# Research projects

## Project CEMAPRE internal

 Title Generic properties of $C^2$ Hamiltonians II Participants João Lopes Dias (Principal Investigator), Filipe Santos Summary A major result in the generic theory of smooth dynamical systems was recently obtained by Bochi, following some ideas by Mane. It says that a $C^1$-generic symplectomorphism on a compact manifold is either partially hyperbolic or else it has zero Lyapunov exponents almost everywhere. This reduces (in the $C^1$-topology) the multitude of dynamics of a generic system. In particular, non-uniformly hyperbolic systems (with positive Lyapunov exponents but not partially hyperbolic) are not generic. We deal with the version of this result for Hamiltonian flows.