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We consider multi-particle systems with linear deterministic hamiltoniandynamics. Besides Liouville measure it has continuum of invariant tori and thuscontinuum of invariant measures. But if one specified particle is subjected toa simple linear deterministic transformation (velocity flip) in random timemoments, we prove convergence to Liouville measure for any initial state. Forthe proof it appeared necessary to study non-linear transformations on theenergy surface.

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