ISRAEL JOURNAL OF MATHEMATICS,Vol. 20, No. 2, 1975 SOFIC SYSTEMS BY ETHAN M. COVEN AND MICHAEL E. PAUL ABSTRACT A symbolic flow is called a sofic system if it is a homomorphic image (factor) of a subshift of finite type. We show that every sofic system can be realized as a finite-to-one factor of a subshift of finite type with the same entropy, From this it follows that sofic systems share many properties with subshifts of finite type. We concentrate especially on the properties of TPPD (transitive with periodic points dense) sofic systems. Introduction Sofic systems, the class of symbolic flows which are factors of subshifts of finite type, were introduced by Weiss in [15]. Weiss found sufficient, but not necessary, conditions for a sofic system to be intrinsically ergodic, i.e., to possess a unique entropy-maximizing measure. This paper is an outgrowth of the authors' search for necessary and sufficient conditions for intrinsic ergodicity. In Section 1 we discuss some basic facts about subshifts of finite type and sofic systems. In Section 2 we show that every sofic system may be realized as a finite-to-one factor of a subshift of finite type with the same topological entropy. In Section 3 we show that a sofic system is transitive with periodic points dense (TPPD) if and only if it is intrinsically ergodic with support (IES). (A flow is IES if it is intrinsically ergodic and the unique entropy-maximizing measure is positive on non-empty, open sets.) In Section 4 we examine some properties of TPPD sofic systems. In Section 5 we raise the question of how to distinguish (dynamically as opposed to combinatorially) between subshifts of finite type and "strictly" sofic systems. Received October 10, 1974 165