On The Dimension Group of Unimodular S-Adic Subshifts

TL;DR

This paper investigates the structure of dimension groups in a specific class of minimal Cantor systems called primitive unimodular proper S-adic subshifts, revealing their algebraic invariants and properties related to balanced functions.

Contribution

It computes the dimension group for these subshifts and explores the relationship between infinitesimal subgroup triviality and measure independence, introducing balanced functions and their topological characterization.

Findings

Dimension groups are computed for primitive unimodular proper S-adic subshifts.

Triviality of the infinitesimal subgroup relates to rational independence of measures.

Balanced functions are characterized topologically within this family.

Abstract

Dimension groups are complete invariants of strong orbit equivalence for minimal Cantor systems. This paper studies a natural family of minimal Cantor systems having a finitely generated dimension group, namely the primitive unimodular proper S-adic subshifts. They are generated by iterating sequences of substitutions. Proper substitutions are such that the images of letters start with a same letter, and similarly end with a same letter. This family includes various classes of subshifts such as Brun subshifts or dendric subshifts, that in turn include Arnoux-Rauzy subshifts and natural coding of interval exchange transformations. We compute their dimension group and investigate the relation between the triviality of the infinitesimal subgroup and rational independence of letter measures. We also introduce the notion of balanced functions and provide a topological characterization of bal-ancedness for primitive unimodular proper S-adic subshifts.