Covers of Non-Almost-Finite Type Sofic Systems

Susan Williams
1988 Proceedings of the American Mathematical Society  
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. 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, (¡>) are covers of S such that qb = ir o 9 for some factor map 9 from T to E, we
more » ... y 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.
doi:10.2307/2047495 fatcat:ulq7ngbxojburkkur4uzzco6uy