  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 phase-space
average with respect to Liouville measure of the corresponding classical
observable. The possible existence of any exceptional subsequences of
eigenstates is an important issue, which until now was unresolved in any
example. The absence of exceptional subsequences is referred to as quantum
unique ergodicity (QUE). We present the first examples of maps which satisfy
QUE: Irrational skew 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 show are optimal. We construct special
examples of these maps for which the rate of convergence is arbitrarily slow.
