On December 3rd, 2009, M.A. Rodriguez-Hertz presented at the seminar of “Topologie et dynamique” at Orsay a new result extending to three dimensional manifolds the well-known theorem announced by Mañé and proved by Bochi in 2002:

**Theorem (M.A. Rodriguez-Hertz).*** Consider the space of all C^1 diffeomorphism preserving the volume on a three-dimensional compact manifold. Then generically, one of the following occurs:*

*all Lyapunov exponents vanish Lebesgue-almost everywhere;**there is a dominated splitting and Lebesgue-almost everywhere it separates positive and negative exponents (and there is no zero Lyapunov exponents) and it is ergodic with respect to volume.*