A partially hyperbolic diffeomorphism f of a closed manifold M splits the tangent bundle TM in three subbundles E^s, E^c and E^u, which have, respectively, a contracting, dominated and expanding behavior by Df. The unstable bundle E^u is known to be tangent to an invariant foliation W^u, and understanding the structure of this foliation is useful for obtaining dynamical consequences for f. For instance, minimality of W^u implies that f is topologically mixing, and a bound in the number of minimal sets of W^u gives a bound in the number of attracting regions for f. In this talk I will discuss ongoing work about the number of minimal sets of the unstable foliation W^u for certain collapsed Anosov flows.

This is a joint work with Sylvain Crovisier and Rafael Potrie.