Sturmian subshifts and their C*-algebras

TL;DR

This paper explores the structure of C*-algebras derived from one-sided Sturmian subshifts, revealing their classification via conjugacy and flow equivalence, and computing their nuclear dimension as one.

Contribution

It provides an explicit construction of the local homeomorphism associated with Sturmian subshifts and establishes their C*-algebra classification and dimension properties using elementary dynamical tools.

Findings

C*-algebras are *-isomorphic if and only if the systems are conjugate.

C*-algebras are Morita equivalent if and only if the defining irrationals are equivalent.

The nuclear dimension of these C*-algebras is exactly one.

Abstract

This paper investigates the structure of C*-algebras built from one-sided Sturmian subshifts. They are parametrised by irrationals in the unit interval and built from a local homeomorphism associated to the subshift. We provide an explicit construction and description of this local homeomorphism. The C*-algebras are *-isomorphic exactly when the systems are conjugate, and they are Morita equivalent exactly when the defining irrationals are equivalent (this happens precisely when the systems are flow equivalent). Using only elementary dynamical tools, we compute the dynamic asymptotic dimension of the (groupoid of the) local homeomorphism to be one, and by a result of Guentner, Willett, and Yu, it follows that the nuclear dimension of the C*-algebras is one.