Study of chaotic maps and complicated invariant sets in topological dynamics by using continuum theory

利用连续统理论研究拓扑动力学中的混沌映射和复杂不变量集

• 批准号: 14540060
• 负责人: KATO Hisao
• 金额: $ 2.24万
• 依托单位: University of Tsukuba
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540060/
• 关键词: chaotic dynamical systems; entropy; minimal sets; expansive dynamical systems; attractors; indecomposable continuum; dimension; ergodic property; カオス; スクランブル集合; Bing map; 分割不可能性; 拡大同相写像; Lelek写像; 拡大力学系; パインね変換; カントール集合; シフト写像; 共役

We investigated several dynamical properties of (continuum-wise) expansive homeomorphisms, invariant sets and minimal sets. R.Man'e proved that minimal sets of expansive homeomorphisms are 0-dimensional. More generally, we showed that minimal sets of continuum-wise expansive homeomorphisms are 0-dimensional. we proved that if f is a continuum-wise expansive homeomorphism of a compactum X with dim X=1, then there is a Cantor set Z in X such that for some natural number N,f^{N}(Z)=Z and f^{N}|Z is semi-conjugate to the shift homeomorphism of the symbolic dynamics. By using this result, we showed that there are many infinite minimal sets in X. Also, we investigated the topological entropy of maps on some fractal sets. Related to this study of dynamical systems, we also proved the following theorems in geometric topology : The sets of Bing maps and Lelek maps are G-delta dense in the mapping space of all maps from compacta to polyhedra. These results will be applied to the research of the theory ofdynamical systems.

我们研究了（连续）膨胀的同构，不变套装和最小套件的几种动力学特性。 r.man'e证明，最少的巨大同构态度是0维的。更普遍的是，我们表明，最小的连续性膨胀同构为0维。我们证明，如果f是具有昏暗x = 1的压实X的连续性膨胀同构，那么x中有一个cantor集z，以至于对于某些自然数字n，f^{n}（z）= z和f^{n} | z是半偶像到同型动力学的同性形态的半偶像。通过使用此结果，我们表明X中有许多无限最小的集合。此外，我们研究了某些分形集中地图的拓扑熵。与这项动力学系统研究相关，我们还证明了几何拓扑中的以下定理：Bing Maps和Lelek Maps的集合在从Compacta到Polyhedra的所有地图的映射空间中都是G-Delta密集的。这些结果将应用于对动力系统理论的研究。

项目成果

酒井克郎: "Hyperspaces of normal linear spaces with Attouch-Wets topology"Set-Valued Analysis. 11. 21-36 (2003)

Katsuro Sakai：“具有 Attouch-Wets 拓扑的正常线性空间的超空间”集值分析。11. 21-36 (2003)

Infinite minimal sets of continuum-wise expansive homeomorphisms of 1-dimensional compacta

一维紧致体的连续体扩展同胚的无限极小集

• 发表时间: 2003
• 期刊: Topology and its Applications 130
• 作者: Hisao Kato;Eiichi;Matsuhashi;加藤久男
• 通讯作者: 加藤久男

川村一宏: "A note on singular homelogy groups of infinite products of compacta"Fund. Math.. 175. 285-289 (2002)

Kazuhiro Kawamura：“关于紧凑的无限乘积的奇异同源群的注释”数学.. 175. 285-289 (2002)。

Hyperspaces with the Hausdorff Metric and Uniform ANR's

• DOI: 10.2969/jmsj/1158242069
• 发表时间: 2005-04
• 期刊: Journal of The Mathematical Society of Japan
• 影响因子: 0.7
• 作者: Masayuki Kurihara;K. Sakai;Masato Yaguchi

保科隆雄: "Relative normality and product spaces"Comment Math.Univ.Carolinae. 44. 515-524 (2003)

Takao Hoshina：“相对正态性和乘积空间”Comment Math.Univ.Carolinae 44. 515-524 (2003)