Journal of Statistical Physics. Vol. 74. Nos. 5/6. 1994. p. 1281-1292.
We consider shift-invariant probability measures on subshift dynamical systems with a transition matrix A which satisfies the Chapman-Kolmogorov equation for some stochastic matrix П compatible with A. We call them Chapman-
Kolmogorov measures. A nonequilibrium entropy is associated to this class of dynamical systems. We show that if A is irreducible and aperiodic, then there are Chapman-Kolmogorov measures distinct from the Markov chain associated with П and its invariant row probability vector q. If, moreover, (q, П) is a reversible chain, then we construct reversible Chapman-Kolmogorov measures on the subshift which are distinct from (q, П).