On Quantum Ergodicity for Linear Maps of the Torus

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 √ N .

Weyl's law and quantum ergodicity for maps with divided phase space (with an appendix Converse quantum ergodicity)

“…Results in this direction have recently been obtained in the case of cat maps [6,19,20] and piecewise affine maps on the torus [12], which neatly complement the proofs of quantum unique ergodicity for cat maps [17,25,26], parabolic maps [31] and the modular surface [42,29] (see the survey [39]). …”

Distributional properties of powers of matrices

“…The case n = 2 seems to be a distinguished one. It was shown in [15] and [14] (see also [13]) to be related with quantum ergodicity on flat tori (an instance of arithmetic quantum chaos). In [23] it is also studied in connection with the order of the reduction of units in quadratic fields.…”

On Quantum Unique Ergodicity for Linear Maps of the Torus

“…In [9], we show equidistribution of all eigenfunctions of U N (A) for almost all integers N : Previously, the only result giving an infinite set of N for which all eigenfunctions of U N (A) become equidistributed is by Degli-Esposti, Graffi and Isola [4], which conditional on the Generalized Riemann Hypothesis give an infinite set of primes.…”