Pär Kurlberg: Quantum ergodicity for higher dimensional cat maps

Abstract: We study eigenfunction localization for higher dimensional cat maps (linear symplectic maps in \(\mathrm{Sp}(2g, \mathbb{Z})\)), a popular model of quantum chaos. Under natural assumptions, we show that there is a density one sequence of integers \(N\) so that as \(N\) tends to infinity along this sequence, all eigenfunctions of the quantized map at the inverse Planck constant \(N\) are uniformly distributed. Compared to the the two dimensional case, higher dimensions requires a new techniques, including additive combinatorics and properties of tensor product structures for the cat map. “One good prime is all you need.”