Joint Quasimodes, Positive Entropy, and Quantum Unique Ergodicity

Details Joint Quasimodes, Positive Entropy, and Quantum Unique Ergodicity Joint Quasimodes, Positive Entropy, and Quantum Unique Ergodicity

2011-12-22

arxiv.org/abs/1112.5311v1

Mathematics

Dynamical Systems

38 pages

Scientific paper

We study joint quasimodes of the Laplacian and one Hecke operator on compact congruence surfaces, and give conditions on the orders of the quasimodes that guarantee positive entropy on almost every ergodic component of the corresponding semiclassical measures. Together with the measure classification result of the second-named author, this implies Quantum Unique Ergodicity for such functions. Our result is optimal with respect to the dimension of the space from which the quasi-mode is constructed. We also study equidistribution for sequences of joint quasimodes of the two partial Laplacians on compact irreducible quotients of $\mathbb{H}\times\mathbb{H}$.

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