Coding of the dimension group

Abstract

We study an aspect of dimension group theory, linked to coding. The dimension group that we consider is built on a given square primitive integer matrix M satisfying the conditions that | det M | ⩾ 2 and that the characteristic polynomial of M is irreducible. The coding is based on iteration of what could be seen as a generalization to Z d of the Euclidean algorithm induced by the matrix M and in a natural way we define a binary operation of addition in the coding group. The set B of symbols is a subset of Z d , and if we denote by ρ the Perron–Frobenius eigenvalue of M and by v a left eigenvector associated to ρ, we define a function Z d × B N * → R which assigns to the element ( p , b 1 , b 2 , … ) the series 〈 v , p 〉 + 1 ρ 〈 v , b 1 〉 + 1 ρ 2 〈 v , b 2 〉 + ⋯ (in case M = ( 10 ) , this is the decimal expansion) and the restriction of this function to finite codes is the classical embedding of the dimension group into R . Finally, and under some suitable conditions, we prove that the last function is surjective and this allows the coding of real numbers and consequently the dimension group embedded into R appears as the set of decimal numbers.