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

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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

Abstract
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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 and 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.
[2]	L. Arnold, N. D. Cong and V. I. Oseledets, Jordan normal form for linear cocycles, Random Oper. Stochastic Equations, 7 (1999), 303-358.
[3]	F. Colonius and W. Kliemann, "The Dynamics of Control," Birkhäuser, Boston, 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), 309-398.
[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, 156. Birkhser Boston, 1998.
[7]	S. Helgason, "Groups and Geometric Analysis," Academic Press, 1984.
[8]	J. Hilgert and G. Ólafsson, "Causal Symmetric Spaces," Geometry and Harmonic Analysis, Perspectives in Mathematics, 18, Academic Press, 1997.
[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.
[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.
[11]	A. W. Knapp, "Lie Groups: Beyond an Introduction," Second Edition, Progress in Mathematics, 140, Birkhäuser, 2004.
[12]	G. Link, "Limit Sets of Discrete Groups Acting on Symmetric Spaces," Ph.D thesis, Fakultät für Mathematik der Universität Karlsruhe, 2002.
[13]	S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," vol. I, Interscience Publishe, 1963.
[14]	M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362.
[15]	D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. Math. IHES, 50 (1979), 27-58. doi: 10.1007/BF00587200.
[16]	L. A. B. San Martin, Maximal semigroups in semi-simple lie groups, Trans. Amer. Math. Soc., 353 (2001), 5165-5184.
[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.
[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.
[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, Birkäuser, 1984.

show all references

References:

[1]	L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, 1998.
[2]	L. Arnold, N. D. Cong and V. I. Oseledets, Jordan normal form for linear cocycles, Random Oper. Stochastic Equations, 7 (1999), 303-358.
[3]	F. Colonius and W. Kliemann, "The Dynamics of Control," Birkhäuser, Boston, 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), 309-398.
[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, 156. Birkhser Boston, 1998.
[7]	S. Helgason, "Groups and Geometric Analysis," Academic Press, 1984.
[8]	J. Hilgert and G. Ólafsson, "Causal Symmetric Spaces," Geometry and Harmonic Analysis, Perspectives in Mathematics, 18, Academic Press, 1997.
[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.
[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.
[11]	A. W. Knapp, "Lie Groups: Beyond an Introduction," Second Edition, Progress in Mathematics, 140, Birkhäuser, 2004.
[12]	G. Link, "Limit Sets of Discrete Groups Acting on Symmetric Spaces," Ph.D thesis, Fakultät für Mathematik der Universität Karlsruhe, 2002.
[13]	S. Kobayashi and K. Nomizu, "Foundations of Differential Geometry," vol. I, Interscience Publishe, 1963.
[14]	M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362.
[15]	D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. Math. IHES, 50 (1979), 27-58. doi: 10.1007/BF00587200.
[16]	L. A. B. San Martin, Maximal semigroups in semi-simple lie groups, Trans. Amer. Math. Soc., 353 (2001), 5165-5184.
[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.
[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.
[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, Birkäuser, 1984.

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