Accumulation points of the Birkhoff average of $m$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T21:31:52Z http://mathoverflow.net/feeds/question/74121 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74121/accumulation-points-of-the-birkhoff-average-of-m Accumulation points of the Birkhoff average of $m$ Pengfei 2011-08-31T01:57:13Z 2011-08-31T15:12:28Z <p>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</p> <p>----------$f^km(A):=m(f^{-k}A)$ for all measurable subset $A\subset M$.</p> <p>Then the Birkhoff averages $\nu_k=\frac{1}{k}\sum_{j=0}^{k-1}f^jm$ are probability measures on $M$ for all $k\ge1$. <em>The question is</em>: </p> <ul> <li>What can we say about the measure(s) in the set $\mathcal{V}(m)$ of accumulation points of $\{\nu_k:k\ge1\}$?</li> </ul> <p>We know that there exists a unique SRB measure $\mu_+$ for $f$ (and a unique SRB measure $\mu_-$ for $f^{-1}$). Do we have $\mathcal{V}(m)\subset\{\mu_+,\mu_-\}$?</p> http://mathoverflow.net/questions/74121/accumulation-points-of-the-birkhoff-average-of-m/74169#74169 Answer by Vaughn Climenhaga for Accumulation points of the Birkhoff average of $m$ Vaughn Climenhaga 2011-08-31T15:12:28Z 2011-08-31T15:12:28Z <p>Yes. In fact, we have <code>$\mathcal{V}(m) = \{\mu_+\}$</code> whenever $m$ is a probability measure on $M$ that is absolutely continuous with respect to volume. This is shown in (0.4) of "<a href="http://www.jstor.org/pss/2373810" rel="nofollow">A measure associated with Axiom-A attractors</a>" by David Ruelle, <em>American Journal of Mathematics</em> <strong>98</strong> (1976), 619--654.</p>