Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus

Abstract

We look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribu- tion in the semiclassical limit. We construct super-scars that are stable under the arithmetic symmetries of the system and localize on invariant manifolds. We show that these super-scars exist only when there are isotropic rational subspaces, invariant under the linear map. In the case where there are no such scars, we com- pute the variance of the fluctuations of the matrix elements for the desymmetrized system, and present a conjecture for their limiting distributions.