Algebraic shift equivalence and primitive matrices

Abstract

Motivated by symbolic dynamics, we study the problem, given a unital subring S S of the reals, when is a matrix A A algebraically shift equivalent over S S to a primitive matrix? We conjecture that simple necessary conditions on the nonzero spectrum of A A are sufficient, and establish the conjecture in many cases. If S 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.