transactions of the american mathematical society Volume 336, Number 1, March 1993 ALGEBRAICSHIFT EQUIVALENCEAND PRIMITIVE MATRICES MIKE BOYLEAND DAVID HANDELMAN Abstract. Motivated by symbolic dynamics, we study the problem, given a unital subring 5 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 real- izing primitive matrices. For Dedekind domains, we prove that algebraic shift equivalence implies algebraic strong shift equivalence. 0. Introduction Unless specified otherwise, we use S to denote an integral domain; it most commonly will be a unital subring of the reals and frequently it will be equipped with the relative ordering, S+ = S n [0, oo). A matrix over S is nonnegative if all of its entries lie in S+ ; a matrix is primitive if it is square, nonnegative, and some power is strictly positive. In this paper we consider the following problem: When is a matrix with entries from S algebraically shift equivalent to a primitive matrix over S ? This is one piece of the fundamental problem of understanding the constraints which ordering places on algebraic structure. Our results in this direction, which we summarize at the end of this section, are particularly relevant to symbolic dynamics and linear algebra. Our methods derive from matrix theory, abstract algebra, and symbolic dynamics. We also give some new results of independent interest for algebraic shift equivalence and matrix forms. The problem involves different areas of mathematics and is new, so we will take some time now to explain it and give some background. Suppose sf and ¿i§ are endomorphisms in some category. Then sf and 3S are shift equivalent (in that category) if sf = 3§ or there exist a positive integer / (the lag) and morphisms X and Y such that the following equations hold: (0.1) XY = s/', YX = 3§1, X3$=sfX, Ysf=3§Y. When the lag is one, the equations sf = XY, ¿¡S = YX imply the others. Shift equivalence is an equivalence relation; in general, lag one shift equiva- lence is not. The transitive closure of lag one shift equivalence is called strong Received by the editors November 28, 1990. 1991 Mathematics Subject Classification. Primary 15A48; Secondary06F25, 58F03, 15A36, 15A33. The first author was partially supported by NSF Grant DMS-860169. The second author was supported in part by an operating grant from NSERC (Canada). ©1993 American Mathematical Society 0002-9947/93 $1.00+ $.25 per page 121 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use