Research on comlex dynamical systems and C*-algebras

2017 Fiscal Year Final Research Report

• Project/Area Number: 25287019
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyushu University
• Principal Investigator: Watatani Yasuo 九州大学, 数理学研究院, 教授 (00175077)
• Co-Investigator(Kenkyū-buntansha): 幸崎 秀樹 九州大学, 数理学研究院, 学術研究者 (20186612)
• Co-Investigator(Renkei-kenkyūsha): Matsui Taku 九州大学, 数理学研究院, 教授 (50199733) ; Ueda Yoshimichi 名古屋大学, 多元数理科学研究科, 教授 (00314724) ; Masuda Toshihiko 九州大学, 数理学研究院, 准教授 (60314978) ; Kajiwara Tsuyoshi 岡山大学, 環境学研究科, 教授 (50169447) ; Matsumoto Kengo 上越教育大学, 学校教育研究科, 教授 (40241864)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Keywords: 作用素環 / 複素力学系 / 自己相似写像 / K-理論

Outline of Final Research Achievements

In this research, we associate C*-algebras with complex dynamical systems of iterations of rational functions R using Hilbert bimodule, and we have found interesting relations between operator algebras and complex dynamical systems. We can recover some properties of complex dynamical systems of R from the associated C*-algebras with the gauge action.　We explain the structure of the singularity of branched points in terms of K-theory and KMS states for gauge actions. We determine the extreme points of the traces on the cores of the C*-algebras associated with self-similar maps. We find their ideal structure. We showed that the dimension group of the tent map is the free abelian infinite group and the shift on it.

Free Research Field

作用素環論