$C^*$-algebras and Fell bundles associated to a textile system

TL;DR

This paper explores the connection between textile systems, two-dimensional shifts of finite type, and associated $C^*$-algebras and Fell bundles, providing new algebraic tools to analyze complex symbolic dynamics.

Contribution

It introduces $C^*$-algebras and Fell bundles linked to textile shifts, extending the algebraic framework for analyzing two-dimensional symbolic dynamical systems.

Findings

Defined families of $C^*$-algebras for textile shifts

Computed specific $C^*$-algebras in key examples

Established Fell bundles encoding shift complexity

Abstract

The notion of textile system was introduced by M. Nasu in order to analyze endomorphisms and automorphisms of topological Markov shifts. A textile system is given by two finite directed graphs $G$ and $H$ and two morphisms $p,q:G\to H$, with some extra properties. It turns out that a textile system determines a first quadrant two-dimensional shift of finite type, via a collection of Wang tiles, and conversely, any such shift is conjugate to a textile shift. In the case the morphisms $p$ and $q$ have the path lifting property, we prove that they induce groupoid morphisms $\pi, \rho:\Gamma(G)\to \Gamma(H)$ between the corresponding \'etale groupoids of $G$ and $H$. We define two families ${\mathcal A}(m,n)$ and $\bar{\mathcal A}(m,n)$ of $C^*$-algebras associated to a textile shift, and compute them in specific cases. These are graph algebras, associated to some one-dimensional shifts of finite type constructed from the textile shift. Under extra hypotheses, we also define two families of Fell bundles which encode the complexity of these two-dimensional shifts. We consider several classes of examples of textile shifts, including the full shift, the Golden Mean shift and shifts associated to rank two graphs.