An introduction to symbolic dynamics and coding
An introduction to symbolic dynamics and coding
The Work of Kim and Roush in Symbolic Dynamics
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
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In the early 1990's, Kim and Roush developed path methods for establishing strong shift equivalence (SSE) of positive matrices over a dense subring $\mathcal{U}$ of 驴. This paper gives a detailed, unified and generalized presentation of these path methods. New arguments which address arbitrary dense subrings $\mathcal{U}$ of 驴 are used to show that for any dense subring $\mathcal{U}$ of 驴, positive matrices over $\mathcal{U}$ which have just one nonzero eigenvalue and which are strong shift equivalent over $\mathcal{U}$ must be strong shift equivalent over $\mathcal{U}_{+}$ . In addition, we show matrices on a path of positive shift equivalent real matrices are SSE over 驴+; positive rational matrices which are SSE over 驴+ must be SSE over 驴+; and for any dense subring $\mathcal{U}$ of 驴, within the set of positive matrices over $\mathcal{U}$ which are conjugate over $\mathcal{U}$ to a given matrix, there are only finitely many SSE- $\mathcal{U}_{+}$ classes.