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）扩张同胚、不变集和极小集的动力学性质。R. Man 'e证明了扩张同胚的极小集是0维的。更一般地，我们证明了连续可扩张同胚的极小集是0维的。证明了：如果f是紧空间X的连续扩张同胚，且dim X=1，则在X中存在一个Cantor集Z，使得对某个自然数N，f^{N}（Z）=Z，且f^{N}| Z与符号动力学的移位同胚是半共轭的。利用这个结果，我们证明了X中存在许多无穷小集。此外，我们还研究了某些分形集上映射的拓扑熵。与动力系统的研究相关，我们还证明了几何拓扑学中的下列定理：Bing映射和Lelek映射的集合在从顶点到多面体的所有映射的映射空间中是G-δ稠密的。这些结果将应用于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)。

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)

加藤久男: "Infinite minimal sets of continuum-wise expansive homeomorphisms of 1-dimensional compacta"Topology and its Applications. (to appear).

Hisao Kato：“一维紧致体的连续体扩展同胚的无限最小集”拓扑及其应用（即将出现）。