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Dynamics of Subgroup Actions on Homogeneous Spaces of Lie Groups and Applications to Number Theory

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Dynamics of Subgroup Actions on Homogeneous Spaces of Lie Groups and Applications to Number Theory hass v.2002/01/24 Prn:31/01/2002; 15:11 F:HASS11.tex; VTEX/ELE p. 1 CHAPTER 11 Dynamics of Subgroup Actions on Homogeneous Spaces of Lie Groups and Applications to Number Theory Dmitry Kleinbock Brandeis University, Waltham, MA 02454-9110, USA E-mail: [email protected] Nimish Shah Tata Institute of Fundamental Research, Mumbai 400005, India E-mail: [email protected] Alexander Starkov Moscow State University, 117234 Moscow, Russia E-mail: [email protected] Contents Introduction . 3 1. Lie groups and homogeneous spaces . 4 1.1. Basics on Lie groups . 5 1.2. Algebraic groups . 9 1.3. Homogeneous spaces . 11 1.4. Homogeneous actions . 16 2. Ergodic properties of flows on homogeneous spaces . 22 2.1. The Mautner phenomenon, entropy and K-property . 22 2.2. Ergodicity and mixing criteria . 27 2.3. Spectrum, Bernoullicity, multiple and exponential mixing . 30 2.4. Ergodic decomposition . 34 2.5. Topological equivalence and time change . 36 2.6. Flows on arbitrary homogeneous spaces . 39 3. Unipotent flows and applications . 41 3.1. Recurrence property . 42 HANDBOOK OF DYNAMICAL SYSTEMS, VOL. 1A Edited by B. Hasselblatt and A. Katok ©2002ElsevierScienceB.V.Allrightsreserved 1 hass v.2002/01/24 Prn:31/01/2002; 15:11 F:HASS11.tex; VTEX/ELE p. 2 2 D. Kleinbock et al. 3.2. Sharper nondivergence results . 45 3.3. Orbit closures, invariant measures and equidistribution . 47 3.4. Techniques for using Ratner’s measure theorem . 52 3.5. Actions of subgroups generated by unipotent elements . 57 3.6. Variations of Ratner’s equidistribution theorem . 59 3.7. Limiting distributions of sequences of measures . 61 3.8. Equivariant maps, ergodic joinings and factors . 64 4. Dynamics of non-unipotent actions . 67 4.1. Partially hyperbolic one-parameter flows . 68 4.2. Quasi-unipotent one-parameter flows . 75 4.3. Invariant sets of one-parameter flows . 77 4.4. On ergodic properties of actions of connected subgroups . 82 5. Applications to number theory . 88 5.1. Quadratic forms . 88 5.2.Linearforms ............................................... 93 5.3. Products of linear forms . 102 5.4. Counting problems . 104 Acknowledgements ................................................ 109 References ..................................................... 109 hass v.2002/01/24 Prn:31/01/2002; 15:11 F:HASS11.tex; VTEX/ELE p. 3 Dynamics of subgroup actions 3 Introduction This survey presents an exposition of homogeneous dynamics, that is, dynamical and ergodic properties of actions on homogeneous spaces of Lie groups. Interest in this area rose significantly during late eighties and early nineties, after the seminal work of Margulis (the proof of the Oppenheim conjecture) and Ratner (conjectures of Raghunathan, Dani and Margulis) involving unipotent flows on homogeneous spaces. Later developments were considerably stimulated by newly found applications to number theory. By the end of the 1990s this rather special class of smooth dynamical systems has attracted widespread attention of both dynamicists and number theorists, and became a battleground for applying ideas and techniques from diverse areas of mathematics. The classical set-up is as follows. Given a Lie group G and a closed subgroup Γ G, ⊂ one considers the left action of any subgroup F G on G/Γ : ⊂ x fx, x G/Γ, f F. "→ ∈ ∈ Usually F is a one-parameter subgroup; the action obtained is then called a homogeneous (one-parameter) flow. One may recall that many concepts of the modern theory of dynamical systems appeared in connection with the study of the geodesic flow on a compact surface of constant negative curvature. While establishing ergodicity and mixing properties of the geodesic flow, Hopf, Hedlund et al., initiated the modern theory of smooth dynamical systems. Essentially, they used the existence of what is now called strong stable and unstable foliations; later this transformed into the theory of Anosov and Axiom A flows. On the other hand, while studying orbits of the geodesic flow, Artin and Morse developed methods that later gave rise to symbolic dynamics. The methods used in the 1920–1940s were of geometric or arithmetic nature. The fact that the geodesic flow comes from transitive G-action on G/Γ , Γ being a lattice in G SL(2, R),wasnoticedandexploredbyGelfandandFominintheearly1950s.This distinguished= the class of homogeneous flows into a subject of independent interest; first results of general nature appeared in the early 1960s in the book [9]byAuslander,Green, and Hahn. Since then, ergodic properties of homogeneousone-parameterflows with respect to Haar measure have been very well studied using the theory of unitary representations, more specifically, the so called Mautner phenomenon developed in papers of Auslander, Dani, and Moore. Applying the structure of finite volume homogeneous spaces, one can now give effective criteria for ergodicity and mixing properties, reduce the study to the ergodic case, calculate the spectrum, etc.; see Sections 1 and 2 for the exposition. The algebraic nature of the phase space and the action itself allows one to obtain much more advanced results as compared to the general theory of smooth dynamical systems. This can be seen on the example of smooth flows with polynomial divergence of trajectories. Not much is known about these in the general case apart from the fact that they have zero entropy. The counterpart in the class of homogeneous flows is the class of the so called unipotent flows. Motivated by certain number theoretic applications (namely, by the Oppenheim conjecture on indefinite quadratic forms), in the late 1970s Raghunathan hass v.2002/01/24 Prn:31/01/2002; 15:11 F:HASS11.tex; VTEX/ELE p. 4 4 D. Kleinbock et al. conjectured that any orbit closure of a unipotent flow on G/Γ must be itself homogeneous, i.e., must admit a transitive action of a subgroup of G.Thisandarelatedmeasure-theoretic conjecture by Dani, giving a classification of ergodic measures, was proved in the early 1990s by Ratner thus really providing a breakthrough in the theory. This was preceded by highly nontrivial results of Margulis–Dani on nondivergence of unipotent trajectories. Dynamics of unipotent action is exposed in Sections 3 of our survey. On the other hand, a lot can be also said about partially hyperbolic homogeneous flows. What makes the study easier, is that the Lyapunov exponents are constant. This (and, of course, the algebraic origin of the flow) makes it possible to prove some useful results: to calculate the Hausdorff dimension of nondense orbits, to classify minimal sets, to prove the exponential and multiple mixing properties, etc. Needless to say that such results are impossible or very difficult to prove for smooth partially hyperbolic flows of general nature. The substantial progress achieved due to Ratner’s results for unipotent actions made it possible to study individual orbits and ergodic measures for certain multi-dimensional subgroups F G.Theseandaforementionedresultsfornon-unipotentactionsareexposed ⊂ in Section 4. As was mentioned above, the interest in dynamical systems of algebraic origin in the last 20 years rose considerably due to remarkable applications to number theory. It was Margulis who proved the long standing Oppenheim conjecture in the mid 1980s settling a special case of Raghunathan’s conjecture. Recently Eskin, Margulis and Mozes (preceded by earlier results of Dani and Margulis) obtained a quantitative version of the Oppenheim conjecture. Kleinbock and Margulis proved Sprindžuk’s conjectures for Diophantine approximations on manifolds. Eskin, Mozes, and Shah obtained new asymptotics for counting lattice points on homogeneous manifolds. Skriganov established atypicalasymptoticsforcountinglatticepointsinpolyhedra.Theseandrelatedresultsare discussed in Section 5. The theory of homogeneous flows has so many diverse applications that it is impossible to describe all the related results, even in such an extensive survey. Thus for various reasons we do not touch upon certain subjects. In particular, all Lie groups throughout the survey are real; hence we are not concerned with dynamical systems on homogeneous spaces over other local fields; we only note that the Raghunathan–Dani conjectures are settled in this setup as well (see [147,148,190,191]). We also do not touch upon the analogy between actions on homogeneous spaces of Lie groups and the action of SL(2, R) on the moduli space of quadratic differentials, with newly found applications to interval exchange transformations and polygonal billiards. See [152]foranextensiveaccountofthesubject. The reader interested in a more detailed exposition of dynamical systems on homogeneous spaces may consult recent books [237]byStarkovand[13]byBekkaandMayer,or surveys [57,59]byDani.Foralistofopenproblemsweaddressthereadertoapaper of Margulis [144]. 1. Lie groups and homogeneous spaces In this section we give an introduction into the theory of homogeneous actions. We start with basic results from Lie groups and algebraic groups (Sections 1.1 and 1.2). Then in hass v.2002/01/24 Prn:31/01/2002; 15:11 F:HASS11.tex; VTEX/ELE p. 5 Dynamics of subgroup actions 5 Section 1.3 we expose briefly the structure of homogeneous spaces of finite volume. For more details one can refer to books by Raghunathan [174], Margulis [141], Zimmer [264], and survey by Vinberg, Gorbatsevich and Shvartzman [249]. Homogeneous actions come into stage in Section 1.4. Basic examples include: rectilinear flow on a torus; solvable flows on a three-dimensional locally Euclidean manifold; suspensions of toral automorphisms; nilflows on homogeneous spaces
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