Factoring Factor Maps

A noninjective bounded-to-one factor map from an irreducible shift of nite type onto a sooc system can be factored as a composition of other such maps in only nitely many ways (up to isomorphism). This generalizes to factor maps from systems with canonical coordinates to nitely presented dynamical systems. The analysis of bounded-to-one and especially of resolving factor maps has played a key role in the study of various coding relations among shifts of nite type and Markov chains 2, 3, 7, 14, 20 ... ]. The classiication of sooc systems up to topo-logical conjugacy reduces to the classiication of such maps, because sooc systems come with canonical resolving factor maps from shifts of nite type 15, 5]. The study of these maps is also at the heart of applications of symbolic dynamical ideas to the problem of eeciently coding data into and out of magnetic media 1, 12, 17]. The problem of determining when there exists a resolving map between two given irreducible shifts of nite type (SFT's) was solved by Ashley 3]. This problem for bounded-to-one maps is still open, but the necessary conditions of 14] are conjectured 4, p.34] to be suucient. The only known general constructions of such maps are as resolving maps and their compositions (e.g. 2, 3, 7]). Let B be the class of bounded-to-one factor maps from irreducible shifts of nite type onto sooc systems. With the operation of composition (when deened), B is a category. One natural goal in this setting is to understand the factorizations of maps in B. Adler and Marcus 2] asked if bounded-to-one factor maps between irreducible SFT's are always compositions of resolving maps, a question answered in the negative by Kitchens 14]. Certain sooc systems T (the AFT systems of Marcus 17]) are the images of factor maps which are \minimal covers" through which every factor map in B onto T must factor, and all irreducible sooc systems have \minimal resolving covers" 5]. Recently Trow 21] proved that a map in B can have only nitely many factorizations (up to isomorphism) as a left resolving map followed by a right resolving map, and such a factorization is unique (up to isomorphism) if the map has degree 1. His proofs rest on nontrivial constructions of accessory factor maps, generalizing constructions of Nasu 19] and built up via magic words in an elaboration of ideas in 14]. It was …