Accumulation points of the Birkhoff average of $m$

2011-08-31T01:57:13Z

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

Answer by Vaughn Climenhaga for Accumulation points of the Birkhoff average of $m$

2011-08-31T15:12:28Z

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" by David Ruelle, American Journal of Mathematics 98 (1976), 619--654.

Accumulation points of the Birkhoff average of $m$ - MathOverflow

Accumulation points of the Birkhoff average of $m$

Answer by Vaughn Climenhaga for Accumulation points of the Birkhoff average of $m$