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  "body": "\n## Statement\n\nLet $(X, \\mathcal{B}, \\mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \\colon X \\to X$ be an [[ergodic|AZRPHS478B]] [[measure-preserving transformation|556WBDAHNP]].\nLet $A \\colon X \\to GL(d, \\R)$ be a measurable map, and let $\\norm{\\cdot}$ be the [[operator norm|06EKW1AKXF]].\nAssume that $\\log^+ \\norm{A}$ and $\\log^+ \\norm{A^{-1}}$ are integrable, where $\\log^+ t = \\max(\\log t, 0)$.\nFor $n \\ge 1$ let $A^{(n)}(x) = A(T^{n-1}x) \\cdots A(Tx) \\, A(x)$.\nThen there are an integer $k$ with $1 \\le k \\le d$ and real numbers $\\lambda_1 \u003e \\lambda_2 \u003e \\cdots \u003e \\lambda_k$ with the following property.\nFor [[almost every|RARTMA0ZEC]] $x \\in X$ there is a chain of linear subspaces\n$$\\R^d = V_1(x) \\supsetneq V_2(x) \\supsetneq \\cdots \\supsetneq V_k(x) \\supsetneq V_{k+1}(x) = \\set{0}$$\nsuch that, for each $i = 1, \\dots, k$:\n\n1. $A(x) V_i(x) = V_i(Tx)$.\n2. For every $v \\in V_i(x) \\setminus V_{i+1}(x)$,\n$$\\lim_{n \\to \\infty} \\frac{1}{n} \\log \\norm{A^{(n)}(x) v} = \\lambda_i.$$\n3. The dimension $\\dim V_i(x) - \\dim V_{i+1}(x)$ does not depend on $x$.\n\n## Remarks\n\nThe numbers $\\lambda_i$ are the [[Lyapunov exponents|NJ1FJSNK6A]] of the cocycle $A$, and the dimensions in clause 3 are their multiplicities.\nThe 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|833XVEEZCJ]], since $\\log \\norm{A^{(n)}}$ is subadditive along orbits.\nApplying the same argument to exterior powers gives the sums $\\lambda_1 + \\cdots + \\lambda_j$ with multiplicity, and linear algebra then builds the subspaces.\nWithout ergodicity the theorem still holds, but $k$, the $\\lambda_i$ and the multiplicities become $T$-invariant functions of $x$.\nExample: if $A$ is a constant diagonal matrix $\\mathrm{diag}(e^{a_1}, \\dots, e^{a_d})$ with $a_1 \u003e \\cdots \u003e a_d$, then $\\lambda_i = a_i$ and $V_i$ is spanned by the last $d - i + 1$ coordinate vectors.\nIf $T$ 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.\n\n## Explanations\n\n### L1 human\n\nMultiply many random matrices together, each chosen by a stationary random process.\nThe theorem says that the product grows at a few well-defined exponential rates, the Lyapunov exponents.\nEach vector grows at one of these rates, and which rate it gets depends only on which of a nested family of subspaces it belongs to.\nThis is the law of large numbers for products of matrices.\n\n### L2 human\n\nThe main use is in smooth ergodic theory: take $A(x) = Df(x)$, the derivative of a diffeomorphism $f$ preserving a measure.\nThen the Lyapunov exponents measure how fast nearby orbits separate in each direction, and positive exponents signal chaos.\nPesin theory builds stable and unstable manifolds from the Oseledets subspaces, and Ruelle's inequality bounds the entropy by the sum of the positive exponents.\nFor $d = 1$ the theorem reduces to [[Birkhoff's ergodic theorem|J6NMBX3E55]] applied to $\\log \\abs{A}$.\n",
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