Oseledets theorem

Remarks

The numbers $\lambda_i$ are the Lyapunov exponents of the cocycle $A$, and the dimensions in clause 3 are their multiplicities. The largest one is $\lambda_1 = \lim_{n} \frac{1}{n} \log \norm{A^{(n)}(x)}$ for almost every $x$, which already follows from Kingman's subadditive ergodic theorem, since $\log \norm{A^{(n)}}$ is subadditive along orbits. Applying the same argument to exterior powers gives the sums $\lambda_1 + \cdots + \lambda_j$ with multiplicity, and linear algebra then builds the subspaces. Without ergodicity the theorem still holds, but $k$, the $\lambda_i$ and the multiplicities become $T$-invariant functions of $x$. Example: if $A$ is a constant diagonal matrix $\mathrm{diag}(e^{a_1}, \dots, e^{a_d})$ with $a_1 > \cdots > a_d$, then $\lambda_i = a_i$ and $V_i$ is spanned by the last $d - i + 1$ coordinate vectors. If $T$ is invertible, the filtration refines to a splitting $\R^d = E_1(x) \oplus \cdots \oplus E_k(x)$ into subspaces on which the growth rate is $\lambda_i$ both forward and backward in time.

Going deeper

For a student

The main use is in smooth ergodic theory: take $A(x) = Df(x)$, the derivative of a diffeomorphism $f$ preserving a measure. Then the Lyapunov exponents measure how fast nearby orbits separate in each direction, and positive exponents signal chaos. Pesin theory builds stable and unstable manifolds from the Oseledets subspaces, and Ruelle's inequality bounds the entropy by the sum of the positive exponents. For $d = 1$ the theorem reduces to Birkhoff's ergodic theorem applied to $\log \abs{A}$.