Covers of Non-Almost-Finite Type Sofic Systems

Abstract

ABSTRACT. 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 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. Propertiesof 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

Flow equivalence of sofic shifts

“…(iv) X has a minimal cover (i.e. an SFT Y and a factor map π : Y → X such that any other factor map ϕ : Y ′ → X (from an SFT Y ′ onto X) must factor through π) [12,41]. (This cover must be conjugate to the left and right Fischer covers.)…”

Resolving Extensions of Finitely Presented Systems

“…In general, there is no minimal extension to a Smale space for a finitely presented system, see [15,16]. However, if (Y, f ) is finitely presented and transitive we have the following result.…”

Reducibility of covers of AFT shifts

“…By [5, Proposition 3] the irreducible strictly sofic shifts of degree 1 are precisely the AFT shifts. Many authors (see [25,8,39,18], for example) have studied AFT shifts and, as a result, we now have the following list of equivalent conditions on a strictly sofic shift. Theorem 2.17.…”