The colloquium will be centered around problems on weakly hyperbolic dynamical systems. One of the main problems is to understand how rigid differentiable phenomenoms and more flexible ergodic structures can co-exist simultaneaously. The problem of existence of (rare or abundant) positive lyapunov exponents is still unknown. The problem of the relationship between some form of weak hyperbolicity, the notions of stably ergodicity and non integrability of the stable foliation also deserves a lot of attention. Weak hyperbolicity is also related to the spectrum of the Ruelle dynamical zeta function (analyticity, decay of correlation), to the existence of the so called physical measures (SRB, Gibbs measures) and their thermodynamical properties (for instance when the temperature goes to zero).