TITLE: When is an Anosov diffeomorphism mixing?
QUESTION [6 upvotes]: Let $M$ be a compact Riemannian manifold and let $T : M \rightarrow M$ be Anosov. I have read here that it is an open problem to prove that $T$ is topologically mixing if $M$ is connected. Katok and Hasselblatt 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 (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?

REPLY [2 votes]: I believe it's true that an Anosov diffeomorphism on a nilmanifold or infranilmanifold has a full non-wandering set. In combination with Vaughn and coudy's answers, the question of whether every Anosov diffeomorphism of a connected compact Riemannian manifold is mixing is therefore linked to the conjecture that every Anosov diffeomorphism takes place on an (infra-)nilmanifold. If you have a specific example in mind, then either it's defined on an infranilmanifold and is mixing, or it's not defined on an infranilmanifold, and hence its very existence is a shoo-in for Annals of Mathematics... 
I don't know a good general survey on Anosov diffeomorphisms, but there are quite a few references to papers on this topic in Gorodnik's survey paper "Open problems in dynamics and related fields" (the conjecture mentioned above is Conjecture 3 in Gorodnik's paper).