Zero-dimensional almost 1–1 extensions of odometers from graph coverings

It is well known that the one- or two-sided Toeplitz flows are characterised as the symbolic almost 1–1 extensions of infinite odometers. The two-sided Toeplitz flows are also characterised as the Bratteli–Vershik systems with the equal path number property. Nevertheless, for one-sided systems, the Bratteli–Vershik way is not suitable as a combinatorial representation. We summarise one- or two-sided almost 1–1 extensions of infinite odometers by the graph covering that Gambaudo and Martens (2006) presented. They used the inverse limit of a certain kind of sequences of finite directed graphs. When Gjerde and Johansen (2000) characterised the two-sided Toeplitz flows, they used the notion of the equal path number property. The notion we employ is the translated equal period property that implies that all the circuits of graphs have equal period in the way of graph coverings of Gambaudo and Martens. In our summary, we also characterise the one-sided Toeplitz flows by these graph coverings. As an application, we show that the natural extension of a one-sided Toeplitz flow is Toeplitz. We also summarise the link between the general Bratteli–Vershik representations and the graph coverings that Gambaudo and Martens gave. If we consider the expansiveness, taking the natural extension is very significant. We also show that the family of natural extensions of inverse limits of their coverings with the equal period property coincides with the two-sided zero-dimensional homeomorphisms that are almost 1–1 extensions of odometers. Gjerde and Johansen's original aim highlighted the difficulty in finding a condition of expansiveness for ordered Bratteli diagrams. Sugisaki (2001) responded by deriving a sufficient condition for expansiveness; we extend this condition using the graph coverings given by Gambaudo and Martens. We also discuss some relation between the expansiveness of the natural extensions of inverse limits of graph coverings and the positive expansiveness of the inverse limits. As an application, we show that the topological rank of a one-sided minimal subshift is not greater than its natural extension.