Liouville ergodicity of linear multi-particle hamiltonian system with one marked particle velocity flips

We consider multi-particle systems with linear deterministic hamiltonian dynamics. Besides Liouville measure it has continuum of invariant tori and thus continuum of invariant measures. But if one specified particle is subjected to a simple linear deterministic transformation (velocity flip) in random time moments, we prove convergence to Liouville measure for any initial state. For the proof it appeared necessary to study non-linear transformations on the energy surface.