Semiclassical measures for higher dimensional quantum cat maps

Semyon Dyatlov; Malo Jézéquel

Paper

Semiclassical measures for higher dimensional quantum cat maps

Abstract

Consider a quantum cat map

associated to a matrix

, which is a common toy model in quantum chaos. We show that the mass of eigenfunctions of

on any nonempty open set in the position-frequency space satisfies a lower bound which is uniform in the semiclassical limit, under two assumptions: (1) there is a unique simple eigenvalue of

of largest absolute value and (2) the characteristic polynomial of

is irreducible over the rationals. This is similar to previous work [arXiv:1705.05019], [arXiv:1906.08923] on negatively curved surfaces and [arXiv:2103.06633] on quantum cat maps with

, but this paper gives the first results of this type which apply in any dimension. When condition (2) fails we provide a weaker version of the result and discuss relations to existing counterexamples. We also obtain corresponding statements regarding semiclassical measures and damped quantum cat maps.