Title：Topological aspects of tree-shifts

Speaker：Prof. Zhihong Zhang（NationalUniversityof Kaohsiung）

Time：4:15-5:15 P.M. Sep 10th, 2015

Location：Room 4318, Building No.4, Wushan Campus

Abstract: The 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, the textile systems, is difficult and only a few results have been obtained so far.

    In this talk, we consider shifts defined on infinite trees, that 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 in between one-sided shifts and multidimensional shifts. We have shown that, not only an irreducible tree-shift of finite type, but a mixing tree-shift are 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.