On uniformly lightness of ring mappings and convergence to a homeomorphism

TL;DR

This paper investigates conditions under which families of homeomorphisms are uniformly light and open, demonstrating that integrability of a majorant ensures these properties and convergence to a homeomorphism.

Contribution

It establishes that integrability of the modulus majorant guarantees uniform lightness, openness, and convergence of homeomorphism families.

Findings

Uniform lightness is ensured by integrable modulus majorants.

Families of homeomorphisms are uniformly open under these conditions.

Such families converge uniformly to a homeomorphism.

Abstract

A family of mappings is called uniformly light if the image of the continuum under these mappings cannot be contracted to a point under the sequence of mappings of the family. In this paper, we are interested in the problem of the uniform lightness of a family of homeomorphisms satisfying upper moduli inequalities. We have shown that a family of such homeomorphisms satisfies the above-mentioned condition of uniform lightness if the majorant participating in the modulus estimate defining the family is integrable over almost all spheres. Under the same conditions, we show that this family of homeomorphisms is uniformly open, i.e., their image contains a ball of fixed radius, independent of each mapping separately. As an application of the results obtained, we have proved the assertion about the uniform convergence of homeomorphisms to a homeomorphism.