovidesanintroductiontothistheory,andprovidesfullproofs of the most fundamental theorems (by Keane, Masur, Veech, Zorich) in the first tensections and an introduction tosome more advancedtopics (Kontsevich-Zorich cocycle,cohomological equation, connectedcomponents of the moduli space, expo- nential mixing of the Teichmu¨ller flow) in the last four sections. Further advanced topics are provided by notes based on the shorter lecture series [6] Chaoticity of the Teichmu¨ller flow by A. Avila giveninthelastweekofschool;inthesenotestheinterestedreadercanfindsurveys of the proof of two recent theorems: the simplicity of the Lyapunov spectrum for the Kontsevich-Zorich cocycle and that a typical interval exchange map with three or more intervals is weak mixing. Flows on homogeneous spaces and applications to arithmetic. Flows on homogeneous spaces concerns the dynamics of group actions on quotient spaces Γ\G, where G is usually taken to be either a (i) Lie group, or (ii) an algebraic INTRODUCTION ix group over R, or (iii) an algebraic group over the p-adic numbers Q , or (iv) a p product of algebraic groups as in (ii) and (iii) above, involving several different fields(sometimescalledanS-algebraicgroup,whereS referstothesetof“primes” p that are used(1).) A simple case is the case ofG = SL(2,R) and Γ a lattice in G, for instance Γ = SL(2,Z). In this case we have discussed (e.g. (1)) the action of two one- parameter subgroups of SL(2,R): the group g corresponding to the geodesic flow t on the unit tangent bundle on Γ\H and u , which corresponds to the horocycle t flow on the same space. These two flows behave very differently: the u -flow is t very rigid, and one can algebraically classify orbit closures, invariant measures, measurablefactors,selfjoinings,andeventheasymptoticdistributionofindividual orbits. The g -flow is very flexible: it is certainly ergodic, but individual orbits can t behave very badly. Moreover, the g -flow is measure-theoretically equivalent to a t Bernoulli shift which has a wealth of measurable factors and self joinings. The group u is an example of a unipotent group. In a fundamental series of t papers published in 1990; 91, M. Ratner proved that the above mentioned rigidity properties of u -flow are shared by all unipotent group actions on homogeneous t spaces, in particular establishing in complete generality Raghunathan’s conjecture about orbit closures for such actions (some cases of which were known previously, notably in the context of the Oppenheim conjecture discussed below). For the g - t flowthesituationisratherdifferent: whileadiagonalizableone-parametergroupin general behaves very much like g , higher-dimensional diagonalizable groups seem t to behave much more rigidly (though not as rigidly as unipotent group actions). The notes by Eskin discuss in detail unipotent flows, with an emphasis on ap- plications, particularly regarding values attained by indefinite quadratic forms and Oppenheim’s Conjecture. This long-standing conjecture was proved by Margulis in the mid-80s using homogeneous dynamics, and in particular unipotent dynam- ics. In dynamical terms, what Margulis has shown is that any bounded orbit of SO(2,1)onSL(3,Z)\SL(3,R)isclosed. Thenotesalsogiveadetailedexpositionof a more delicate result giving precise asymptotics to the distribution of these values by Eskin, Margulis and Mozes (under certain assumptions on the signature of a quadratic form). Some of the ideas and methods used in the theory of unipotent flows, and in particular some of the ideas used by Ratner in her proof of the Mea- sure Classification Theorem are also described in these notes. Eskin’s notes also contain other interesting applications of unipotent rigidity as well as connections to dynamics of rational billiards. Kleinbock’s notes focus on a method originally introduced by Margulis and developed significantly since, to show that orbits of unipotent group actions do not diverge to infinity. In particular, a quantitative version of the non-divergence statement due to S. G. Dani is an important ingredient in the proof of various versions of orbit closure and equidistribution theorems, including Ratner’s Orbit Closure Theorem. However these techniques are more widely applicable and, in particular,wereusedbyKleinbockandMargulistoproveaconjectureofSprindˇzuk on Diophantine approximations; this connection is also carefully discussed. ThenotesbyEinsiedlerandLindenstraussdiscussdiagonalizablegroupactions, based mostly on work by the authors and by A. Katok in various combinations. A crucial role in current analysis of these actions is played by the concept of entropy. (1)Forthispurpose∞isaprimeandQ∞=R.