A lambda-graph system for the Dyck shift and its \(K\)-groups
Let \(\Sigma\) be a finite alphabet. On the \(\Sigma^{\mathbb Z}\) one has the left-shift. A Shannon graph is some directed graph \(G\) with edges labelled by the symbols in \(\Sigma\). In some sense a Shannon graph \(G\) presents a subshift \(X\subset\Sigma^{\mathbb Z}\). A \(\lambda\)-graph system is a directed labelled Bratteli diagram with an additional structure. Given a subshift \(X\subset\Sigma^{\mathbb Z}\) there is a one-to-one correspondence between the compact forward separated Shannon graphs that present \(X\), and the forward separated Shannon \(\lambda\)-graph systems that present \(X\). A class of subshifts \(X\subset\Sigma^{\mathbb Z}\) with an intrinsically defined shift invariant dense set \(P_{\text{Ch}}(X)\) of periodic points is introduced. Here the record of an infinite sequence of events can be viewed as a point in a Cantor discontinuum. Therefore subshifts in this class called Cantor horizon subshifts. The symbolic matrix system \((M^{\text{Ch}(D_N)}, I^{\text{Ch}(D_N)})\) of the Cantor horizon \(\lambda\)-graph systems of the Dyck shift \(D_N\) is defined. Denote the group of all \(\mathbb Z\)-valued continuous functions on the Cantor discontinuum \(\mathbb C\) by \(C({\mathbb Z},{\mathbb Z})\). For such systems the \(K\)-groups \(K_i (M,I)\), \(i=0,1\), and Bowen-Franks groups \(\text{BF}^i (M,I)\), \(i=0,1\), are computed. The main result of the paper: \[ \begin{aligned} K_0 (M,I)&\approxeq {\mathbb Z}/N{\mathbb Z} \oplus C({\mathbb C},{\mathbb Z}),\\ K_1 (M,I) &\approxeq 0,\\ \text{BF}^0 (M,I) &\approxeq {\mathbb Z}/N{\mathbb Z},\\ \text{BF}^i (M,I) &\approxeq \Hom_{\mathbb Z}(C({\mathbb C},{\mathbb Z}),{\mathbb Z}).\end{aligned} \] {}.
scientific article; zbMATH DE number 1985400