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

Abstract

A λ-graph bisystem $\mathcal{L}$ consists of a pair $({\mathcal{L}}^{-},{\mathcal{L}}^{+})$ of two labelled Bratteli diagrams, that presents a two-sided subshift ${\mathrm{\Lambda }}_{\mathcal{L}}$. 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 $R_{\mathcal{L}}$ with a natural shift homeomorphism ${\sigma }_{\mathcal{L}}$ coming from the shift homeomorphism ${\sigma }_{{\mathrm{\Lambda }}_{\mathcal{L}}}$ on the subshift ${\mathrm{\Lambda }}_{\mathcal{L}}$. The equivalence relation $R_{\mathcal{L}}$ yields an AF-algebra ${\mathcal{F}}_{\mathcal{L}}$ with an automorphism ${\rho }_{\mathcal{L}}$ induced by ${\sigma }_{\mathcal{L}}$. We will study invariance of the étale equivalence relation $R_{\mathcal{L}}$, the groupoid ${\mathcal{G}}_{\mathcal{L}}=R_{\mathcal{L}}{⋊}_{{\sigma }_{\mathcal{L}}}\mathbb{Z}$ and the groupoid $C^{\ast }$-algebras $C^{\ast }\left(R_{\mathcal{L}}\right)$, $C^{\ast }\left({\mathcal{G}}_{\mathcal{L}}\right)$ under topological conjugacy of the presenting two-sided subshifts.

Keywords:

• Subshift
• Bratteli diagram
• equivalence relation
• groupoid
• $C^{\ast }$-algebra

Mathematics Subject Classifications:

• Primary 37A55
• Secondary 37A20
• 37B10
• 46L55

Acknowledgments

The author would like to thank the referee for his/her helpful comments and suggestions in the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by JSPS KAKENHI [grant numbers 15K04896, 19K03537].