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Discrete and continuous spectra on laminations over AubryMather sets
1.  LAGA, CNRS UMR 7539, Université Paris 13, Villetaneuse 93430 
The answer will essentially depend on the number of orbits of gaps in the AubryMather set. More precisely, if the AubryMather 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 AubryMather 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.
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