Oseledets multiplicative ergodic theorem

Statement

Let $T$ be an invertible ergodic transformation of a standard probability space and let $A:X\to GL_d(\mathbb R)$ be measurable with

\log^+\|A\|,\ \log^+\|A^{-1}\|\in L^1, \qquad\log^+t=\max(0,\log t).

There are real numbers $\lambda_1>\cdots>\lambda_r$ and, on an invariant conull set, a measurable direct sum

\mathbb R^d=E_1(x)\oplus\cdots\oplus E_r(x),\qquad A(x)E_i(x)=E_i(Tx),

such that for every nonzero $v\in E_i(x)$ ,

\lim_{n\to\pm\infty}\frac1n\log\|A^{(n)}(x)v\|=\lambda_i.

This is the invertible finite-dimensional form of the multiplicative ergodic theorem.

Applications and hypotheses

The theorem makes Lyapunov exponents and invariant growth directions available in smooth dynamics and random matrix products. Integrability of the inverse is used for the two-sided finite-exponent splitting stated here. Versions with weaker assumptions give filtrations or allow infinite exponents.

References

Simion Filip, Notes on the Multiplicative Ergodic Theorem, Theorem 2.2.6 and Variant 2.2.10.