Let $f:M \to M$ be a $C^{1}$ diffeomorphism on a compact Riemannian manifold with a normalized Riemannian volume $\mathrm{Leb}$. Given an $f$-invariant Borel probability $\mu$ in $M$, we call the basin of attraction of $\mu$ the set $B(\mu)$ of the points $x \in M$ such that the averages of Dirac measures along the orbit of $x$ converge to $\mu$ in the weak* sense:
$$\lim _{n \rightarrow+\infty} \frac{1}{n} \sum_{j=0}^{n-1} \varphi\left(f^{j}(x)\right)=\int \varphi d \mu$$ for any continuous $\varphi: M \rightarrow \mathbb{R}$. Then we say that $\mu$ is a physical measure for $f$ if the basin of attraction $B(\mu)$ has positive Lebesgue measure in $M$.
A particular type of physical measures are the so-called Sinai–Ruelle–Bowen, or SRB,
measures which have the property of having nonzero Lyapunov exponents $\mu$-almost
everywhere and admitting a system of conditional measures such that the conditional measures on unstable manifolds are absolutely continuous with respect to the Lebesgue measures $\mathrm{Leb}$ on these manifolds induced by the restriction of the Riemannian structure.
An ergodic SRB measure is physical. The figure-eight attractor has
a physical measure with a positive Lyapunov exponent which is not an
SRB measure. I want to know whether there is an example of an SRB measure which is not physical.
Best Answer
You can just take an Anosov map on $T^2$ and multiply by identity on the circle. Then, you will have SRB measures supported on $T^2 \times pt$ which are not physical.