Zero-dimensional almost 1-1extensions of odometers from graph coverings
Zero-dimensional almost 1-1extensions of odometers from graph coverings
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来自图覆盖的里程表的零维几乎 1-1 扩展
DOI：
10.1016/j.topol.2016.05.018
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发表时间：
2016
影响因子：
0.6
通讯作者：
Takashi Shimomura
中科院分区：
文献类型：
--
作者：
Ayumu Hoshino;Jun'ichi Shiraishi;星野歩，白石潤一;Ayumu Hoshino and Jun'ichi Shiraishi;Ayumu Hoshino;星野歩;星野歩，白石潤一;星野歩，白石潤一;星野歩;星野歩;星野歩;Takashi Shimomura;Takashi Shimomura;Takashi Shimomura;Takashi Shimomura;Takashi Shimomura;Takashi Shimomura
众所周知，单边或双面托普利兹流的特征是无限里程的象征性几乎1-1扩展。双面Toeplitz流也被描述为具有等径数性质的brattelli - vershik系统。然而，对于单侧系统，brattelli - vershik方法不适合作为组合表示。我们通过Gambaudo和Martens（2006）提出的图表总结了无限里程计的单向或双向几乎1-1扩展。他们用了有限有向图序列的逆极限。当Gjerde和Johansen（2000）描述双面Toeplitz流时，他们使用了等径数属性的概念。我们使用的概念是平移等周期性质，这意味着所有图的回路都有相等的周期，以Gambaudo和Martens的图覆盖的方式。在我们的总结中，我们也通过这些图覆盖来描述片面的Toeplitz流。作为一个应用，我们证明了单侧Toeplitz流的自然扩展是Toeplitz流。我们还总结了一般brattelli - vershik表示与Gambaudo和Martens给出的图形覆盖之间的联系。如果我们考虑扩张性，采取自然延伸是非常重要的。我们还证明了具有等周期性质的它们的复盖逆极限的自然扩展族与几乎是里程表的1-1扩展的双边零维同胚是一致的。Gjerde和Johansen最初的目标强调了寻找有序Bratteli图的可扩展性条件的困难。Sugisaki（2001）给出了可扩展性的充分条件；我们利用Gambaudo和Martens给出的图覆盖扩展了这个条件。讨论了图覆盖的逆极限的自然展开性与逆极限的正展开性之间的关系。作为一个应用，我们证明了单侧极小子移的拓扑秩不大于它的自然扩展。
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.