On Quantum Ergodicity for Higher Dimensional Cat Maps

Abstract

We study eigenfunction localization for higher dimensional cat maps, a popular model of quantum chaos. These maps are given by linear symplectic maps in \(\operatorname {Sp}(2g,{{\mathbb {Z}}})\), which we take to be ergodic. Under some 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. For the two-dimensional case (\(g=1\)), this was proved by Kurlberg and Rudnick (Duke Math J 103:47–78, 2000). The higher dimensional case offers several new features and requires a completely different set of tools, including from additive combinatorics, such as a bound of Bourgain (J Am Math Soc 18:477–499, 2005) for Mordell sums, and a study of tensor product structures for the cat map, which has never been exploited in this context.