Asymptotic Enlargement of the States of Certain Stochastic Automata

Let a homogeneous Markov chain having a finite number of states and describing a stochastic automaton depend on a parameter $\varepsilon$ in such a way that the transition probabilities are continuous functions of $\varepsilon$ for $\varepsilon=\varepsilon_0$ and the set of states of the chain for $\varepsilon=\varepsilon_0$ decomposes into the union of $k>1$ ergodic sets $X_1,\dots,X_k$. A family of Markov processes describing a random walk of the original Markov process on the sets $X_1,\dots,X_k$ as $\varepsilon\to\varepsilon_0$ is constructed.