Santiago Martinchich

Unstable minimal sets for certain collapsed Anosov flows

A diffeomorphism of a closed manifold M is called partially hyperbolic if the tangent bundle TM splits into 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 f-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 give a brief panorama of the study of partially hyperbolic diffeomorphisms in dimension 3 and 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.