Lower entropy factors of sofic systems

A mixing subshift of finite type T is a factor of a sofic shift S of greater entropy if and only if the period of any periodic point of S is divisible by the period of some periodic point of T . Mixing sofic shifts T satisfying this theorem are characterized, as are those mixing sofic shifts for which Krieger's Embedding Theorem holds. These and other results rest on a general method for extending shift-commuting continuous maps into mixing subshifts of finite type.