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

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

Luciana A. Alves 1, and  Luiz A. B. San Martin 2,

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.
Keywords: Lyapunov exponents, multiplicative ergodic theorem, reductive Lie groups, flag manifolds., semi-simple Lie groups.
Mathematics Subject Classification: Primary: 37H15, 22E46, 37B5.
Citation: Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247
References:
[1]

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

[2]

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

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

[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), 309-398.  Google Scholar

[5]

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

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

[7]

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

[8]

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

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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.  Google Scholar

[18]

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

[19]

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

[20]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[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), 309-398.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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.  Google Scholar

[18]

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

[19]

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

[20]

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

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