Equidistribution of Eigenfunctions of Quantum Cat Maps

TL;DR

This paper proves that certain quantum eigenfunctions of cat maps become uniformly distributed on the torus, despite showing strong localization in some coordinates, contrasting with previous scarring phenomena.

Contribution

It establishes equidistribution of short-period eigenfunctions of quantum cat maps and reveals coexistence of localization and equidistribution.

Findings

Eigenfunctions equidistribute on the torus in the semiclassical limit

Logarithmically large norms concentrate on few coordinates

Results confirm numerical evidence and contrast with scarring phenomena

Abstract

We prove that the short-period eigenfunctions of quantum cat maps constructed by Kim and the author equidistribute on $\mathbb{T}^2$ in the sense of semiclassical measures. We also show that their logarithmically large $\ell^\infty$-norm is asymptotically concentrated on a bounded number of coordinates. Thus, for this explicit family, strong coordinate localization coexists with semiclassical equidistribution. These results confirm the behavior suggested by earlier numerical evidence of Kim and the author, and contrast with the scarring phenomena for short-period eigenfunctions observed by Faure, Nonnenmacher, and De Bi\`evre.