Path methods for strong shift equivalence of positive matrices

Abstract

In the early 1990's, Kim and Roush developed path methods for establishing strong shift equivalence (SSE) of positive matrices over a dense subring U of the real numbers R. This paper gives a detailed, unified and generalized presentation of these path methods. New arguments which address arbitrary dense subrings U of R are used to show that for any dense subring U of R, positive matrices over U which have just one nonzero eigenvalue and which are strong shift equivalent over U must be strong shift equivalent over U_+. In addition, we show positive real matrices on a path of shift equivalent positive real matrices are SSE over R_+; positive rational matrices which are SSE over R_+ must be SSE over Q_+; and for any dense subring U of R, within the set of positive matrices over U which are conjugate over U to a given matrix, there are only finitely many SSE-U_+ classes.

This version adds a 3-part program for studying SEE over the reals. One part is handled by the arxiv post "Strong shift equivalence and algebraic K-theory". This version is the author version of the paper published in the Kim memorial volume. From that, my short lifestory of Kim (and more) is on my web page http://www.math.umd.edu/~mboyle/papers/index.html