  Nonlinear Markov chains with finite state space have been introduced in
Kolokoltsov (2010). The characteristic property of these processes is that the
transition probabilities do not only depend on the state, but also on the
distribution of the process. In this paper we provide first results regarding
their invariant distributions and long-term behaviour. We will show that under
a continuity assumption an invariant distribution exists. Moreover, we provide
a sufficient criterion for the uniqueness of the invariant distribution that
relies on the Brouwer degree. Thereafter, we will present examples of peculiar
limit behaviour that cannot occur for classical linear Markov chains. Finally,
we present for the case of small state spaces sufficient (and easy-to-verify)
criteria for the ergodicity of the process.
