Entropy versus orbit equivalence for minimal homeomorphisms
Entropy versus orbit equivalence for minimal homeomorphisms
复制标题
最小同胚的熵与轨道等效
DOI：
10.2140/pjm.1994.164.1
复制
发表时间：
1994
影响因子：
0.6
通讯作者：
D. Handelman
中科院分区：
文献类型：
--
作者：
M. Boyle;D. Handelman
1.导言和定义。我们使用Vershik的字典序或“adic”映射（如赫尔曼，Putnam和Skau [P]，[HPS]，[Sk]所修改的），构造了所有可能的（拓扑）熵（包括无穷大）的最小同胚的例子，这些熵是“强”轨道等价于并矢加法机。特别是，这回答了一个决定性的负面方式的问题，是否轨道等价同胚有相同的熵。我们还证明了康托集的任何极小同胚都是强轨道等价于零熵的一个。我们回顾一些定义。这些数据的来源是Skau [Sk]的调查文章。Bratteli图B是一个有向图，其顶点集分解为有限子集，“层”或“行”，Bk（k = 0，1，.），以及从Bk中的顶点到Bk+X中的顶点的边。此外，Bk的每个顶点都连接到Bk+X的一个顶点。布拉泰利图是简单的，如果对所有k，存在k1> k，使得对于Bk中的每个顶点v和Bk>中的每个顶点v'，都有一条从v到v'的路。Bratteli图是指向的，如果|2f o| = 1，即存在一个可分辨的顶顶点。无限路的集合，通常记为X，称为Bratteli紧集，是一个紧零
1. Introduction and definitions. We construct, using the lexicographic or "adic" maps of Vershik (as modified by Herman, Putnam, and Skau [P], [HPS], [Sk]), examples of minimal homeomorphisms of all possible (topological) entropies (including infinite) which are "strongly" orbit equivalent to the dyadic adding machine. In particular, this answers in a decisively negative way the question as to whether orbit equivalent homeomorphisms have the same entropy. We also show that any minimal homeomorphism of the Cantor set is strongly orbit equivalent to one of zero entropy. We recall some definitions. A source for these is the survey article of Skau [Sk]. A Bratteli diagram B is a directed graph whose vertex set decomposes into finite subsets, "levels" or "rows", Bk (k = 0,1,...), together with edges from vertices in Bk to vertices in Bk+X additionally, every vertex of Bk is joined to a vertex of Bk+X. The Bratteli diagram is simple if for all k, there exists k1> k such that for every vertex v in Bk and every vertex v' in Bk>, there is a path from v Xo v'. The Bratteli diagram is pointed if |2f o| = 1, that is, there is a distinguished top vertex. The set of infinite paths, usually denoted X, is called the Bratteli compactum, and is a compact zero