  We investigate convergence properties of discrete-time semigroup quantum
dynamics, including asymptotic stability, probability and speed of convergence
to pure states and subspaces. These properties are of interest in both the
analysis of uncontrolled evolutions and the engineering of controlled dynamics
for quantum information processing. Our results include two Hilbert space
decompositions that allow for deciding the stability of the subspace of
interest and for estimating of the speed of convergence, as well as a formula
to obtain the limit probability distribution for a set of orthogonal invariant
subspaces.
