Ergodicity, mixing and KAM

To prove the ergodicity for Hamiltonian systems of big or infinite dimension is a notoriously complicated problem of high importance. But what we often have in physics are not Hamiltonian systems, but systems of the form <Hamiltonian system> + <small dissipation> + <small random forcing>, where the random forcing may be very degenerate (i.e. it affects only a few modes). For such systems an analogy of the ergodicity is called the mixing. In my talk I will remind the definition of the mixing and explain how the KAM-theory provides a powerful tool to prove the mixing for the systems above. The talk is based on a joint work with Armen Shirikyan and Vahagn Nersesyan [1]