A companion to the Oseledec multiplicative ergodic theorem

Abstract

Let F 1 , F 2 , … {F_1},{F_2}, \ldots be a stationary sequence of continuously differentiable mappings from [ 0 , 1 ] [0,1] into the set of d × d d \times d matrices. Assume F k ( 0 ) = I {F_k}(0) = I for each k k and E [ sup 0 ≤ p ≤ 1 | | F k ′ ( p ) | | ] > ∞ E[{\sup _{0 \leq p \leq 1}}||{F’_k}(p)||] > \infty . Let I \mathcal {I} denote the invariant sigma field for the sequence. Then \[ lim n → ∞ F n ( 1 n ) ⋯ F 2 ( 1 n ) F 1 ( 1 n ) = exp ⁡ E [ F 1 ′ ( 0 ) | I ] \lim \limits _{n \to \infty } {F_n}\left ( {\frac {1}{n}} \right ) \cdots {F_2}\left ( {\frac {1}{n}} \right ){F_1}\left ( {\frac {1}{n}} \right ) = \exp E[{F’_1}(0)|\mathcal {I}] \] with probability one.