When is an Anosov diffeomorphism mixing?

Question

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?

Answer 1

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?")

Answer 2

Some author assume in their definition of Anosov diffeomorphisms that the diffeo is transitive. In that case, connectedness of M implies mixing.

Also it is known that any Anosov diffeomorphism with no wandering vector is transitive. It is a conjecture that an Anosov diffeomorphism has a full non-wandering set.

For Axiom A diffeomorphisms, the situation is a bit different. These diffeos admit a spectral decomposition. The non-wandering set can be written as a finite union of compact basic pieces, and on each piece, some power of the diffeo is mixing. This is an analog of a classical result on Markov chains. Let me mention a criterium to prove that an Axiom A diffeo is mixing.

Theorem Let f be an Axiom A diffeomorphism. The following are equivalent

• the diffeo is mixing in restriction to the non-wandering set

• the diffeo has a stable leaf dense in the non-wandering set

• the diffeo is transitive and the set of periods of periodic orbits generate Z.

Answer 3

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).