Covers of non-almost-finite type sofic systems

An almost finite type (AFT) sofic system S has a cover which intercepts every other cover of S (BKM), We show that if an irreducible sofic system S is not AFT, it has an infinite collection of covers such that no two are intercepted by a common cover of S. 0. Introduction. If ir is a factor map from a subshift of finite type E onto a sofic system S, we call (E, it)— or, for brevity, -k— a cover of S. If (E, n) and (r, (i>) are covers of S such that qb = ir o 9 for some factor map 9 from T to E, we say n intercepts 0; if 9 is invertible we say the covers 0 and tt are conjugate over S. It was shown in (BKM) that an irreducible sofic system S is almost finite type (AFT) if and only if it has a cover which intercepts all other covers of S. The authors asked whether a non-AFT sofic system has a finite collection C of covers such that every cover is intercepted by a cover in C. In (W) we proved by example that this is not always the case. In this paper we show that there is never such a collection for a non-AFT sofic system. The example in (W) uses an extremely simple sofic system taken from (BKM). A feeling that this example is somehow archetypal for non-AFT systems led to the general result. This feeling is made concrete in a characterization of AFT systems which we state as Theorem 2. From this point we work to mimic the construction in ( W) of an infinite collection of covers such that no two have a common intercepting cover. To do this in a general setting requires some technical groundwork, found in §3. The construction itself is given in §4. 1. Properties of the joint cover. For background on subshifts of finite type (SFT) and sofic systems we refer the reader to (AM, BKM, and BMT). We regard a subshift of finite type E as being determined by a directed graph G(E), with the alphabet of E equal to the edge set of G(E). Every cover of a sofic system S is conjugate over 5 to a 1-block cover; that is, a cover (E, rr) where -k is generated by a labeling of the edges of G(E) with elements of the alphabet of S. Such a cover is right resolving if distinct edges in G(E) with the same initial vertex have distinct labels, and right closing if for some n, any two paths in G(E) with the same initial vertex and same labeling must coincide except on their final n edges. Left resolving and left closing are defined analogously. A word w of 5 is a