Systèmes codés. (Coded systems)
By definition, coded systems are those subshifts on a finite alphabet A which are generated by a strongly connected graph with a denumerable set of vertices (whereas sofic systems are generated by a finite graph). Now, letting F be any set of finite sequences on A, it is proved that the closure S of the set \(F^{\infty}\) of doubly infinite sequences that may be cut into blocks belonging to F is always coded - the converse being obvious. Therefore, coded systems are stable under factor maps (homomorphisms). Synchronizing systems are coded systems for which a canonical family of generating sets F exists. The structure of the set of ergodic measures on \(F^{\infty}\) is roughly the same as that of the set of all ergodic measures on a sofic system, and some further results are proved for synchronizing systems: but in many instances, no measure on \(F^{\infty}\) reaches the topological entropy of S. Various examples are discussed in the end. Since this paper was written down, W. Krieger has found out a simple topological characterization of coded systems, and proved that synchronizing systems are stable under conjugacy maps; A. Bertrand-Mathis has used results on coded systems for the study of \(\beta\)-shifts.
scientific article; zbMATH DE number 3970131
scientific article