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Title: Tree-shifts: Irreducibility, mixing, and the chaos of tree-shifts
Authors: 班榮超
Ban, Jung-Chao
Chang, Chih-Hung
Contributors: 應數系
Date: 2017-05
Issue Date: 2020-05-27 09:03:03 (UTC+8)
Abstract: Topological behavior, such as chaos, irreducibility, and mixing of a one-sided shift of finite type, is well elucidated. Meanwhile, the investigation of multidimensional shifts, for instance, textile systems, is difficult and only a few results have been obtained so far.
This paper studies shifts defined on infinite trees, which are called tree-shifts. Infinite trees have a natural structure of one-sided symbolic dynamical systems equipped with multiple shift maps and constitute an intermediate class between one-sided shifts and multidimensional shifts. We have shown not only an irreducible tree-shift of finite type but also a mixing tree-shift that is chaotic in the sense of Devaney. Furthermore, the graph and labeled graph representations of tree-shifts are revealed so that the verification of irreducibility and mixing of a tree-shift is equivalent to determining the irreducibility and mixing of matrices, respectively. This extends the classical results of one-sided symbolic dynamics.
A necessary and sufficient condition for the irreducibility and mixing of tree-shifts of finite type is demonstrated. Most important of all, the examination can be done in finite steps with an upper bound.
Relation: Transactions of the American Mathematical Society, Vol.369, No.12, pp.8389-8407
Data Type: article
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