Let $M$ be a closed manifold, $m$ be the normalized volume measure on $M$, and $f:M\to M$ be a $C^2$ transitive Anosov diffeomorphism. Consider the pushforward $f^km$ defined by

----------$f^km(A):=m(f^{-k}A)$ for all measurable subset $A\subset M$.

Then the Birkhoff averages $\nu_k=\frac{1}{k}\sum_{j=0}^{k-1}f^jm$ are probability measures on $M$ for all $k\ge1$. *The question is*:

- What can we say about the measure(s) in the set $\mathcal{V}(m)$ of accumulation points of $\{\nu_k:k\ge1\}$?

We know that there exists a unique SRB measure $\mu_+$ for $f$ (and a unique SRB measure $\mu_-$ for $f^{-1}$). Do we have $\mathcal{V}(m)\subset\{\mu_+,\mu_-\}$?