Characterizing the support of semiclassical measures for higher-dimensional cat maps

TL;DR

This paper investigates the support of semiclassical measures for higher-dimensional quantum cat maps, establishing conditions under which these measures have full support and providing examples of measures supported on specific subtori.

Contribution

It proves that irreducibility of characteristic polynomials implies full support of semiclassical measures in higher dimensions, extending previous results and providing new examples.

Findings

Full support for semiclassical measures under irreducibility conditions.

Generic irreducibility of characteristic polynomials for most matrices.

Existence of semiclassical measures supported on unions of symplectic subtori.

Abstract

Quantum cat maps are toy models in quantum chaos associated to hyperbolic symplectic matrices $A\in \operatorname{Sp}(2n,\mathbb{Z})$. The macroscopic limits of sequences of eigenfunctions of a quantum cat map are characterized by semiclassical measures on the torus $\mathbb{R}^{2n}/\mathbb{Z}^{2n}$. We show that if the characteristic polynomial of every power $A^k$ is irreducible over the rationals, then every semiclassical measure has full support. The proof uses an earlier strategy of Dyatlov-J\'ez\'equel [arXiv:2108.10463] and the higher-dimensional fractal uncertainty principle of Cohen [arXiv:2305.05022]. Our irreducibility condition is generically true, in fact we show that asymptotically for $100\%$ of matrices $A$, the Galois group of the characteristic polynomial of $A$ is $S_2 \wr S_n$. When the irreducibility condition does not hold, we show that a semiclassical measure cannot be supported on a finite union of parallel non-coisotropic subtori. On the other hand, we give examples of semiclassical measures supported on the union of two transversal symplectic subtori for $n=2$, inspired by the work of Faure-Nonnenmacher-De Bi\`evre [arXiv:nlin/0207060] in the case $n=1$. This is complementary to the examples by Kelmer [arXiv:math-ph/0510079] of semiclassical measures supported on a single coisotropic subtorus.