Asymptotic equipartition property for a Markov source having ambiguous alphabet

We propose a generalization of the asymptotic equipartition property to discrete sources with an ambiguous alphabet, and prove that it holds for irreducible stationary Markov sources with an arbitrary distinguishability relation. Our definition is based on the limiting behavior of graph parameters appearing in a recent dual characterization of the Shannon capacity, evaluated at subgraphs of strong powers of the confusability graph induced on high-probability subsets. As a special case, our results give an information-theoretic interpretation of the graph entropy rate of such sources.