Random attractors on countable state spaces

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 a sufficient condition 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 size of the random attractor.