The Number of Phases of Inhomogeneous Markov Fields with Finite State Spaces on N and Z and their Behaviour at Infinity

We consider the set of nonhomogeneous Markov fields on T = N or T = Z with finite state spaces E n , n ϵ T , with fixed local characteristics. For T = N we show that has at most \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_\infty = \mathop {\lim \inf}\limits_{n \to \infty} \left| {\mathop E\nolimits_n} \right| $\end{document} phases. If T = Z , has at most N ‐∞ · N ∞; phases, where \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_{-\infty} = \mathop {\lim \inf}\limits_{n \to -\infty} \left| {\mathop E\nolimits_n} \right| $\end{document} . We give examples, that for T = N for any number k , 1 ≦ k ≦ N ∞ , there are local characteristics with k phases, whereas for T = Z every number l · k , 1 ≦ l ≦ N ‐∞ , 1 ≦ k ≦ N ∞ occurs. We describe the inner structure of , the behaviour at infinity and the connection between the one‐sided and the two‐sided tail‐fields. Simple examples of Markov fields which are no Markov processes are given.