Augmenting dimension group invariants for substitution dynamics

Abstract

Abstract. We present new invariants for substitutional dynamical systems. Our main contribution is a flow invariant which is strictly finer than, but related and akin to, the dimension groups of Herman, Putnam and Skau. We present this group as a stationary inductive limit of a system associated to an integer matrix defined from combinatorial data based on the class of special words of the dynamical system.

Aperiodic substitution systems and their Bratteli diagrams

“…We recall that Bratteli-Vershik models of substitutional dynamical systems were constructed for primitive substitutions in the papers [For] and [DHS]. We also refer the reader to the papers [CE1], [CE2], [Yua1], and [Yua2], where related topics such as various dimension groups and invariant measures for substitutional dynamical systems are considered. It is worthwhile to mention the pioneering paper by Ferenczi [Fer] where the study of substitutions on infinite alphabets was initiated.…”

Embedding tiling spaces in surfaces

“…This played a crucial role in [BDH03] and [CE03]. It shows up in this work as a verifiable necessary condition for a tiling space to embed in a surface.…”

Ordered K-groups associated to substitutional dynamics

“…Since the group on the right hand side may be described directly in terms of the shift space, this characterization allows a more direct analysis of the group K 0 (O X ) which in this context is often denoted simply K 0 (X). A wide ranging analysis along these lines is carried out in Matsumoto's work, and under extra assumptions such as properties ( * ), ( * * ) or the shift space being substitutional, we have contributed in [12,13].…”

“…[34]) associated to any shift space in a fashion closely related to the way the groups of Bowen and Franks are associated to shifts of finite type [3]. This group is not always easily computable, but in a previous paper [12], we have given a concrete inductive limit description of the Matsumoto K 0 -groups associated to substitutional shift spaces, proved that they contain the dimension groups of the system (cf. [18]), and demonstrated by examples that the Matsumoto K 0 -group often carries more information than the dimension group.…”

“…This is in contrast to our description of the Matsumoto K 0 -group as a group [12], which could employ a more direct description of these groups using the past equivalence structure (or the closely related lambda-graph [31]) of the shift spaces in question.…”