Quantum unique ergodicity for parabolic maps

We study the ergodic properties of quantized ergodic maps of the torus. It is known that these satisfy quantum ergodicity: For almost all eigenstates, the expectation values of quantum observables converge to the classical ph ase-space average with respect to Liouville measure of the corresponding cl assical observable. The possible existence of any exceptional subsequences of eigenstates is an important issue, which until now was unresolved in any example. The absenc e of exceptional subsequences is referred to as quantum unique ergodicity ( QUE). We present the first examples of maps which satisfy QUE: Irrational s kew translations of the two-torus, the parabolic analogues of Arnold's cat maps. These maps are classically uniquely ergodic and not mixing. A crucial step is to find a quantization recipe which respects the quantum-classical correspondence principle. In addition to proving QUE for these maps, we also give results on the rate of convergence to the phase-space average. We give upper bounds which we s how are optimal. We construct special examples of these maps for which the rate of convergence is arbitrarily slow.