Discrete and continuous spectra on laminations over Aubry-Mather sets

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August 2008, 21(3): 823-834. doi: 10.3934/dcds.2008.21.823

Discrete and continuous spectra on laminations over Aubry-Mather sets

1.	LAGA, CNRS UMR 7539, Université Paris 13, Villetaneuse 93430

Received July 2007 Revised December 2007 Published April 2008

Abstract
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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 Poincaré 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.

Keywords: Aubry-Mather sets, weak mixing, time change, perturbations of completely integrable systems..

Mathematics Subject Classification: Primary: 37A20, 37J40; Secondary: 37E9.

Citation: Bassam Fayad. Discrete and continuous spectra on laminations over Aubry-Mather sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 823-834. doi: 10.3934/dcds.2008.21.823

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