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

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-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 $S_2 \wr S_n$. 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 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.

💡 Research Summary

This paper presents a comprehensive analysis of the support properties of semiclassical measures associated with higher-dimensional quantum cat maps. Quantum cat maps are canonical toy models in quantum chaos, corresponding to the quantization of hyperbolic linear symplectomorphisms (given by a matrix A in Sp(2n, ℤ)) on the 2n-dimensional torus. A semiclassical measure describes the limiting spatial distribution of a sequence of eigenfunctions of the quantized map in the high-frequency limit (Planck’s constant tending to zero).

The central achievement of the work is a precise characterization of which subsets of the torus can support such limiting measures. The main results are threefold. First, Theorem 1.1 establishes a sharp algebraic condition guaranteeing that all semiclassical measures must have full support on the entire torus. Specifically, if the characteristic polynomial of every power A^k (k ∈ ℕ) is irreducible over the rationals ℚ, then any semiclassical measure μ satisfies supp(μ) = T^(2n). The proof innovatively generalizes the strategy of Dyatlov-Jézéquel to higher dimensions without requiring a spectral gap condition, crucially employing Cohen’s recently developed higher-dimensional fractal uncertainty principle. The authors, with an appendix by Anderson and Lemke Oliver, further demonstrate that this irreducibility condition is generic: asymptotically, 100% of matrices A in Sp(2n, ℤ) satisfy it, as their characteristic polynomials have the maximal Galois group S_2 ≀ S_n.

Second, the paper investigates the situation where the irreducibility condition fails. Theorem 1.3 proves a key geometric obstruction: if A is hyperbolic and diagonalizable over ℂ, then a semiclassical measure cannot be supported on a finite union of parallel subtori whose common tangent space V is non-coisotropic (i.e., its symplectic orthogonal complement V^⊥ is not a subspace of V). This result is complementary to the well-known examples by Kelmer of measures supported on single coisotropic subtori. The proof leverages the hyperbolicity of A and the non-coisotropic nature of V to construct invariant symplectic subspaces with specific contraction/expansion properties, leading to a contradiction via a fundamental uncertainty principle.

Third, Theorem 1.4 constructs explicit examples showing that when the irreducibility condition is violated, more intricate support structures are possible. For n=2, the authors demonstrate the existence of a hyperbolic A and a sequence of eigenfunctions whose associated semiclassical measure is supported on the union of two transversal symplectic subtori. The constructed measure is μ = (1/2)