An étale equivalence relation on a tiling space arising from a two-sided subshift and associated C*-algebras. Issue 3 (3rd July 2021)

• Record Type: Journal Article
• Authors: Matsumoto, Kengo
• Is Part Of: Dynamical systems. Volume 36:Issue 3(2021)
• Page Start: 445
• Page End: 482
• Publication Date: 2021-07-03
• Subjects: Subshift -- Bratteli diagram -- equivalence relation -- groupoid -- C∗-algebra; Primary 37A55 -- Secondary 37A20 -- 37B10 -- 46L55; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2021.1928605
• Languages: English
• ISSNs: 1468-9367

Abstract

Abstract : A λ -graph bisystem L consists of a pair ( L −, L + ) of two labelled Bratteli diagrams, that presents a two-sided subshift Λ L . We will construct a compact totally disconnected metric space consisting of tilings of a two-dimensional half plane from a λ -graph bisystem. The tiling space has a certain AF-equivalence relation written R L with a natural shift homeomorphism σ L coming from the shift homeomorphism σ Λ L on the subshift Λ L . The equivalence relation R L yields an AF-algebra F L with an automorphism ρ L induced by σ L . We will study invariance of the étale equivalence relation R L, the groupoid G L = R L ⋊ σ L Z and the groupoid C ∗ -algebras C ∗ ( R L ), C ∗ ( G L ) under topological conjugacy of the presenting two-sided subshifts.