Finite asymptotic clusters of metric spaces

TL;DR

This paper studies the structure of unbounded metric spaces at infinity by analyzing their asymptotic clusters of pretangent spaces, representing these clusters as weighted graphs and characterizing those with finite structures.

Contribution

It introduces a framework for understanding the asymptotic behavior of metric spaces through finite clusters of pretangent spaces and characterizes the corresponding weighted graphs.

Findings

Finite asymptotic clusters are characterized by specific graph structures.

Weighted graphs representing clusters are isomorphic to certain finite metric spaces.

The paper provides criteria for when a weighted graph corresponds to an asymptotic cluster.

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.