度量空间的有限渐近簇
Finite asymptotic clusters of metric spaces
摘要：
设$(X, d)$是无界度量空间，$\tilde r=(r_n)_{n\in\mathbb N}$是趋于无穷大的正实数序列.一个在无穷远处的预切空间$Ω_{\infty, \tilde r}^{X}$到$(X, d)$是重标度序列$\left(X, \frac{1}{r_n}d\right).$的极限所有预切空间$Ω_{\infty, \tilde r}^{X}$的集合称为预切空间的渐近簇。这样的聚类可以被认为是加权图$(G_{X, \tilde r}, ρ_{X})$，其最大团与$Ω_{\infty, \tilde r}^{X}$一致，并且权重$ρ_{X}$由$Ω_{\infty, \tilde r}^{X}$上的度量定义。本文描述了具有预切空间的有限渐近簇的度量空间的结构，并刻画了与这些簇同构的有限权图。
英文摘要：
Let $(X, d)$ be an unbounded metric space and let $\tilde r=(r_n)_{n\in\mathbb N}$ be a sequence of positive real numbers tending to infinity. A pretangent space $Ω_{\infty, \tilde r}^{X}$ to $(X, d)$ at infinity is a limit of the rescaling sequence $\left(X, \frac{1}{r_n}d\right).$ The set of all pretangent spaces $Ω_{\infty, \tilde r}^{X}$ is called an asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph $(G_{X, \tilde r}, ρ_{X})$ whose maximal cliques coincide with $Ω_{\infty, \tilde r}^{X}$ and the weight $ρ_{X}$ is defined by metrics on $Ω_{\infty, \tilde r}^{X}$. We describe the structure of metric spaces having finite asymptotic clusters of pretangent spaces and characterize the finite weighted graphs which are isomorphic to these clusters.
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