On equicontinuous families of mappings in metric spaces

TL;DR

This paper investigates the conditions under which families of mappings with finite distortion in metric spaces are equicontinuous, extending classical results on quasiregular mappings and introducing criteria involving mean oscillation and omitted continua.

Contribution

It provides new criteria for equicontinuity of finite distortion mappings in metric spaces, including generalized quasiisometries on Riemannian manifolds, based on mean oscillation and omitted sets.

Findings

Families are equicontinuous if the characteristic has finite mean oscillation.

Equicontinuity holds for generalized quasiisometries on Riemannian manifolds.

Results extend classical quasiregular mapping theory to metric spaces.

Abstract

The article is devoted to the study of mappings with finite distortion in metric spaces. Analogues of results relating to equicontinuity and normality of families of quasiregular mappings are obtained. It is proved that the indicated families are equicontinuous if the characteristic of the mappings has a finite mean oscillation at each inner point, and the maps omit a certain fixed continuum. An equicontinuity of generalized quasiisometries on Riemannian manifolds is also obtained.