Remarks on Martio’s conjecture

Abstract

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