  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.
