A companion to the Oseledec multiplicative ergodic theorem

Abstract:

Let ${F_1},{F_2}, \ldots$ be a stationary sequence of continuously differentiable mappings from $[0,1]$ into the set of $d \times d$ matrices. Assume ${F_k}(0) = I$ for each $k$ and $E[{\sup _{0 \leq p \leq 1}}||{F’_k}(p)||] < \infty$. Let $\mathcal {I}$ denote the invariant sigma field for the sequence. Then \[ \lim \limits _{n \to \infty } {F_n}\left ( {\frac {1}{n}} \right ) \cdots {F_2}\left ( {\frac {1}{n}} \right ){F_1}\left ( {\frac {1}{n}} \right ) = \exp E[{F’_1}(0)|\mathcal {I}]\] with probability one.