Abstract: Shub gave the first examples of robustly transitive diffeomorphisms on the 4-torus which are not uniformly hyperbolic (they are though partially hyperbolic). We will show that these examples have further interesting properties: under some bunching conditions, there exists an open dense set of diffeomorphisms in this class exhibiting a unique homoclinic class of index 2 supported on the whole torus. This has important consequences in the study of equilibrium states. Joint work with C. Liang, F. Yang and J. Yang.
