Logarithmic-scale quasimodes that do not equidistribute

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Given any compact hyperbolic surface M, and a closed geodesic on M, we construct of a sequence of quasimodes on M whose microlocal lifts concentrate positive mass on the geodesic. Thus, the quantum unique ergodicity (QUE) property does not hold for these quasimodes. This is analogous to a construction of Faure-Nonnenmacher-De Bievre in the context of quantized cat maps and lends credence to the suggestion that large multiplicities play a role in the known failure of QUE for certain "toy models" of quantum chaos. We moreover conjecture a precise threshold for the order of quasimodes needed for QUE to hold-the result of the present paper shows that this conjecture, if true, is sharp.

Original languageEnglish
Pages (from-to)11934-11960
Number of pages27
JournalInternational Mathematics Research Notices
Issue number22
StatePublished - 2015

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