Subshifts and C*-algebras from one-counter codes

TL;DR

This paper introduces standard one-counter subshifts, explores their topological and algebraic properties, and analyzes associated C*-algebras, revealing cases where the algebra's simplicity differs between a shift and its inverse.

Contribution

It defines a new class of subshifts called standard one-counter shifts and investigates their conjugacy, flow equivalence, and associated C*-algebras, including K-group computations.

Findings

Certain standard one-counter shifts have simple C*-algebras while their inverses do not.

Examples of subshifts not flow equivalent to their inverses are provided.

K-groups are explicitly computed for a family of structured shifts.

Abstract

We introduce a class of subshifts under the name of "standard one-counter shifts". The standard one-counter shifts are the Markov coded systems of certain Markov codes that belong to the family of one-counter languages. We study topological conjugacy and flow equivalence of standard one-counter shifts. To subshifts there are associated C*-algebras by their $\lambda$-graph systems. We describe a class of standard one-counter shifts with the property that the C*-algebra associated to them is simple, while the C*-algebra that is associated to their inverse is not. This gives examples of subshifts that are not flow equivalent to their inverse. For a family of highly structured standard one-counter shifts we compute the K-groups.