The Relationship between Entropy and Strong Orbit Equivalence for the Minimal Homeomorphisms (II)
The Relationship between Entropy and Strong Orbit Equivalence for the Minimal Homeomorphisms (II)
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最小同胚的熵与强轨道等价的关系（二）
DOI：
10.3836/tjm/1270041818
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发表时间：
1998
影响因子：
0.6
通讯作者：
Fumiaki Sugisaki
中科院分区：
文献类型：
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作者：
Fumiaki Sugisaki
在[1]中，M.波义耳和D.汉德曼表明，每一个康托系统，即一个最小的动力系统的康托集，是强烈的轨道相当于一个零（拓扑）熵和并矢添加机变换是强烈的轨道相当于康托系统的所有可能的熵。他们提出了一个猜想，其表述如下：“康托集的每一个极小同胚都是与所有熵的同胚强轨等价的。这个猜想是Dye拓扑版[3]结果的一个类似.在[8]中，N. Ormes给出了强轨道实现定理，它推广了Dye定理[3]和朱厄特-Krieger定理[6，7].利用强轨道实现定理，他还表明，这一猜想是真实的情况下，唯一遍历康托系统。在本文中，我们证明了猜想是正确的一般康托系统的情况下，有限熵。我们在另一篇论文中已经证明，这也适用于无穷熵的情况[9]。结果表明，康托系统之间的两个等价关系，一个是强轨道等价关系，另一个是具有相同的熵，是相互独立的。
In [1], M. Boyle and D. Handelman showed that every Cantor system, namely, a minimal dynamical system on a Cantor set, is strongly orbit equivalent to one of zero (topological) entropy and that the dyadic adding machine transformation is strongly orbit equivalent to Cantor systems of all possible entropies. They made a conjecture which is stated as follows: “Every minimal homeomorphism of Cantor set is strongly orbit equivalent to homeomorphisms of all entropies.” This conjecture is an analogue of Dye’s result in topological version [3]. In [8], N. Ormes showed strong orbit realization theorem which generalize both Dye’s theorem [3] and the Jewett– Krieger theorem [6, 7]. Using strong orbit realization theorem, he also showed that this conjecture is true in the case of uniquely ergodic Cantor systems. In this paper we show that the conjecture is true for a general Cantor system for the case of finite entropies. We have shown in another paper that this holds also for the case of infinite entropy [9]. The result implies that among Cantor systems two equivalence relations, the one for strong orbit equivalence and the other for having the same entropy, are independent of each other.