On lower distance estimates of mappings in metric spaces

TL;DR

This paper investigates lower bounds for distance distortion in mappings within metric spaces, providing new theorems on the discreteness of limit mappings and analyzing cases in Euclidean and general metric spaces.

Contribution

It introduces novel lower bounds for distance distortion in mappings and applies these results to establish discreteness of limit mappings in various spaces.

Findings

Established lower bounds for distance distortion in mappings.

Proved theorems on the discreteness of limit mappings.

Analyzed mappings in Euclidean and general metric spaces.

Abstract

We consider mappings satisfying an upper bound for the distortion of families of curves. We establish lower bounds for the distortion of distances under such mappings. As applications, we obtain theorems on the discreteness of the limit mapping of a sequence of mappings converging locally uniformly. We separately consider cases when mappings are defined in Euclidean $n$-dimensional space and in a metric space.