Invariant and stationary measures for the SL ( 2 , ) action on Moduli space
Publications Mathématiques de l'IHÉS, Tome 127 (2018), pp. 95-324.

We prove some ergodic-theoretic rigidity properties of the action of SL ( 2 , ) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL ( 2 , ) is supported on an invariant affine submanifold.

The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.

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     title = {Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on {Moduli} space},
     journal = {Publications Math\'ematiques de l'IH\'ES},
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Eskin, Alex; Mirzakhani, Maryam. Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on Moduli space. Publications Mathématiques de l'IHÉS, Tome 127 (2018), pp. 95-324. doi : 10.1007/s10240-018-0099-2. http://archive.numdam.org/articles/10.1007/s10240-018-0099-2/

[ABEM] Athreya, J.; Bufetov, A.; Eskin, A.; Mirzakhani, M. Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J., Volume 161 (2012), pp. 1055-1111 | DOI | MR | Zbl

[At] Atkinson, G. Recurrence of co-cycles and random walks, J. Lond. Math. Soc. (2), Volume 13 (1976), pp. 486-488 | DOI | MR | Zbl

[Ath] Athreya, J. Quantitative recurrence and large deviations for Teichmüller geodesic flow, Geom. Dedic., Volume 119 (2006), pp. 121-140 | DOI | Zbl

[AthF] Athreya, J.; Forni, G. Deviation of ergodic averages for rational polygonal billiards, Duke Math. J., Volume 144 (2008), pp. 285-319 | DOI | MR | Zbl

[ACO] Arnold, L.; Cong, N.; Jordan, V. O. Normal form for linear cocycles, Random Oper. Stoch. Equ., Volume 7 (1999), pp. 301-356 | MR

[AEZ] Athreya, J.; Eskin, A.; Zorich, A. Rectangular billiards and volumes of spaces of quadratic differentials on P 1 , Ann. Sci. Éc. Norm. Supér. (4), Volume 49 (2016), pp. 1311-1386 (with an appendix by Jon Chaika) | DOI | MR | Zbl

[ANW] Aulicino, D.; Nguyen, D.; Wright, A. Classification of higher rank orbit closures in odd ( 4 ) , J. Eur. Math. Soc., Volume 18 (2016), pp. 1855-1872 | DOI | MR | Zbl

[AEM] Avila, A.; Eskin, A.; Moeller, M. Symplectic and isometric SL ( 2 , R ) invariant subbundles of the Hodge bundle, J. Reine Angew. Math., Volume 732 (2017), pp. 1-20 | DOI | MR | Zbl

[AG] Avila, A.; Gouëzel, S. Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow, Ann. Math. (2), Volume 178 (2013), pp. 385-442 | DOI | MR | Zbl

[AGY] Avila, A.; Gouëzel, S.; Yoccoz, J-C. Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., Volume 104 (2006), pp. 143-211 | DOI | Zbl

[AV2] Avila, A.; Viana, M. Extremal Lyapunov exponents: an invariance principle and applications, Invent. Math., Volume 181 (2010), pp. 115-189 | DOI | MR | Zbl

[ASV] Avila, A.; Santamaria, J.; Viana, M. Cocycles over partially hyperbolic maps, Astérisque, Volume 358 (2013), pp. 1-12 | MR | Zbl

[AV1] Avila, A.; Viana, M. Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture, Acta Math., Volume 198 (2007), pp. 1-56 | DOI | MR | Zbl

[Ba] Bainbridge, M. Billiards in L -shaped tables with barriers, Geom. Funct. Anal., Volume 20 (2010), pp. 299-356 | DOI | MR | Zbl

[BaM] Bainbridge, M.; Möller, M. Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3, Acta Math., Volume 208 (2012), pp. 1-92 | DOI | MR | Zbl

[BoM] Bouw, I.; Möller, M. Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. Math. (2), Volume 172 (2010), pp. 139-185 | DOI | MR | Zbl

[BQ] Benoist, Y.; Quint, J-F. Mesures Stationnaires et Fermés Invariants des espaces homogènes, Ann. Math. (2), Volume 174 (2011), pp. 1111-1162 (French) [Stationary measures and invariant subsets of homogeneous spaces] | DOI | MR | Zbl

[BG] Bufetov, A.; Gurevich, B. Existence and uniqueness of the measure of maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials, Mat. Sb., Volume 202 (2011), pp. 3-42 (Russian), translation in Sb. Math. 202 (2011), 935–970 | DOI | MR | Zbl

[Ca] Calta, K. Veech surfaces and complete periodicity in genus two, J. Am. Math. Soc., Volume 17 (2004), pp. 871-908 | DOI | MR | Zbl

[CK] V. Climenhaga and A. Katok, Measure theory through dynamical eyes, | arXiv

[CW] Calta, K.; Wortman, K. On unipotent flows in H 1 , 1 , Ergod. Theory Dyn. Syst., Volume 30 (2010), pp. 379-398 | DOI | MR | Zbl

[Dan1] Dani, S. G. On invariant measures, minimal sets and a lemma of Margulis, Invent. Math., Volume 51 (1979), pp. 239-260 | DOI | MR | Zbl

[Dan2] Dani, S. G. Invariant measures and minimal sets of horoshperical flows, Invent. Math., Volume 64 (1981), pp. 357-385 | DOI | MR | Zbl

[Dan3] Dani, S. G. On orbits of unipotent flows on homogeneous spaces, Ergod. Theory Dyn. Syst., Volume 4 (1984), pp. 25-34 | DOI | MR | Zbl

[Dan4] Dani, S. G. On orbits of unipotent flows on homogenous spaces II, Ergod. Theory Dyn. Syst., Volume 6 (1986), pp. 167-182 | Zbl

[De] Deza, M.; Deza, E. Encyclopaedia of Distances, Springer, Berlin, 2014 | Zbl

[DM1] Dani, S. G.; Margulis, G. A. Values of quadratic forms at primitive integral points, Invent. Math., Volume 98 (1989), pp. 405-424 | DOI | MR | Zbl

[DM2] Dani, S. G.; Margulis, G. A. Orbit closures of generic unipotent flows on homogeneous spaces of SL ( 3 , ) , Math. Ann., Volume 286 (1990), pp. 101-128 | DOI | MR | Zbl

[DM3] Dani, S. G.; Margulis, G. A. Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Indian Acad. Sci. J., Volume 101 (1991), pp. 1-17 | MR | Zbl

[DM4] Dani, S. G.; Margulis, G. A. Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gelfand Seminar, Am. Math. Soc., Providence, 1993, pp. 91-137 | DOI

[Ef] Effros, E. G. Transformation groups and C * -algebras, Ann. Math. (2), Volume 81 (1965), pp. 38-55 | DOI | MR | Zbl

[EKL] Einsiedler, M.; Katok, A.; Lindenstrauss, E. Invariant measures and the set of exceptions to Littlewood’s conjecture, Ann. Math. (2), Volume 164 (2006), pp. 513-560 | DOI | MR | Zbl

[EL] Einsiedler, M.; Lindenstrauss, E. Diagonal actions on locally homogeneous spaces, Homogeneous Flows, Moduli Spaces and Arithmetic, Am. Math. Soc., Providence, 2010, pp. 155-241 | Zbl

[EMa] Eskin, A.; Masur, H. Asymptotic formulas on flat surfaces, Ergod. Theory Dyn. Syst., Volume 21 (2001), pp. 443-478 | DOI | MR | Zbl

[EMM] Eskin, A.; Marklof, J.; Morris, D. Unipotent flows on the space of branched covers of Veech surfaces, Ergod. Theory Dyn. Syst., Volume 26 (2006), pp. 129-162 | DOI | MR | Zbl

[EMM1] Eskin, A.; Margulis, G.; Mozes, S. Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. Math. (2), Volume 147 (1998), pp. 93-141 | DOI | MR | Zbl

[EMM2] Eskin, A.; Margulis, G.; Mozes, S. Quadratic forms of signature (2,2) and eigenvalue spacings on flat 2 -tori, Ann. Math. (2), Volume 161 (2005), pp. 679-725 | DOI | MR | Zbl

[EMiMo] Eskin, A.; Mirzakhani, M.; Mohammadi, A. Isolation, equidistribution, and orbit closures for the SL ( 2 , ) action on moduli space, Ann. Math. (2), Volume 182 (2015), pp. 673-721 | DOI | MR | Zbl

[EMR] A. Eskin, M. Mirzakhani and K. Rafi, Counting closed geodesics in strata, 2012, | arXiv

[EMS] Eskin, A.; Masur, H.; Schmoll, M. Billiards in rectangles with barriers, Duke Math. J., Volume 118 (2003), pp. 427-463 | DOI | MR | Zbl

[EMZ] Eskin, A.; Masur, H.; Zorich, A. Moduli spaces of Abelian differentials: the principal boundary, counting problems and the Siegel–Veech constants, Publ. Math. Inst. Hautes Études Sci., Volume 97 (2003), pp. 61-179 | DOI | MR | Zbl

[EMat] A. Eskin and C. Matheus, Semisimplicity of the Lyapunov spectrum for irreducible cocycles, preprint.

[Fo] Forni, G. Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. Math., Volume 155 (2002), pp. 1-103 | DOI | MR | Zbl

[Fo2] Forni, G. On the Lyapunov exponents of the Kontsevich-Zorich cocycle, Handbook of Dynamical Systems, vol. 1B, Elsevier, Amsterdam, 2006, pp. 549-580 | DOI | Zbl

[FoM] G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, 2008, | arXiv

[FoMZ] Forni, G.; Matheus, C.; Zorich, A. Lyapunov spectrum of invariant subbundles of the Hodge bundle, Ergod. Theory Dyn. Syst., Volume 34 (2014), pp. 353-408 | DOI | MR | Zbl

[Fu] Furman, A. Random walks on groups and random transformations, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2002, pp. 931-1014

[F1] Furstenberg, H. A Poisson formula for semi-simple Lie groups, Ann. Math., Volume 77 (1963), pp. 335-386 | DOI | MR | Zbl

[F2] Furstenberg, H. Non commuting random products, Trans. Am. Math. Soc., Volume 108 (1963), pp. 377-428 | DOI | Zbl

[Fi1] Filip, S. Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, Invent. Math., Volume 205 (2016), pp. 617-670 | DOI | MR | Zbl

[Fi2] Filip, S. Splitting mixed Hodge structures over affine invariant manifolds, Ann. Math. (2), Volume 183 (2016), pp. 681-713 | DOI | MR | Zbl

[GM] Gol’dsheid, I. Ya.; Margulis, G. A. Lyapunov indices of a product of random matrices, Russ. Math. Surv., Volume 44 (1989), pp. 11-71 | DOI

[GR1] Guivarc’h, Y.; Raugi, A. Frontiere de Furstenberg, propriotes de contraction et theoremes de convergence, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 69 (1985), pp. 187-242 | DOI | Zbl

[GR2] Guivarc’h, Y.; Raugi, A. Propriétés de contraction d’un semi-groupe de matrices inversibles. Coefficients de Liapunoff d’un produit de matrices aléatoires indépendantes, Isr. J. Math., Volume 65 (1989), pp. 165-196 (French) [Contraction properties of an invertible matrix semigroup. Lyapunov coefficients of a product of independent random matrices] | DOI | Zbl

[HLM] Hubert, P.; Lanneau, E.; Möller, M. GL 2 + ( ) -orbit closures via topological splittings, Geometry of Riemann Surfaces and Their Moduli Spaces, International Press, Somerville, 2009, pp. 145-169

[HST] Hubert, P.; Schmoll, M.; Troubetzkoy, S. Modular fibers and illumination problems, Int. Math. Res. Not., Volume 2008 (2008) | MR | Zbl

[Ka] Kac, M. On the notion of recurrence in discrete stochastic processes, Bull. Am. Math. Soc., Volume 53 (1947), pp. 1002-1010 | DOI | MR | Zbl

[Ke] Kesten, H. Sums of stationary sequences cannot grow slower than linearly, Proc. Am. Math. Soc., Volume 49 (1975), pp. 205-211 | DOI | MR | Zbl

[Kn] Knapp, A. Lie Groups, Beyond an Introduction, Birkhäuser, Boston, 2002 | Zbl

[KS] Kalinin, B.; Sadovskaya, V. Cocycles with one exponent over partially hyperbolic systems, Geom. Dedic., Volume 167 (2013), pp. 167-188 | DOI | MR | Zbl

[KH] Katok, A.; Hasselblat, B. Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995 | DOI

[KuSp] Kurdyka, K.; Spodzieja, S. Separation of real algebraic sets and the Łojasiewicz exponent, Proc. Am. Math. Soc., Volume 142 (2014), pp. 3089-3102 | DOI | Zbl

[LN1] Lanneau, E.; Nguyen, D. Teichmueller curves generated by Weierstrass Prym eigenforms in genus three and genus four, J. Topol., Volume 7 (2014), pp. 475-522 | DOI | MR | Zbl

[LN2] Lanneau, E.; Nguyen, D. Complete periodicity of Prym eigenforms, Ann. Sci. Éc. Norm. Supér. (4), Volume 49 (2016), pp. 87-130 | DOI | MR | Zbl

[LN3] Lanneau, E.; Nguyen, D. GL + ( 2 , R ) -orbits in Prym eigenform loci, Geom. Topol., Volume 20 (2016), pp. 1359-1426 | DOI | MR | Zbl

[L] Ledrappier, F. Positivity of the exponent for stationary sequences of matrices, Lyapunov Exponents (1986), pp. 56-73 | DOI

[LS] Ledrappier, F.; Strelcyn, J. M. A proof of the estimation from below in Pesin’s entropy formula, Ergod. Theory Dyn. Syst., Volume 2 (1982), pp. 203-219 | DOI | MR | Zbl

[LY] Ledrappier, F.; Young, L. S. The metric entropy of diffeomorphisms. I, Ann. Math., Volume 122 (1985), pp. 503-539 | Zbl

[M1] Mañé, R. A proof of Pesin’s formula, Ergod. Theory Dyn. Syst., Volume 1 (1981), pp. 95-102 | DOI | MR | Zbl

[M2] Mañé, R. Ergodic Theory and Differentiable Dynamics, Springer, Berlin, 1987 | DOI | Zbl

[Mar1] Margulis, G. A. On the action of unipotent groups in the space of lattices, Lie Groups and Their Representations, Proc. of Summer School in Group Representations (1975), pp. 365-370

[Mar2] Margulis, G. A. Formes quadratiques indèfinies et flots unipotents sur les spaces homogènes, C. R. Acad. Sci. Paris Ser. I, Volume 304 (1987), pp. 247-253 | Zbl

[Mar3] Margulis, G. A. Discrete subgroups and ergodic theory, Number Theory, Trace Formulas and Discrete Subgroups, a Symposium in Honor of a Selberg, Academic Press, Boston, 1989, pp. 377-398

[Mar4] Margulis, G.A. Indefinite quadratic forms and unipotent flows on homogeneous spaces, Dynamical Systems and Ergodic Theory, Banach Center Publ., PWN—Polish Scientific Publ., Warsaw, 1989, pp. 399-409

[MaT] Margulis, G. A.; Tomanov, G. M. Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., Volume 116 (1994), pp. 347-392 | DOI | MR | Zbl

[Mas1] Masur, H. Interval exchange transformations and measured foliations, Ann. Math. (2), Volume 115 (1982), pp. 169-200 | DOI | MR | Zbl

[Mas2] Masur, H. The growth rate of trajectories of a quadratic differential, Ergod. Theory Dyn. Syst., Volume 10 (1990), pp. 151-176 | DOI | MR | Zbl

[Mas3] Masur, H. Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential (Drasin, D., ed.), Holomorphic Functions and Moduli, Springer, New York, 1988, pp. 215-228 | DOI

[MW] Matheus, C.; Wright, A. Hodge-Teichmueller planes and finiteness results for Teichmueller curves, Duke Math. J., Volume 164 (2015), pp. 1041-1077 | DOI | MR | Zbl

[Mc1] McMullen, C. Billiards and Teichmüller curves on Hilbert modular surfaces, J. Am. Math. Soc., Volume 16 (2003), pp. 857-885 | DOI | Zbl

[Mc2] McMullen, C. Teichmüller geodesics of infinite complexity, Acta Math., Volume 191 (2003), pp. 191-223 | DOI | MR | Zbl

[Mc3] McMullen, C. Teichmüller curves in genus two: discriminant and spin, Math. Ann., Volume 333 (2005), pp. 87-130 | DOI | MR | Zbl

[Mc4] McMullen, C. Teichmüller curves in genus two: the decagon and beyond, J. Reine Angew. Math., Volume 582 (2005), pp. 173-200 | DOI | MR | Zbl

[Mc5] McMullen, C. Teichmüller curves in genus two: torsion divisors and ratios of sines, Invent. Math., Volume 165 (2006), pp. 651-672 | DOI | MR | Zbl

[Mc6] McMullen, C. Dynamics of SL 2 ( ) over moduli space in genus two, Ann. Math. (2), Volume 165 (2007), pp. 397-456 | DOI | MR | Zbl

[Mö1] Möller, M. Variations of Hodge structures of a Teichmüller curve, J. Am. Math. Soc., Volume 19 (2006), pp. 327-344 | DOI | Zbl

[Mö2] Möller, M. Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math., Volume 165 (2006), pp. 633-649 | DOI | MR | Zbl

[Mö3] Möller, M. Finiteness results for Teichmüller curves, Ann. Inst. Fourier (Grenoble), Volume 58 (2008), pp. 63-83 | DOI | MR | Zbl

[Mö4] Möller, M. Linear manifolds in the moduli space of one-forms, Duke Math. J., Volume 144 (2008), pp. 447-488 | DOI | MR | Zbl

[Mor] Morris, D. W. Ratner’s Theorems on Unipotent Flows, University of Chicago Press, Chicago, 2005 (arXiv:math/0310402 [math.DS]) | Zbl

[Moz] Mozes, S. Epimorphic subgroups and invariant measures, Ergod. Theory Dyn. Syst., Volume 15 (1995), pp. 1207-1210 | DOI | MR | Zbl

[MoSh] Mozes, S.; Shah, N. On the space of ergodic invariant measures of unipotent flows, Ergod. Theory Dyn. Syst., Volume 15 (1995), pp. 149-159 | MR | Zbl

[MZ] Zimmer, R.; Witte Morris, D. Ergodic Theory, Groups, and Geometry, Am. Math. Soc., Providence, 2008 (x+87 pp. Published for the Conference Board of the Mathematical Sciences, Washington, DC) | Zbl

[NW] Nguyen, D.; Wright, A. Non-Veech surfaces in hyp ( 4 ) are generic, Geom. Funct. Anal., Volume 24 (2014), pp. 1316-1335 | DOI | MR | Zbl

[NZ] Nevo, A.; Zimmer, R. Homogeneous projective factors for actions of semisimple Lie groups, Invent. Math., Volume 138 (1999), pp. 229-252 | DOI | MR | Zbl

[Ra1] Ratner, M. Rigidity of horocycle flows, Ann. Math., Volume 115 (1982), pp. 597-614 | DOI | MR | Zbl

[Ra2] Ratner, M. Factors of horocycle flows, Ergod. Theory Dyn. Syst., Volume 2 (1982), pp. 465-489 | DOI | MR | Zbl

[Ra3] Ratner, M. Horocycle flows, joinings and rigidity of products, Ann. Math., Volume 118 (1983), pp. 277-313 | DOI | MR | Zbl

[Ra4] Ratner, M. Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math., Volume 101 (1990), pp. 449-482 | DOI | MR | Zbl

[Ra5] Ratner, M. On measure rigidity of unipotent subgroups of semisimple groups, Acta Math., Volume 165 (1990), pp. 229-309 | DOI | MR | Zbl

[Ra6] Ratner, M. On Raghunathan’s measure conjecture, Ann. Math., Volume 134 (1991), pp. 545-607 | DOI | MR | Zbl

[Ra7] Ratner, M. Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J., Volume 63 (1991), pp. 235-280 | DOI | MR | Zbl

[R] Rokhlin, V. A. Lectures on the theory of entropy of transformations with invariant measures, Russ. Math. Surv., Volume 22 (1967), pp. 1-54 | DOI | Zbl

[Sch] Schmidt, K. Amenability, Kazhdan’s property T , strong ergodicity and invariant means for ergodic group-actions, Ergod. Theory Dyn. Syst., Volume 1 (1981), pp. 223-236 | DOI | MR | Zbl

[Ve1] Veech, W. Gauss measures for transformations on the space of interval exchange maps, Ann. Math., Volume 15 (1982), pp. 201-242 | DOI | MR | Zbl

[Ve2] Veech, W. Siegel measures, Ann. Math., Volume 148 (1998), pp. 895-944 | DOI | MR | Zbl

[Wr1] Wright, A. The field of definition of affine invariant submanifolds of the moduli space of Abelian differentials, Geom. Topol., Volume 18 (2014), pp. 1323-1341 | DOI | MR | Zbl

[Wr2] Wright, A. Cylinder deformations in orbit closures of translation surfaces, Geom. Topol., Volume 19 (2015), pp. 413-438 | DOI | MR | Zbl

[WWF] Wang, L.; Wang, X.; Feng, J. Subspace distance analysis with application to adaptive Bayesian algorithm for face recognition, Pattern Recognit., Volume 39 (2006), pp. 456-464 | DOI | Zbl

[Zi1] Zimmer, R. J. Induced and amenable ergodic actions of Lie groups, Ann. Sci. Éc. Norm. Supér., Volume 11 (1978), pp. 407-428 | DOI | MR | Zbl

[Zi2] Zimmer, R. J. Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984 | DOI | Zbl

[Zo] Zorich, A. Flat Surfaces, Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, pp. 437-583 | DOI

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