Discrete and continuous spectra on laminations over Aubry-Mather sets

Abstract

We study a class of completely integrable Hamiltonian system with two degrees of freedom for which the perturbed flow displays, on some energy level, invariant sets that are laminations over Aubry-Mather sets of a Poincare section of the flow. Each one of these laminations carries a unique invariant probability measure for the flow and it is interesting therefore to understand the statistical properties of this measure. From a result of Kocergin in [13], we know that mixing is a priori impossible. In this paper, we investigate on the possible occurrence of weak mixing. The answer will essentially depend on the number of orbits of gaps in the Aubry-Mather set. More precisely, 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.