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

Abstract

Let $Q\rightarrow X$ be a principal bundle having as structuralgroup $G$ areductive Lie group in the Harish-Chandra class that includes the case when $G$ is semi-simple with finite center. A semiflow $\phi _{k}$ ofendomorphisms of $Q$ induces a semiflow $\psi _{k}$ on theassociated bundle $\mathbb{E}=Q\times _{G}\mathbb{F}$, where$\mathbb{F}$ is the maximal flag bundle of $G$. The $A$-componentof the Iwasawa decomposition $G=KAN$ yields an additive vectorvalued 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 ofOseledets for this cocycle: If $\nu $ is a probability measureinvariant by the semiflow on $X$ then the $\mathfrak{a}$-Lyapunovexponent $\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 setsof $\lambda \left( \cdot \right) $ on the fibers are described inalgebraic terms. When $\phi _{k}$ is a flow the description of thelevel sets is sharpened. We relate the cocycle $\mathsf{a}\left(k,\xi \right) $ with the Lyapunov exponents of a linear flow on avector bundle and other growth rates.