Logarithmic-scale Quasimodes that do not Equidistribute

Abstract

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 Bi\`evre 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.