Skip to Main Content

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Oseledec multiplicative ergodic theorem for laminations

About this Title

Viêt-Anh Nguyên, Université Paris-Sud, Laboratoire de Mathématique, UMR 8628, Bâtiment 425, F-91405 Orsay, France

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 246, Number 1164
ISBNs: 978-1-4704-2253-0 (print); 978-1-4704-3637-7 (online)
Published electronically: December 1, 2016
Keywords: Lamination, foliation, harmonic measure, Wiener measure, Brownian motion, Lyapunov exponents, multiplicative ergodic theorem, Oseledec decomposition, holonomy invariant
MSC: Primary 37A30, 57R30; Secondary 58J35, 58J65, 60J65

View full volume PDF

View other years and numbers:

Table of Contents


  • Acknowledgement
  • 1. Introduction
  • 2. Background
  • 3. Statement of the main results
  • 4. Preparatory results
  • 5. Leafwise Lyapunov exponents
  • 6. Splitting subbundles
  • 7. Lyapunov forward filtrations
  • 8. Lyapunov backward filtrations
  • 9. Proof of the main results
  • A. Measure theory for sample-path spaces
  • B. Harmonic measure theory and ergodic theory for sample-path spaces


Given a $n$-dimensional lamination endowed with a Riemannian metric, we introduce the notion of a multiplicative cocycle of rank $d,$ where $n$ and $d$ are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as well as its tensor powers provide examples of multiplicative cocycles. Next, we define the Lyapunov exponents of such a cocycle with respect to a harmonic probability measure directed by the lamination. We also prove an Oseledec multiplicative ergodic theorem in this context. This theorem implies the existence of an Oseledec decomposition almost everywhere which is holonomy invariant. Moreover, in the case of differentiable cocycles we establish effective integral estimates for the Lyapunov exponents. These results find applications in the geometric and dynamical theory of laminations. They are also applicable to (not necessarily closed) laminations with singularities. Interesting holonomy properties of a generic leaf of a foliation are obtained. The main ingredients of our method are the theory of Brownian motion, the analysis of the heat diffusions on Riemannian manifolds, the ergodic theory in discrete dynamics and a geometric study of laminations.

References [Enhancements On Off] (What's this?)

  • Christian Bonatti, Xavier Gómez-Mont, and Marcelo Viana, Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 4, 579–624 (French, with English and French summaries). MR 1981401, DOI 10.1016/S0294-1449(02)00019-7
  • Alberto Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 4, 489–516. MR 1235439
  • Alberto Candel, The harmonic measures of Lucy Garnett, Adv. Math. 176 (2003), no. 2, 187–247. MR 1982882, DOI 10.1016/S0001-8708(02)00036-1
  • Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 1732868
  • Alberto Candel and Lawrence Conlon, Foliations. II, Graduate Studies in Mathematics, vol. 60, American Mathematical Society, Providence, RI, 2003. MR 1994394
  • A. Candel and X. Gómez-Mont, Uniformization of the leaves of a rational vector field, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 4, 1123–1133 (English, with English and French summaries). MR 1359843
  • C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
  • Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
  • Gustave Choquet, Lectures on analysis. Vol. II: Representation theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Edited by J. Marsden, T. Lance and S. Gelbart. MR 0250012
  • I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433
  • Bertrand Deroin, Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 1, 57–75 (French, with English and French summaries). MR 2136481, DOI 10.1016/j.ansens.2004.10.004
  • T.-C. Dinh, V.-A. Nguyên, and N. Sibony, Heat equation and ergodic theorems for Riemann surface laminations, Math. Ann. 354 (2012), no. 1, 331–376. MR 2957629, DOI 10.1007/s00208-011-0730-8
  • Dinh T.-C.; Nguyên V.-A.; Sibony N., Entropy for hyperbolic Riemann surface laminations I, Frontiers in Complex Dynamics: a volume in honor of John Milnor’s 80th birthday, (A. Bonifant, M. Lyubich, S. Sutherland, editors), (2014), 569-592, Princeton University Press.
  • Dinh T.-C.; Nguyên V.-A.; Sibony N., Entropy for hyperbolic Riemann surface laminations II, Frontiers in Complex Dynamics: a volume in honor of John Milnor’s 80th birthday, (A. Bonifant, M. Lyubich, S. Sutherland, editors), (2014), 593-621, Princeton University Press.
  • Richard M. Dudley, Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989. MR 982264
  • J. E. Fornæss and N. Sibony, Harmonic currents of finite energy and laminations, Geom. Funct. Anal. 15 (2005), no. 5, 962–1003. MR 2221156, DOI 10.1007/s00039-005-0531-x
  • John Erik Fornæss and Nessim Sibony, Riemann surface laminations with singularities, J. Geom. Anal. 18 (2008), no. 2, 400–442. MR 2393266, DOI 10.1007/s12220-008-9018-y
  • John Erik Fornæss and Nessim Sibony, Unique ergodicity of harmonic currents on singular foliations of $\Bbb P^2$, Geom. Funct. Anal. 19 (2010), no. 5, 1334–1377. MR 2585577, DOI 10.1007/s00039-009-0043-1
  • Steven Hurder, Classifying foliations, Foliations, geometry, and topology, Contemp. Math., vol. 498, Amer. Math. Soc., Providence, RI, 2009, pp. 1–65. MR 2664589, DOI 10.1090/conm/498/09741
  • Lucy Garnett, Foliations, the ergodic theorem and Brownian motion, J. Functional Analysis 51 (1983), no. 3, 285–311. MR 703080, DOI 10.1016/0022-1236(83)90015-0
  • Étienne Ghys, Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997) Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. ix, xi, 49–95 (French, with English and French summaries). MR 1760843
  • Alexander Grigor′yan, Heat kernel of a noncompact Riemannian manifold, Stochastic analysis (Ithaca, NY, 1993) Proc. Sympos. Pure Math., vol. 57, Amer. Math. Soc., Providence, RI, 1995, pp. 239–263. MR 1335475, DOI 10.1090/pspum/057/1335475
  • V. A. Kaĭmanovich, Brownian motion on foliations: entropy, invariant measures, mixing, Funktsional. Anal. i Prilozhen. 22 (1988), no. 4, 82–83 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 4, 326–328 (1989). MR 977003, DOI 10.1007/BF01077429
  • Shizuo Kakutani, Random ergodic theorems and Markoff processes with a stable distribution, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, pp. 247–261. MR 0044773
  • Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374
  • Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411
  • F. Ledrappier, Quelques propriétés des exposants caractéristiques, École d’été de probabilités de Saint-Flour, XII—1982, Lecture Notes in Math., vol. 1097, Springer, Berlin, 1984, pp. 305–396 (French). MR 876081, DOI 10.1007/BFb0099434
  • A. Lins Neto, Uniformization and the Poincaré metric on the leaves of a foliation by curves, Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 3, 351–366. MR 1817093, DOI 10.1007/BF01241634
  • V.-A., Nguyên, Geometric characterization of Lyapunov exponents for Riemann surface laminations, math.CV, math.DS, arXiv:1503.05231, 35 pp.
  • V.-A., Nguyên, Singular holomorphic foliations by curves I: Integrability of holonomy cocycle in dimension 2, math.DS, math.CV, math.DG, arXiv:1403.7688.
  • V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
  • Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791
  • David Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 27–58. MR 556581
  • Dennis Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225–255. MR 433464, DOI 10.1007/BF01390011
  • PawełWalczak, Dynamics of foliations, groups and pseudogroups, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 64, Birkhäuser Verlag, Basel, 2004. MR 2056374
  • Peter Walters, A dynamical proof of the multiplicative ergodic theorem, Trans. Amer. Math. Soc. 335 (1993), no. 1, 245–257. MR 1073779, DOI 10.1090/S0002-9947-1993-1073779-7
  • Richard L. Wheeden and Antoni Zygmund, Measure and integral, Marcel Dekker, Inc., New York-Basel, 1977. An introduction to real analysis; Pure and Applied Mathematics, Vol. 43. MR 0492146