Finite Asymptotic Clusters of Metric Spaces

Authors

Viktoriia Bilet, Institute of Applied Mathematics and Mechanics of NAS of UkraineFollow
Oleksiy Dovgoshey, Institute of Applied Mathematics and Mechanics of NAS of UkraineFollow

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.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Bilet, Viktoriia and Dovgoshey, Oleksiy (2018) "Finite Asymptotic Clusters of Metric Spaces," Theory and Applications of Graphs: Vol. 5: Iss. 2, Article 1.
DOI: 10.20429/tag.2018.050201
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol5/iss2/1