Metrics transformations preserving the types of one-dimensional minimal fillings

Abstract

Given a class $F$ of metric spaces and a family of transformations $T$ of a metric, one has to describe a family of transformations $ T'\subset T$ that transfer $F$ into itself and preserve some types of minimal fillings.
The article considers four cases.
First, when $F$ is the class of all finite metric spaces, $T=\{(M,\r)\to(M,\,f\c\r):f\:\R_{>0}\to\R_{>0}\}$, and the elements of $T'$ preserve all non-degenerate types of minimal fillings of four-point metric spaces and finite non-degenerate stars, and we prove that $T'=\{(M,\r)\to(M,\,\l\r+a)\:a>\l a_\r\}$.
Second, when $F$ is the class of all finite metric spaces, the class $T$ consists of the maps $\r\to N\r$, where the matrix $N$ is the sum of a positive diagonal matrix $A$ and a matrix with the same rows of non-negative elements. The elements of $T'$ preserve all minimal fillings of the type of non-degenerate stars.
It has been proven that $T'$ consists of maps $\r\to N\r$, where $A$ is scalar.
Third, when $F$ is the class of all finite additive metric spaces, $T$ is the class of all linear mappings given by matrices, and the elements of $T'$ preserve all non-degenerate types of minimal fillings, and we proved that for metric spaces consisting of at least four points $T'$ is the set of transformations given by scalar matrices.
Fourth, when $F$ is the class of all finite ultrametric spaces, $T$ is the class of all linear mappings given by matrices, and we proved that for three-point spaces the matrices have the form $A=R(B+\l E)$, where $B$ is a matrix of identical rows of positive elements, and $R$ is a permutation of the points $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$.