On the Lebesgue component of semiclassical measures for abelian quantum actions
Abstract
For a large class of symplectic integer matrices, the action on the torus extends to a symplectic \mathbb{Z}^{r} -action with r\geq 2 . We apply this to the study of semiclassical measures for joint eigenfunctions of the quantization of the symplectic matrices of the \mathbb{Z}^{r} -action. In the irreducible setting, we prove that the resulting probability measures are convex combinations of the Lebesgue measure with weight \geq 1/2 and a zero entropy measure. We also provide a general theorem in the reducible case showing that the Lebesgue components along isotropic and symplectic invariant subtori must have total weight \geq 1/2 .
Fields of Study & Subjects
Random Matrices and Applications
Spectral Theory in Mathematical Physics
Geometry and complex manifolds
Publication details
- Journal / source
- Journal of Spectral Theory
- Volume
- 16
- Issue
- 4
- Pages
- 1473–1513
- Publication date
- 10 June 2026
- Language
- EN