Continuous spectrum on laminations over Aubry-Mather sets

Abstract

If we perturb a completely integrable Hamiltonian system with two degrees of freedom, the perturbed flow might display, on every energy level, invariant sets that are laminations over Aubry-Mather sets of a Poincar\'e section of the flow. Each one of these laminations carry a unique invariant probability measure for the flow on which mixing is impossible in this low dimensional frame. We prove that if the Aubry-Mather set has exactly one orbit of gaps and is hyperbolic then the special flow over it with any smooth ceiling function will be conjugate to a suspension with a constant ceiling function, failing hence to be weak mixing or even topologically weak mixing. To the contrary, if the Aubry-Mather set has more than one orbit of gaps with at least two in a general position then the special flow over it will in general be weak mixing.