  We introduce synchronizing objectives for Markov decision processes (MDP).
Intuitively, a synchronizing objective requires that eventually, at every step
there is a state which concentrates almost all the probability mass. In
particular, it implies that the probabilistic system behaves in the long run
like a deterministic system: eventually, the current state of the MDP can be
identified with almost certainty.
  We study the problem of deciding the existence of a strategy to enforce a
synchronizing objective in MDPs. We show that the problem is decidable for
general strategies, as well as for blind strategies where the player cannot
observe the current state of the MDP. We also show that pure strategies are
sufficient, but memory may be necessary.
