Uniqueness of spaces pretangent to metric spaces at infinity

We find the necessary and sufficient conditions under which an unbounded metric space X has, at infinity, a unique pretangent space Ωˣ∞, ř for every scaling sequence ř. In particular, it is proved that Ωˣ∞, ř is unique and isometric to the closure of X for every logarithmic spiral X and every ř. It is also shown that the uniqueness of pretangent spaces to subsets of a real line is closely related to the “asymptotic asymmetry” of these subsets

Finite Asymptotic Clusters of Metric Spaces

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

Finite Asymptotic Clusters of Metric Spaces

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

Pseudometrics and partitions

Pseudometric spaces $(X, d)$ and $(Y, \rho)$ are combinatorially similar if there are bijections $\Psi \colon Y \to X$ and $f \colon d(X^2)~\to~\rho(Y^2)$ such that $$\rho(x, y) = f(d(\Psi(x), \Psi(y)))$$ for all $x$, $y \in X$. Let us denote by $\mathcal{IP}$ the class of all pseudometric spaces $(X, d)$ for which every combinatorial self-similarity $\Phi\colon~X~\to~X$ satisfies the equality $d(x, \Phi(x))=0,$ but all permutations of metric reflection of $(X, d)$ are combinatorial self-similarities of this reflection. The structure of $\mathcal{IP}$ spaces is fully described.Comment: 28 page