Algebraic shift equivalence and primitive matrices

Boyle, Mike; Handelman, David

doi:10.1090/S0002-9947-1993-1102219-4

Algebraic shift equivalence and primitive matrices
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by Mike Boyle and David Handelman

Trans. Amer. Math. Soc. 336 (1993), 121-149

DOI: https://doi.org/10.1090/S0002-9947-1993-1102219-4

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Abstract:

Motivated by symbolic dynamics, we study the problem, given a unital subring $S$ of the reals, when is a matrix $A$ algebraically shift equivalent over $S$ to a primitive matrix? We conjecture that simple necessary conditions on the nonzero spectrum of $A$ are sufficient, and establish the conjecture in many cases. If $S$ is the integers, we give some lower bounds on sizes of realizing primitive matrices. For Dedekind domains, we prove that algebraic shift equivalence implies algebraic strong shift equivalence.

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Algebraic shift equivalence and primitive matrices
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