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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 $C^2$ [[diffeomorphism|0KZZ70ZHXP]].\nLet $\\mu$ be an $f$-[[invariant|B232EZ1FMB]] Borel probability measure on $M$.\nThe measure $\\mu$ is a *Sinai–Ruelle–Bowen measure* (an *SRB measure*) if both conditions hold:\n\n1. For $\\mu$-almost every $x$, at least one [[Lyapunov exponent|NJ1FJSNK6A]] of $f$ at $x$ is positive.\n2. The conditional measures of $\\mu$ on the local unstable manifolds given by [[Pesin theory|ZHKS81A7QN]] are [[absolutely continuous|P0P21KDV6E]] with respect to their Riemannian volume.\n\n## Remarks\n\nExample: let $f$ be the map of $\\R^2 / \\Z^2$ given by the matrix $\\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$ and $\\mu$ Lebesgue measure.\nThe exponent $\\log \\frac{3 + \\sqrt 5}{2}$ is positive at every point.\nThe unstable manifolds are the lines of the expanding eigendirection, and the conditional measures of $\\mu$ on them are multiples of length.\nSo $\\mu$ is an SRB measure.\n\nNon-example: for the same map, the Dirac measure at the fixed point $0$ is invariant and has a positive exponent.\nIts conditional measure on the unstable line through $0$ is a point mass, which is not absolutely continuous with respect to length, so clause 2 fails.\n\nSome authors use a different definition: $\\mu$ is SRB, or *physical*, if the set of points $x$ with $\\frac{1}{n} \\sum_{k=0}^{n-1} \\varphi(f^k x) \\to \\int \\varphi \\, d\\mu$ for every continuous $\\varphi$ has positive volume.\nFor an Axiom A attractor of a $C^2$ diffeomorphism, there is a unique SRB measure on each basic piece of the attractor, and it is physical in this sense.\nLedrappier and Young proved a characterization for $C^2$ diffeomorphisms.\nAn invariant measure $\\mu$ that satisfies clause 1 is SRB in the sense above if and only if its entropy equals the integral of the sum of its positive exponents.\n",
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