Noncommutative duality in dynamical systems (07rit162)

Objectives

The main goal is to further develop our understanding of the role of
hyperbolicity in all of these examples. More specifically we intend to
the the following:

1. Completion of the paper proving duality for the Ruelle algebras
associated to a Smale space.

About ten years ago, the KP published a paper proving
duality for shifts of finite type (Comm. Math. Phys. 187 (1997),
509-522). This proved the existence of the required K-theory
isomorphisms between the Cuntz-Krieger algebras O_A and O_A^T, where A
is a 0-1 matrix defining the shift. Smale spaces are a broad
generalization of shifts of finite type which include the basic sets
for Smale's Axiom A systems. Generally, they are factors of shifts of
finite type. We also had an outline for a proof in the
more general case using very different methods. This was never
published because the technical details were substantial. In the past
few years better approaches have been developed and the proof can be
simplified considerably. The first aim of the proposal is to complete
this program. Much of it is already written, but there remain a few
hurdles.

2. Extending the result to more general hyperbolic systems.

It is likely the result of Part 1 above could be done in other settings
where some form of hyperbolicity is present. The most natural case
to consider is the geodesic flow on a compact manifold of
negative curvature.

3. Relations with boundary actions of discrete hyperbolic groups and
self-similar groups.

As mentioned above, other cases where duality
exists are closely linked with groups which are hyperbolic in some
sense and their actions. More specifically, recent work of
Nekrashevych and his co-workers (including Grigorchuk) have
established links between self-similar groups, Smale spaces and
C*-algebras. The third aim of the project is to try to understand
these connections better and especially the aspects of duality.

4. Transverse groupoids

Finally, there seems to be a general context in which to study this
type of duality. KP have started developing a notion of
transverse groupoids, which would be basic geometric input to
duality. Some known results fit into this
setting--e.g. the Baum-Connes map relating the K-homology of the
classifying space of a group and the K-theory of its group C*-algebra,
the Fourier-Mukai transform relating derived categories associated to
an abelian variety and its dual, and others. The goal of this part of
the work would be to establish the definition of transverse groupoids
and show how it leads to setting up a K-theoretic duality map, which
will usually require additional conditions (such as hyperbolicity) in
order to be an isomorphism.