  We study the synchronization behavior of discrete-time Markov chains on
countable state spaces. Representing a Markov chain in terms of a random
dynamical system, which describes the collective dynamics of trajectories
driven by the same noise, allows for the characterization of synchronization
via random attractors.
  We establish the existence and uniqueness of a random attractor under mild
conditions and show that forward and pullback attraction are equivalent in our
setting. Additionally, we provide necessary and sufficient conditions for
reaching the random attractor, or synchronization respectively, in a time of
finite mean.
  By introducing insulated and synchronizing sets, we structure the state space
with respect to the synchronization behavior and characterize the shape of the
random attractor.
