# American Institute of Mathematical Sciences

April  2013, 33(4): 1247-1273. doi: 10.3934/dcds.2013.33.1247

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

 1 Faculdade de Matemática - Universidade Federal de Uberlândia, Campus Santa Mônica, Av. João Naves de Ávila - 2121 38.408-100, Uberlândia - MG, Brazil 2 Imecc - Unicamp, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz 13083-859, Campinas - SP, Brazil

Received  September 2011 Revised  January 2012 Published  October 2012

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.
Citation: Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247
##### References:
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##### References:
 [1] L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998). [2] L. Arnold, N. D. Cong and V. I. Oseledets, Jordan normal form for linear cocycles,, Random Oper. Stochastic Equations, 7 (1999), 303. [3] F. Colonius and W. Kliemann, "The Dynamics of Control,", Birkhäuser, (2000). doi: 10.1007/978-1-4612-1350-5_3. [4] 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. [5] R. Feres, "Dynamical Systems and Semisimple Groups: An Introduction,", Cambridge University Press, (1998). doi: 10.5840/ancientphil199818243. [6] Y. Guivarch, L. Ji and J. C. Taylor, "Compactifications of Symmetric Spaces,", Progress in Mathematics, (1998). [7] S. Helgason, "Groups and Geometric Analysis,", Academic Press, (1984). [8] J. Hilgert and G. Ólafsson, "Causal Symmetric Spaces,", Geometry and Harmonic Analysis, 18 (1997). [9] V. A. Kaimanovich, Lyapunov exponents, symmetric spaces and multiplicative ergodic theorem for semisimple lie groups,, J. Soviet Math., 47 (1989), 2387. doi: 10.1007/BF01840421. [10] A. Karlsson and G. A. Margulis, A multiplicative ergodic theorem and nonpositively curved spaces,, Communications in Mathematical Physics, 208 (1999), 107. doi: 10.1007/s002200050750. [11] A. W. Knapp, "Lie Groups: Beyond an Introduction,", Second Edition, 140 (2004). [12] G. Link, "Limit Sets of Discrete Groups Acting on Symmetric Spaces,", Ph.D thesis, (2002). [13] S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry,", vol. I, (1963). [14] M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem,, Israel J. Math., 32 (1979), 356. [15] D. Ruelle, Ergodic theory of differentiable dynamical systems,, Publ. Math. IHES, 50 (1979), 27. doi: 10.1007/BF00587200. [16] L. A. B. San Martin, Maximal semigroups in semi-simple lie groups,, Trans. Amer. Math. Soc., 353 (2001), 5165. [17] L. A. B. San Martin, Nonexistence of invariant semigroups in affine symmetric spaces,, Math. Ann., 321 (2001), 587. doi: 10.1016/S0039-6028(01)00812-3. [18] L. A. B. San Martin and L. Seco, Morse and Lyapunov spectra and dynamics on flag bundles,, Ergodic Theory & Dynamical Systems, 30 (2010), 893. doi: 10.1017/S0143385709000285. [19] G. Warner, "Harmonic Analysis on Semi-simple Lie Groups I,", Springer-Verlag, (1972). doi: 10.1007/978-3-642-50275-0. [20] R.Zimmer, Ergodic theory and semisimple groups,, Monographs in Mathematics, 81 (1984).
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