Ultrametric preserving functions and weak similarities of ultrametric spaces

Abstract

Let $WS(X,d)$ be the class of ultrametric spaces which are weakly similar to ultrametric space $(X,d)$. The main results of the paper completely describe the ultrametric spaces $(X,d)$ for which the equality $$\rho (x,y)=f(d(\Phi (x),\Phi (y)))$$ holds for every $(Y,\rho )\in WS(X,d)$, every weak similarity $\Phi :Y\to X$, and all $x$, $y\in Y$ with some ultrametric (pseudoultrametric) preserving function $f$ depending on $\Phi$.

18 pages, 1 figure