An étale equivalence relation on a tiling space arising from a two-sided subshift and associated C*-algebras

A λ-graph bisystem consists of a pair of two labelled Bratteli diagrams, that presents a two-sided subshift . 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 with a natural shift homeomorphism coming from the shift homeomorphism on the subshift . The equivalence relation yields an AF-algebra with an automorphism induced by . We will study invariance of the étale equivalence relation , the groupoid and the groupoid -algebras , under topological conjugacy of the presenting two-sided subshifts.