On quantum ergodicity for higher dimensional cat maps

TL;DR

This paper investigates eigenfunction distribution for higher-dimensional quantum cat maps, extending known results from two dimensions and employing advanced tools like additive combinatorics and tensor product analysis.

Contribution

It proves quantum ergodicity for higher-dimensional cat maps, introducing new methods and addressing complexities absent in the two-dimensional case.

Findings

Eigenfunctions are uniformly distributed along a density one sequence of integers

Higher-dimensional cases require novel tools like Bourgain's bounds and tensor product analysis

Results extend quantum ergodicity to more complex symplectic 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 ${\mathrm{Sp}}(2g,\mathbb Z)$, 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 P. Kurlberg and Z. Rudnick (2001). The higher dimensional case offers several new features and requires a completely different set of tools, including from additive combinatorics, in particular Bourgain's bound (2005) for Mordell sums, and a study of tensor product structures for the cat map.