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

Elena Kim; Theresa C. Anderson; Robert J. Lemke Oliver

Paper

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

Abstract

Quantum cat maps are toy models in quantum chaos associated to hyperbolic symplectic matrices

. The macroscopic limits of sequences of eigenfunctions of a quantum cat map are characterized by semiclassical measures on the torus

. We show that if the characteristic polynomial of every power

is irreducible over the rationals, then every semiclassical measure has full support. The proof uses an earlier strategy of Dyatlov-Jézéquel [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

of matrices

, the Galois group of the characteristic polynomial of

. 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

, inspired by the work of Faure-Nonnenmacher-De Bièvre [arXiv:nlin/0207060] in the case

. This is complementary to the examples by Kelmer [arXiv:math-ph/0510079] of semiclassical measures supported on a single coisotropic subtorus.