Partially hyperbolic dynamics
Marcelo Viana (IMPA)

The concept of a hyperbolic dynamical system was introduced by S. Smale, D. Anosov and Ya. Sinai in the 1960’s. One goal was to characterize (structurally) stable systems, that is, whose qualitative behavior is not affected by small perturbations of the system. That was eventually achieved in a tour de force by R. Mañé and others. Another goal was to provide a paradigm for the behavior of "most"dynamical systems. This turned out to be too optimistic and, in response, various generalizations of hyperbolicity have been proposed. Among them, the notion of partial hyperbolicity proved to be particularly fruitful, and has been at the heart of the research in the field for the last 30 years or so. I will briefly discuss some of the surprising features exhibited by partially hyperbolic systems, especially concerning rigidity, Lyapunov exponents and pathological foliations.