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.

一λ -图bisystem$\mathcal{升}$ 由一对组成 $({\mathcal{升}}^{-},{\mathcal{升}}^{+})$ 两个带标签的 Bratteli 图，呈现了一个两侧的子位移 ${\mathrm{\Lambda }}_{\mathcal{升}}$. 我们将构造一个紧凑的完全断开的度量空间，它由来自λ -graph双系统的二维半平面的平铺组成。平铺空间有一定的AF-等价关系写成${\mathrm{电阻}}_{\mathcal{升}}$ 具有自然移位同胚 ${\sigma }_{\mathcal{升}}$ 来自移位同胚 ${\sigma }_{{\mathrm{\Lambda }}_{\mathcal{升}}}$ 在班次 ${\mathrm{\Lambda }}_{\mathcal{升}}$. 等价关系${\mathrm{电阻}}_{\mathcal{升}}$ 产生一个 AF 代数 ${\mathcal{F}}_{\mathcal{升}}$ 带有自同构 ${\rho }_{\mathcal{升}}$ 由...介绍 ${\sigma }_{\mathcal{升}}$. 我们将研究 étale 等价关系的不变性${\mathrm{电阻}}_{\mathcal{升}}$, 群像 ${\mathcal{G}}_{\mathcal{升}}={\mathrm{电阻}}_{\mathcal{升}}{⋊}_{{\sigma }_{\mathcal{升}}}\mathbb{Z}$ 和 groupoid $C^{＊}$-代数 $C^{＊}\left({\mathrm{电阻}}_{\mathcal{升}}\right)$, $C^{＊}\left({\mathcal{G}}_{\mathcal{升}}\right)$ 在呈现的两侧子位移的拓扑共轭下。