  We prove the existence of uniform limits for certain sequences of products of
contractions and elements of a family of uniformly continuous propagators
acting on a Hilbert or a Banach space. From the point of view of Quantum
Physics, the considered sequences can represent the evolution of a system whose
dynamics, described by a continuous propagator, is disturbed by a sequence of
generic quantum operations (e.g., projective measurements or unitary pulses).
This includes and also generalizes the so-called quantum Zeno dynamics. The
time-evolution obtained from the limits of the considered sequences are
described by propagators generated by ergodic means. The notion of such ergodic
means is generalized in this paper to also include propagators of
time-dependent generators, and thus our results can be used to develop new
forms of active decoherence suppression. In a similar way, we also consider the
effective time-evolution obtained from adding to the possibly time-dependent
generator of the original propagator a generator of a uniformly continuous
contraction semi-group with asymptotically increasing weight in the sum. By
proving a generalized adiabatic theorem, it is shown that also for this set-up
the resulting time-evolution is governed by a suitable ergodic mean.
