On quantum ergodicity for linear maps of the torus

1999/10/27 by P. Kurlberg, Kurlberg, P., Z. Rudnick +1 · 1 citation
Mathematics · Physics and Astronomy · #11 #Chaotic Dynamics (nlin.CD) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT) #chao-dyn #math-ph #math.MP #math.NT #msc:11 #nlin.CD

paper · pdf · doi:10.48550/arxiv.math/9910145

32 pages

arxiv created 1999/10/27 · arxiv updated 2009/11/30

Abstract

We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (``cat maps''). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed. A key step in the argument is to show that for a hyperbolic matrix in the modular group, there is a density one sequence of integers N for which its order (or period) modulo N is somewhat larger than the square root of N.

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