Remarks on Martio's conjecture

TL;DR

This paper establishes a sharp integrability condition ensuring local homeomorphism of quasiregular mappings with small inner dilatation and analyzes branch points as Lebesgue points of the differential matrix.

Contribution

It introduces a novel integrability condition for the Jacobian's reciprocal that guarantees local homeomorphism in planar quasiregular mappings with small inner dilatation.

Findings

The integrability condition is sharp in the planar case.

Branch points are Lebesgue points of the differential matrix.

Conditions ensure local homeomorphism for small inner dilatation.

Abstract

We introduce a certain integrability condition for the reciprocal of the Jacobian determinant which guarantees the local homeomorphism property of quasiregular mappings with a small inner dilatation. This condition turns out to be sharp in the planar case. We also show that every branch point of a quasiregular mapping with a small inner dilatation is a Lebesgue point of the differential matrix of the mapping.