Let $M$ be a compact Riemannian manifold and let $T : M \rightarrow M$ be Anosov. I have read [here](http://books.google.com/books?id=hFN6oiecbrYC&pg=PA7) that it is an open problem to prove that $T$ is topologically mixing if $M$ is connected. [Katok and Hasselblatt](http://books.google.com/books?id=9nL7ZX8Djp4C&dq&pg=PA151) point out that if $(T, \mu)$ is mixing, then the restriction of $T$ to the support of $\mu$ is topologically mixing. Yet I have also read [here](http://www.jstor.org/pss/2373810) (among other places) that Axiom A diffeomorphisms (and hence Anosov diffeomorphisms) satisfy exponential decay of correlations, which seems like it implies mixing (take characteristic functions). Still, I see references in the physics literature (which is where I am coming from) to "mixing Anosov maps". 


While the Anosov alternative for *flows* is clear enough, I haven't found a characterization of mixing properties for Anosov diffeomorphisms that sorts out these apparently contradictory statements.


With that in mind, my question is (sort of): under what circumstances is an Anosov diffeomorphism guaranteed to be mixing?



When is an Anosov diffeomorphism mixing?
I don't have a proper answer to your main question beyond pointing to the list of equivalent properties in Pesin's book (your first reference), which you've obviously seen already. However, I'll point out that in Ruelle's paper (your third reference), the first main theorem (on page 3), which contains a statement on exponential decay of correlations, is proved under the hypothesis that the unstable manifolds are dense in the attractor. As in Pesin's book, that will imply mixing (on the attractor), so there's no mystery in the coexistence of the result stated by Ruelle and the open problem stated by Pesin.


As far as I know the best that you can say in general is that Anosov diffeos are mixing on the non-wandering set, or on the closure of an unstable manifold. So an equivalent question is, "Under what circumstances is every point non-wandering for an Anosov diffeo?" (Or, "Under what circumstances are unstable manifolds dense?")


