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