An \'etale equivalence relation on a configuration space arising from a subshift and related $C^*$-algebras

A $\lambda$-graph bisystem $\mathcal{L}$ consists of two labeled Bratteli diagrams $(\mathcal{L}^-,\mathcal{L}^+)$, that presents a two-sided subshift $\Lambda_\mathcal{L}$. We will construct a compact totally disconnected metric space with a shift homeomorphism consisting of two-dimensional configurations from a $\lambda$-graph bisystem. The configuration space has a certain \'etale AF-equivalence relation written $R_{\mathcal{L}}$ with a natural shift homeomorphism $\sigma_{\mathcal{L}}$ coming from the shift homeomorphism $\sigma_{\Lambda_{\mathcal{L}}}$ on the subshift $\Lambda_{\mathcal{L}}$. The equivalence relation $R_{\mathcal{L}}$ yields an AF-algebra ${\mathcal{F}}_{\mathcal{L}}$ with an automorphism $\rho_{\mathcal{L}}$ on it. We will study invariance of the \'etale equivalence relation $R_{\mathcal{L}}$, the groupoid $\mathcal{G}_{\mathcal{L}}=R_{\mathcal{L}}\rtimes_{\sigma_{\mathcal{L}}}{\mathbb{Z}}$ and the groupoid $C^*$-algebras $C^*(R_{\mathcal{L}})$, $C^*(\mathcal{G}_{\mathcal{L}})$ under topological conjugacy of the presenting two-sided subshifts.