2024/10/17 by Elena Kim, Kim, Elena, Theresa C. Anderson +3 · 2 citations
Computer Science · Mathematics · #Topological and Geometric Data Analysis #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2410.13449
Quantum cat maps are toy models in quantum chaos associated to hyperbolic symplectic matrices A∈ Sp(2n,ℤ). The macroscopic limits of sequences of eigenfunctions of a quantum cat map are characterized by semiclassical measures on the torus ℝ2n/ℤ2n. We show that if the characteristic polynomial of every power Ak 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 100% of matrices A, the Galois group of the characteristic polynomial of A is S2 \wr Sn. 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 Bièvre [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.