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  "body": "\n## Definition\n\nLet $M$ be a compact Riemannian [[manifold|F45VQ47GPV]] and let $f \\colon M \\to M$ be a $C^{1+\\alpha}$ [[diffeomorphism|0KZZ70ZHXP]] for some $\\alpha \u003e 0$.\nLet $\\mu$ be an $f$-[[invariant|B232EZ1FMB]] Borel probability measure on $M$.\nLet $x \\in M$ and let $v \\in T_x M$ be nonzero.\nThe [[Lyapunov exponent|NJ1FJSNK6A]] of $v$ is\n$$\\lambda(x, v) = \\lim_{n \\to \\infty} \\frac{1}{n} \\log \\norm{D f^n_x \\, v}$$\nwhen the limit exists.\nBy the [[Oseledets theorem|NJ1PDJ3S4R]], for $\\mu$-almost every $x$ the limit exists for every $v$ and takes finitely many values.\nThe measure $\\mu$ is *hyperbolic* if, for $\\mu$-almost every $x$, the number $0$ is not among the values $\\lambda(x, v)$.\n*Pesin theory* is the theory of the pairs $(f, \\mu)$ of this kind with $\\mu$ hyperbolic, also called *nonuniformly hyperbolic* systems.\n\n## Remarks\n\nExample: let $f$ be the linear map of the torus $\\R^2 / \\Z^2$ given by the matrix $\\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$, and let $\\mu$ be Lebesgue measure.\nThe exponents are $\\log \\frac{3 + \\sqrt 5}{2} \u003e 0$ and $\\log \\frac{3 - \\sqrt 5}{2} \u003c 0$ at every point, so $\\mu$ is hyperbolic.\n\nNon-example: an irrational rotation of the torus $\\R^2/\\Z^2$ with Lebesgue measure is an isometry, so both exponents are $0$ everywhere and the measure is not hyperbolic.\n\nThe main results of the theory, for hyperbolic $\\mu$ and $C^{1+\\alpha}$ maps, include the following.\nLocal stable and unstable manifolds exist at $\\mu$-almost every point.\nIf $\\mu$ is absolutely continuous with respect to volume, the entropy of $\\mu$ equals the integral of the sum of the positive exponents, counted with multiplicity (the Pesin entropy formula).\nThe hypothesis $\\alpha \u003e 0$ is needed for the stable manifold theorem: the theory does not extend in general to $C^1$ maps.\n",
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