Joint Quasimodes, Positive Entropy, and Quantum Unique Ergodicity

TL;DR

This paper investigates joint quasimodes of the Laplacian and Hecke operators on compact surfaces, establishing conditions that ensure positive entropy and quantum unique ergodicity, with optimal results related to the quasimodes' dimensionality.

Contribution

It introduces new conditions on joint quasimodes that guarantee positive entropy and quantum unique ergodicity, extending measure classification results to these cases.

Findings

Conditions for positive entropy on ergodic components.

Quantum unique ergodicity established for certain joint quasimodes.

Optimality of results with respect to quasimode dimension.

Abstract

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 $\mathbb{H}\times\mathbb{H}$.