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

• DOI: 10.48550/arxiv.2410.13449
• Publication Date: 2024-10-17
• Authors: Elena Kim; Theresa C. Anderson; Robert J. Lemke O...
• Keywords: Semiclassical physics

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