C*-algebras, foliations and dynamical systems

Question

I am a Ph.D student involved in topics like integrability of foliations arising from center stable bundles of partially hyperbolic dynamical systems. These are generally only continuous bundles, so one either has to use the dynamics of the map, find some meaningful approximations, or try to generalize the theorems of smooth integrability and foliation theory to continuous case (it is more doable in the case of foliation theory).

In studying these topics, I have come to see that there are a lot of interactions with geometry of foliations, which I became interested with. Where smoothness fails and you can not use Frobenius theorem, you can make some intersting connections to foliation theory (for instance Novikov theorem) to see whether a system is integrable or not.

Following this. I have been hearing the "new applications of C-algebras to dynamical systems" by finding "C-algebra of a dynamical system", and I have also seen in Connes book that C*-algebras give a new way of studying foliation spaces. So I became interested in these, although I have only recently met them and did not have much chance to read about them.

My first question would be, what direction (books, questions ideas) would you propose to someone who wants to learn this/these field(s)? My second question is, what is vaguely the use of C-algebras about foliations and dynamical systems? What kind of their properties is studied through their use? Finally, do C-algebras study the case of continuous foliations?

Answer 1

A very good place to start is Connes' book "Noncommutative Geometry", available for free on his website. It's a huge book, but it's possible to skip around quite a bit to get what you need.

To begin, I'll remark that the foliations which are accessible to C*-algebraic techniques are generally smooth in the horizontal direction and integrable. By now it may be possible to relax these assumptions; I'm not really an expert. But I think the techniques are mainly useful for handling cases where the transversal behavior is very bad, e.g. the irrational rotation folation on the torus.

To associate a C*-algebra to a foliated manifold, one can use the foliation groupoid construction. The objects in this groupoid are just the points of the manifold, and there is a morphism between two points if and only if they lie on the same leaf. If the leaves are smooth then one can define a convolution product on the space of smooth compactly supported functions on the foliation groupoid, and this can be completed to form a (possibly noncommutative) C*-algebra. If the foliation comes from a group action (e.g. the irrational rotation action on the torus) then this generalizes the "crossed product" construction in the theory of C* dynamical systems.

With the C*-algebra of a foliated manifold in hand, the idea is to relate invariants of the C*-algebra (e.g. K-theory, cyclic homology) to the geometry of the foliation. Many of the best results that I know about are organized around index theory leafwise elliptic operators. For instance, Connes used these ideas together with the Lichnerowicz vanishing theorem to produce nontrivial topological obstructions to the existence of leafwise positive scalar curvature metrics. That said, I'm not sure how much contact there is with problems that are of interest to experts in dynamics and foliation theory.