On sofic systems I

A new algebraic invariant for weak equivalence of sofic subshifts

“…In particular, if S(X ) is finite then X is sofic. Moreover, it is also known ( [17]; see also [18,Exercise 3.2.8]) that the number of ∼ ω -classes is finite if and only if X is sofic (that number is the number of states of the graph defining the left Krieger cover [17]); the same result holds for the ∼ω-classes.…”

“…The Krieger cover of a sofic subshift X is the cover associated with the essential labeled graph obtained from the minimal automaton of L(X ) by deleting all the states that are not K-states [19,Section 5]. Krieger proved that two sofic subshifts are conjugate if and only if their Krieger covers are conjugate [17]. If the sofic subshift X is irreducible then the labeled graph representing its Krieger cover has a unique terminal strongly connected component which is an essential labeled graph presenting X [7].…”

The Syntactic Graph of a Sofic Shift

“…If X is a sofic shift, the equivalence classes of κ are finitely many [15]. The right Krieger cover of X is defined as the automaton labeled by A in which the states are the κ-classes [x] with x ∈ X − , and there is a an edge labeled a from [x] to [xa] if xa ∈ X − .…”

“…We refer to [18] or [14] for more details. See also [15], [16], [5] about sofic shifts. Basic definitions and properties related to finite semigroups and their structure are given Section 2.2.…”

Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems

“…References can be found in his 1986 paper. Analogous constructions for (two-sided) sofic systems are carried out in [Krieger, 1984].…”