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arXiv 0705.40802007-05-28math.DS

非周期置换系统及其Bratteli图

Aperiodic substitutional systems and their Bratteli diagrams

S. Bezuglyi, J. Kwiatkowski, K. Medynets

备注：: 42 pages

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摘要：

本文研究由非本原置换引起的非周期置换动力系统。证明了定常有序Bratteli图的Vershik同胚$ϕ$与非周期置换系统同胚的充要条件是$ϕ$对极小分支的约束不与里程表同胚.我们还证明了每一个由具有嵌套性质的替换所生成的非周期替换系统都同胚于一个平稳有序Bratteli图的Vershik映射。证明了每一个非周期置换系统都是可识别的。研究了$m$ -本原置换类及其派生置换类.我们还讨论了有限秩Cantor动力系统的可扩张性的概念。

英文摘要：

In the paper we study aperiodic substitutional dynamical systems arisen from non-primitive substitutions. We prove that the Vershik homeomorphism $ϕ$ of a stationary ordered Bratteli diagram is homeomorphic to an aperiodic substitutional system if and only if no restriction of $ϕ$ to a minimal component is homeomorphic to an odometer. We also show that every aperiodic substitutional system generated by a substitution with nesting property is homeomorphic to the Vershik map of a stationary ordered Bratteli diagram. It is proved that every aperiodic substitutional system is recognizable. The classes of $m$-primitive substitutions and associated to them derivative substitutions are studied. We discuss also the notion of expansiveness for Cantor dynamical systems of finite rank.

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