可收缩群与线性伸缩结构
Contractible groups and linear dilatation structures
摘要：
度量空间上的伸缩结构arXiv：math/0608536 v4，是介于群和微分结构之间的概念，用于解释度量空间的近似自相似性。伸缩结构的基本对象是伸缩（或收缩）。伸缩结构的公理规定了不同伸缩结构之间相互作用的规则。线性也是一种可以用膨胀结构来解释的性质。在本文中，我们表明，我们可以谈论两种线性：两个伸缩结构之间的函数的线性和伸缩结构本身的线性。我们在这里的主要结果是一个特征的收缩群的伸缩结构。对于赋范锥群（赋范可收缩群），我们可以自然地将线性扩张结构联系起来。相反，任何线性的强扩张结构都来自于赋范可收缩群的扩张结构。
英文摘要：
A dilatation structure on a metric space, arXiv:math/0608536v4, is a notion in between a group and a differential structure, accounting for the approximate self-similarity of the metric space. The basic objects of a dilatation structure are dilatations (or contractions). The axioms of a dilatation structure set the rules of interaction between different dilatations. Linearity is also a property which can be explained with the help of a dilatation structure. In this paper we show that we can speak about two kinds of linearity: the linearity of a function between two dilatation structures and the linearity of the dilatation structure itself. Our main result here is a characterization of contractible groups in terms of dilatation structures. To a normed conical group (normed contractible group) we can naturally associate a linear dilatation structure. Conversely, any linear and strong dilatation structure comes from the dilatation structure of a normed contractible group.
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