On the Rate of Quantum Ergodicity for Quantised Maps

We study the distribution of expectation values and transition amplitudes for quantised maps on the torus. If the classical map is ergodic then the variance of the distribution of expectation values will tend to zero in the semiclassical limit by the quantum ergodicity theorem. Similarly the variance of transition amplitude goes to zero if the map is weak mixing. In this paper we derive estimates on the rate by which these variances tend to zero. For a class of hyperbolic maps we derive a rate which is logarithmic in the semiclassical parameter, and then show that this bound is sharp for cat maps. For a parabolic map we get an algebraic rate which again is sharp.