Strong shift equivalence of matrices over a ring

Let R be a ring. Two square matrices A,B are elementary strong shift equivalent (ESSE-R) over R if there are matrices U,V over R such that A = RS and B = SR. Strong shift equivalence over (SSE-R) is the equivalence relation generated by ESSE-R. Shift equivalence over R (SE-R) is a tractable equivalence relation which is refined by SSE-R. The refinement is trivial if R = Z (Williams), a principal ideal domain (Effros 1981) or a Dedekind domain (Boyle-Handelman 1993). No results have appeared since 1993. It turns out that this refinement is captured precisely by a certain quotient group of the group NK1(R) of algebraic K-theory. It follows that for very many (not all) rings R, the relations SE-R and SSE-R are the same. For the class of nilpotent matrices over R (nilpotent matrices are those shift equivalent to [0]), this quotient group is NK1(R) itself. When NK1(R) is not trivial, it is not finally generated (Farrell, 1977). I will try to explain background and motivation for SSE in terms of classification of symbolic dynamical systems and inverse problems for nonnegative matrices.Non UBCUnreviewedAuthor affiliation: University of MarylandFacult