A shape theoretical approach to fractal geometry

分形几何的形状理论方法

• 批准号: 23540086
• 负责人: MIYATA Takahisa
• 金额: $ 1.41万
• 依托单位: Kobe University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2011
• 资助国家: 日本
• 起止时间: 2011 至 2013
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-23540086/
• 关键词: shape theory; inverse system; metric space; normal sequence; asymptotic dimensiom; Assouad-Nagata dimension; フラクタル幾何学; シェイプ理論; 近似的逆システム; フラクタル; 距離空間; asymptotic 次元; Assouad-Nagata 次元; 正規列; 逆システム

Shape theory is known as a homotopy theory for topological spaces with locally bad behavior. Fractal, on the other hand, is one of the spaces carrying the notion of distance, and is known as a metric space with locally complicated structures. In this project, we apply the theory of shape, in particular, its categorical aspects, to find the properties of fractals, or even general metric spaces. In particular, we have shown that the notion of inverse system, which is one of the most important approaches to shape theory, and the notion of normal sequence, which is one of the most important notions in the theory of topological spaces, have strong connections to metric spaces.

形状理论被称为具有局部不良行为的拓扑空间的同伦理论。 另一方面，分形是承载距离概念的空间之一，被称为具有局部复杂结构的度量空间。 在这个项目中，我们应用形状理论，特别是其分类方面，来寻找分形甚至一般度量空间的属性。 特别是，我们已经证明，逆系统的概念（形状理论最重要的方法之一）和正态序列的概念（拓扑空间理论中最重要的概念之一）与度量空间有很强的联系。

项目成果

Assouad -Nagata dimension and finite-to-one Lipschitz maps

Assouad -Nagata 维数和有限对一 Lipschitz 映射

• 发表时间: 2013
• 期刊: Topology Proceedings
• 作者: T. Miyata;Z. Virk;T. Miyata and T. Yoshimura
• 通讯作者: T. Miyata and T. Yoshimura

Dimension raising mapping theorem for Assouad-Nagada dimension

Assouad-Nagada 维数的升维映射定理

• 发表时间: 2012
• 作者: T. Miyata;T. Yoshimura
• 通讯作者: T. Yoshimura

Approximate extension property of mappings

映射的近似扩展属性

• 发表时间: 2012
• 期刊: Topology and its Applications
• 影响因子: 0.6
• 作者: T. Miyata;T. Yoshimura;T. Miyata
• 通讯作者: T. Miyata

Dimension-raising maps in a large scale

大比例尺地图升维

• DOI: 10.4064/fm223-1-6
• 发表时间: 2013
• 期刊: Fundamenta Mathematicae
• 影响因子: 0.6
• 作者: T. Miyata;Z. Virk
• 通讯作者: Z. Virk

Assouad-Nagata dimension and finite-to-one Lipschitz maps

Assouad-Nagata 维数和有限对一 Lipschitz 映射

• 发表时间: 2013
• 期刊: Topology Proceedings
• 作者: T. Miyata;T. Yoshimura
• 通讯作者: T. Yoshimura