{
  "id": "2410.13449",
  "version": "v1",
  "published": "2024-10-17T11:22:13.000Z",
  "updated": "2024-10-17T11:22:13.000Z",
  "title": "Characterizing the support of semiclassical measures for higher-dimensional cat maps",
  "authors": [
    "Elena Kim",
    "Theresa C. Anderson",
    "Robert J. Lemke Oliver"
  ],
  "comment": "64 pages, 2 figures; with an appendix by Theresa C. Anderson and Robert J. Lemke Oliver",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-17T11:22:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semiclassical measure",
      "higher-dimensional cat maps",
      "quantum cat map",
      "higher-dimensional fractal uncertainty principle",
      "characteristic polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 64,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}