Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows
Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows
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DOI：
10.1017/s0143385700000948
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发表时间：
2000-12
影响因子：
0.9
通讯作者：
Richard Gjerde;Ø. Johansen
中科院分区：
文献类型：
--
作者：
Richard Gjerde;Ø. Johansen
我们构造了Toeplitz流的Bratteli-Vershik模型，并刻画了一类与这些流对应的真序Bratteli图。我们利用这一结果推广了一种新的方法--使用维群的基本理论--一个关于Toeplitz流的有趣且非平凡的结果，该结果首先由Downarowicz证明。(威廉姆斯此前在这个方向上取得了初步结果。)结果表明，对于任何Choquite单纯形$K$，都存在一个$0$-$1$Toeplitz流$(Y，\psi)$，使得$(Y，\psi)$的不变概率测度集仿射同胚于$K$.我们不仅给出了这一结果的一个新的概念证明，我们还证明了我们可以选择$(Y，\psi)$来具有零熵和全有理谱。此外，对于给定的Toeplitz流，我们的Bratteli-Vershik模型显式地展示了因子到最大等距连续(里程计)因子的映射。利用这一点，我们给出了零熵的唯一遍历0-1 Toeplitz流的存在的简单证明，其中给定的里程表是它的最大等度连续因子，并且强轨道等价于该因子。用同样的方法，我们证明了0-1 Toeplitz流的存在性，它的最大等度连续因子是2-里程计，是强轨等价的，并且假设在$[0，\ln2)$中有任意的熵值。最后，我们通过一个明确的例子，使用Bratteli图，证明了Toeplitz流在Kakutani等价下(实际上，在诱导下)是不保持的--这与置换极小系统的情况形成了对比。事实上，我们展示的例子是一个0-1 Toeplitz流的诱导系统，它与Chacon代换系统共轭，因此它是素数的，即它没有非平凡因子。本文的主旨是论证Bratteli-Vershik模型和维群理论在最小符号系统研究中的相关性和实用性。这在福雷斯特和杜兰德、霍斯特和斯科乌最近的论文中也得到了例证，这些论文处理的是置换极小系统，博伊尔、汉德尔曼和奥姆斯的论文也是如此。
We construct Bratteli–Vershik models for Toeplitz flows and characterize a class of properly ordered Bratteli diagrams corresponding to these flows. We use this result to extend by a novel approach—using basic theory of dimension groups—an interesting and non-trivial result about Toeplitz flows, first shown by Downarowicz. (Williams had previously obtained preliminary results in this direction.) The result states that to any Choquet simplex $K$, there exists a $0$–$1$ Toeplitz flow $(Y,\psi)$, so that the set of invariant probability measures of $(Y,\psi)$ is affinely homeomorphic to $K$. Not only do we give a conceptually new proof of this result, we also show that we may choose $(Y,\psi)$ to have zero entropy and to have full rational spectrum. Furthermore, our Bratteli–Vershik model for a given Toeplitz flow explicitly exhibits the factor map onto the maximal equicontinuous (odometer) factor. We utilize this to give a simple proof of the existence of a uniquely ergodic 0–1 Toeplitz flow of zero entropy having a given odometer as its maximal equicontinuous factor and being strongly orbit equivalent to this factor. By the same token, we show the existence of 0–1 Toeplitz flows having the 2-odometer as their maximal equicontinuous factor, being strong orbit equivalent to the same, and assuming any entropy value in $[0,\ln 2)$. Finally, we show by an explicit example, using Bratteli diagrams, that Toeplitz flows are not preserved under Kakutani equivalence (in fact, under inducing)—contrasting what is the case for substitution minimal systems. In fact, the example we exhibit is an induced system of a 0–1 Toeplitz flow which is conjugate to the Chacon substitution system, thus it is prime, i.e. it has no non-trivial factors. The thrust of our paper is to demonstrate the relevance and usefulness of Bratteli–Vershik models and dimension group theory for the study of minimal symbolic systems. This is also exemplified in recent papers by Forrest and by Durand, Host and Skau, treating substitution minimal systems, and by papers by Boyle, Handelman and by Ormes.