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 $\mathcal{V}(m)\subset\{\mu\_+,\mu\_-\}$?



Accumulation points of the Birkhoff average of $m$
Yes. In fact, we have $\mathcal{V}(m) = \{\mu\_+\}$ whenever $m$ is a probability measure on $M$ that is absolutely continuous with respect to volume. This is shown in (0.4) of "[A measure associated with Axiom-A attractors](http://www.jstor.org/pss/2373810)" by David Ruelle, *American Journal of Mathematics* **98** (1976), 619--654.


