Sinai–Ruelle–Bowen measure

Remarks

Example: let $f$ be the map of $\R^2 / \Z^2$ given by the matrix $\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ and $\mu$ Lebesgue measure. The exponent $\log \frac{3 + \sqrt 5}{2}$ is positive at every point. The unstable manifolds are the lines of the expanding eigendirection, and the conditional measures of $\mu$ on them are multiples of length. So $\mu$ is an SRB measure.

Non-example: for the same map, the Dirac measure at the fixed point $0$ is invariant and has a positive exponent. Its 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.

Some 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. For 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. Ledrappier and Young proved a characterization for $C^2$ diffeomorphisms. An 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.