Hierarchical Algebraic Structure of the Standard Binary Expansion

The standard binary expansion map is a fundamental mathematical object whose topological, algebraic, and dynamical aspects appear scattered across the literature in topology, operator algebras, and ergodic theory. The contribution of this paper is to provide a unified constructive framework: within the theory of Bratteli diagrams, we show how these scattered classical results arise naturally from a single hierarchical structure. Specifically, (1) using the nested structure of cylinder sets, we give an interval-nesting definition of , rigorously verify that the quotient space is homeomorphic to , and prove that the non-injective points are precisely the dyadic rationals; (2) we show that the hierarchical structure of corresponds to a complete binary Bratteli diagram, whose associated AF algebra is the Glimm algebra with group ; (3) we strictly distinguish the doubling map induced by the shift on the path space from the versal minimal system of the Bratteli diagram (the 2-adic odometer acting on the 2-adic integers ), and give a recursive construction of the finite-level automorphism groups—the -fold iterated wreath product of , of order , which is non-abelian for . In particular, this paper extracts two combinatorial-algebraic tools of independent interest: the standard multiplication notation for semidirect products in Proposition 3.5, and the explicit formula for the boundary map of group actions in Proposition 3.8. These tools serve as a benchmark reference system for future studies of non-binary or weighted Bratteli diagrams. Keywords: Cantor space; binary expansion; Bratteli diagram; AF algebra; iterated wreath product; odometer 标准二进制展开映射 是一个基本的数学对象,其拓扑、代数与动力学面貌分别散见于拓扑学、算子代数与遍历论文献中。本文的贡献在于提供一个统一的构造性框架:在Bratteli图理论下,展示这些分散的经典结果如何从同一层级结构中自然导出。具体而言:(1)利用cylinder set的嵌套结构给出 的区间套定义,严格验证商空间同胚于 ,并证明非单射点恰为二进有理数集;(2)证明 的层级结构对应于完全二叉Bratteli图,其关联的AF代数为Glimm代数 , 群为 ;(3)严格区分路径空间移位诱导的倍增映射与Bratteli图的标准versal极小系统(2-adic odometer作用于2-adic整数群 ),并给出有限层自同构群的递归构造—— 的 次迭代圈积,阶为 , 时非阿贝尔。特别地,本文提取了两个具有独立应用价值的组合-代数工具:命题3.5中半直积的标准乘法记号,以及命题3.8中群作用边界映射的显式公式。这些工具为后续研究非二叉或带权Bratteli图提供了基准对照系。 关键词:康托尔空间;二进制展开;Bratteli图;AF代数;迭代圈积;odometer