Finite asymptotic clusters of metric spaces

Abstract

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 ${\Omega }_{\infty ,\tilde{r}}^X$ 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 ${\Omega }_{\infty ,\tilde{r}}^X$ is called an asymptotic cluster of pretangent spaces. Such a cluster can be considered as a weighted graph $(G_{X,\tilde{r}},{\rho }_X)$ whose maximal cliques coincide with ${\Omega }_{\infty ,\tilde{r}}^X$ and the weight ${\rho }_X$ is defined by metrics on ${\Omega }_{\infty ,\tilde{r}}^X$. 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.

38 pages, 3 figures. arXiv admin note: text overlap with arXiv:1708.05235

References in corpus (1)

• Asymptotic structure of general metric spaces at infinity