Advanced Search
Article Contents
Article Contents

Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups

Abstract Related Papers Cited by
  • Let $Q\rightarrow X$ be a principal bundle having as structural group $G$ a reductive Lie group in the Harish-Chandra class that includes the case when $G$ is semi-simple with finite center. A semiflow $\phi _{k}$ of endomorphisms of $Q$ induces a semiflow $\psi _{k}$ on the associated bundle $\mathbb{E}=Q\times _{G}\mathbb{F}$, where $\mathbb{F}$ is the maximal flag bundle of $G$. The $A$-component of the Iwasawa decomposition $G=KAN$ yields an additive vector valued cocycle $\mathsf{a}\left( k,\xi \right) $, $\xi \in \mathbb{E}$, over $\psi _{k}$ with values in the Lie algebra $\mathfrak{a}$ of $A$. We prove the Multiplicative Ergodic Theorem of Oseledets for this cocycle: If $\nu $ is a probability measure invariant by the semiflow on $X$ then the $\mathfrak{a}$-Lyapunov exponent $\lambda \left( \xi \right) =\lim \frac{1}{k}\mathsf{a}\left( k,\xi \right) $ exists for every $\xi $ on the fibers above a set of full $\nu $-measure. The level sets of $\lambda \left( \cdot \right) $ on the fibers are described in algebraic terms. When $\phi _{k}$ is a flow the description of the level sets is sharpened. We relate the cocycle $\mathsf{a}\left( k,\xi \right) $ with the Lyapunov exponents of a linear flow on a vector bundle and other growth rates.
    Mathematics Subject Classification: Primary: 37H15, 22E46, 37B55.


    \begin{equation} \\ \end{equation}
  • [1]

    L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, 1998.


    L. Arnold, N. D. Cong and V. I. Oseledets, Jordan normal form for linear cocycles, Random Oper. Stochastic Equations, 7 (1999), 303-358.


    F. Colonius and W. Kliemann, "The Dynamics of Control," Birkhäuser, Boston, 2000.doi: 10.1007/978-1-4612-1350-5_3.


    J. J. Duistermat, J. A. C. Kolk and V. S. Varadarajan, Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups, Compositio Math., 49 (1983), 309-398.


    R. Feres, "Dynamical Systems and Semisimple Groups: An Introduction," Cambridge University Press, 1998.doi: 10.5840/ancientphil199818243.


    Y. Guivarch, L. Ji and J. C. Taylor, "Compactifications of Symmetric Spaces," Progress in Mathematics, 156. Birkhser Boston, 1998.


    S. Helgason, "Groups and Geometric Analysis," Academic Press, 1984.


    J. Hilgert and G. Ólafsson, "Causal Symmetric Spaces," Geometry and Harmonic Analysis, Perspectives in Mathematics, 18, Academic Press, 1997.


    V. A. Kaimanovich, Lyapunov exponents, symmetric spaces and multiplicative ergodic theorem for semisimple lie groups, J. Soviet Math., 47 (1989), 2387-2398.doi: 10.1007/BF01840421.


    A. Karlsson and G. A. Margulis, A multiplicative ergodic theorem and nonpositively curved spaces, Communications in Mathematical Physics, 208 (1999), 107-123.doi: 10.1007/s002200050750.


    A. W. Knapp, "Lie Groups: Beyond an Introduction," Second Edition, Progress in Mathematics, 140, Birkhäuser, 2004.


    G. Link, "Limit Sets of Discrete Groups Acting on Symmetric Spaces," Ph.D thesis, Fakultät für Mathematik der Universität Karlsruhe, 2002.


    S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," vol. I, Interscience Publishe, 1963.


    M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362.


    D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. Math. IHES, 50 (1979), 27-58.doi: 10.1007/BF00587200.


    L. A. B. San Martin, Maximal semigroups in semi-simple lie groups, Trans. Amer. Math. Soc., 353 (2001), 5165-5184.


    L. A. B. San Martin, Nonexistence of invariant semigroups in affine symmetric spaces, Math. Ann., 321 (2001), 587-600.doi: 10.1016/S0039-6028(01)00812-3.


    L. A. B. San Martin and L. Seco, Morse and Lyapunov spectra and dynamics on flag bundles, Ergodic Theory & Dynamical Systems, 30 (2010), 893-922.doi: 10.1017/S0143385709000285.


    G. Warner, "Harmonic Analysis on Semi-simple Lie Groups I," Springer-Verlag, 1972.doi: 10.1007/978-3-642-50275-0.


    R.Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, 81, Birkäuser, 1984.

  • 加载中

Article Metrics

HTML views() PDF downloads(140) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint