C*-algebras, foliations and dynamical systems

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?

• You could have a look to the book by Alberto Candel and Lawrence Conlon Foliations I and II. Commented May 29, 2013 at 22:52
• Volume II treats also $C^*$-algebras of foliated spaces. Here's a preview: books.google.it/… Commented May 29, 2013 at 22:55
• Thanks alot on top of this I found some papers on this topic. Commented May 30, 2013 at 8:27
• I would also very much welcome answers in the dynamical systems part and from anyone who has worked in this area. Commented May 30, 2013 at 8:32
• C* algebra of a dynamical system, C* dynamical system, is it the same thing?
– user34645
Commented Jun 4, 2013 at 9:39