Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on Moduli space

Alex Eskin; Maryam Mirzakhani

doi:10.1007/s10240-018-0099-2

Invariant and stationary measures for the

SL (2, ℝ)

action on Moduli space

Alex Eskin; Maryam Mirzakhani

Publications Mathématiques de l'IHÉS, Volume 127 (2018), pp. 95-324

Abstract

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

Received: 2014-02-17
Accepted: 2018-02-12
Online First: 2018-04-17
Published online: 2018-06-07

DOI: 10.1007/s10240-018-0099-2

@article{PMIHES_2018__127__95_0,
     author = {Alex Eskin and Maryam Mirzakhani},
     title = {Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on {Moduli} space},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {95--324},
     year = {2018},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {127},
     doi = {10.1007/s10240-018-0099-2},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-018-0099-2/}
}

TY  - JOUR
AU  - Alex Eskin
AU  - Maryam Mirzakhani
TI  - Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on Moduli space
JO  - Publications Mathématiques de l'IHÉS
PY  - 2018
SP  - 95
EP  - 324
VL  - 127
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-018-0099-2/
DO  - 10.1007/s10240-018-0099-2
LA  - en
ID  - PMIHES_2018__127__95_0
ER  -

%0 Journal Article
%A Alex Eskin
%A Maryam Mirzakhani
%T Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on Moduli space
%J Publications Mathématiques de l'IHÉS
%D 2018
%P 95-324
%V 127
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-018-0099-2/
%R 10.1007/s10240-018-0099-2
%G en
%F PMIHES_2018__127__95_0

Alex Eskin; Maryam Mirzakhani. Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on Moduli space. Publications Mathématiques de l'IHÉS, Volume 127 (2018), pp. 95-324. doi: 10.1007/s10240-018-0099-2

References
Cited by

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

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

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

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

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

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

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

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

[AG] A. Avila; S. Gouëzel 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 | MR | Zbl | DOI

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

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

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

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

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

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

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

[BQ] Y. Benoist; J-F. Quint 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] | MR | Zbl | DOI

[BG] A. Bufetov; B. Gurevich 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 | MR | Zbl | DOI

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

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

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

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

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

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

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

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

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

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

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

[DM4] S. G. Dani; G. A. Margulis 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] E. G. Effros Transformation groups and $C^{*}$ -algebras, Ann. Math. (2), Volume 81 (1965), pp. 38-55 | MR | Zbl | DOI

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

[EL] M. Einsiedler; E. Lindenstrauss Diagonal actions on locally homogeneous spaces, Homogeneous Flows, Moduli Spaces and Arithmetic, 10, Am. Math. Soc., Providence, 2010, pp. 155-241 | Zbl

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

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

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

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

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

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

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

[EMZ] A. Eskin; H. Masur; A. Zorich 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 | MR | Zbl | DOI

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

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

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

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

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

[Fu] A. Furman Random walks on groups and random transformations, Handbook of Dynamical Systems, 1A, North-Holland, Amsterdam, 2002, pp. 931-1014

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

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

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

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

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

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

[GR2] Y. Guivarc’h; A. Raugi 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] | Zbl | DOI

[HLM] P. Hubert; E. Lanneau; M. Möller ${GL}_{2}^{+} (ℝ)$ -orbit closures via topological splittings, Geometry of Riemann Surfaces and Their Moduli Spaces, 14, International Press, Somerville, 2009, pp. 145-169

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

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

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

[Kn] A. Knapp Lie Groups, Beyond an Introduction, 140, Birkhäuser, Boston, 2002 | Zbl

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

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

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

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

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

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

[L] F. Ledrappier Positivity of the exponent for stationary sequences of matrices, Lyapunov Exponents, Volume 1186 (1986), pp. 56-73 | DOI

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

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

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

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

[Mar1] G. A. Margulis 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] G. A. Margulis 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] G. A. Margulis 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] G.A. Margulis Indefinite quadratic forms and unipotent flows on homogeneous spaces, Dynamical Systems and Ergodic Theory, 23, Banach Center Publ., PWN—Polish Scientific Publ., Warsaw, 1989, pp. 399-409

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

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

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

[Mas3] H. Masur Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential (D. Drasin, ed.), Holomorphic Functions and Moduli, 1, Springer, New York, 1988, pp. 215-228 | DOI

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[MZ] R. Zimmer; D. Witte Morris Ergodic Theory, Groups, and Geometry, 109, Am. Math. Soc., Providence, 2008 (x+87 pp. Published for the Conference Board of the Mathematical Sciences, Washington, DC) | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cited by Sources: