Symbolic Dynamics from Signed Matrices

We consider a method for assigning a sofic shift to a (not necessarily nonnegative integer) matrix by associating to it a directed graph with some vertices labelled 1 and the rest 2 (the decomposition of the vertices is arbitrary in applications the choice should be natural). We can detect positive topological entropy for this sofic shift by comparing the characteristic polynomial of the original matrix to those for the matrices for the restrictions of the shifts to each piece (1 and 2). Our main application is to the use of the Conley index to detect symbolic dynamics in isolated invariant sets, and is an extension of a result by Carbinatto, Kwapisz, and Mischaikow.