Study of dynamical systems by use of general topology and geometric topology

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400079
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: University of Tsukuba
• Principal Investigator: KATO HISAO 筑波大学, 数理物質系, 教授 (70152733)
• Co-Investigator(Renkei-kenkyūsha): KAWAMURA Kazuhiro 筑波大学, 数理物質系, 教授 (40204771) ; ISHII Atsushi 筑波大学, 数理物質系, 講師 (00531451) ; TANGE Motoo 筑波大学, 数理物質系, 助教 (70452422)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 彩色数 / カオス / エントロピー / 位相力学系 / 幾何学的トポロジー / アトラクター / フラクタル次元 / 連続体論

Outline of Final Research Achievements

In this research, we studied eventually coloring numbers of maps and evaluated the numbers. We obtained some important theorems of the numbers. Moreover, as applications of the theorems, we proved that for any map on any locally compact space, there exist a nice compactification of the space and a nice extension of the map which preserves the conditions of the periodic point set. Also, we introduced the notion of dark space like dark matters of physics, and we proved that if an n-dimensional system has the zero-dimensional periodic point set, then the system can be decomposed into an (n-1)-dimensional dark space and a dense zero-dimensional bright space. These results are very important in the theories of geometric topology and chaotic dynamical systems.

Free Research Field

数物系科学