Boundary extension of μ-homeomorphisms

In this paper we provide local sufficient and necessary conditions for a mu-homeomorphism of the open upper half-plane onto itself to induce (or not) a homeomorphic extension to the boundary. The conditions are in terms of modules of semiannular regions or involve integral expressions depending on the behavior of the complex dilatation (in particular, angular and radial dilatations) near the boundary. We relate our results to the special case of boundary extension of mappings with subexponeritially integrable finite distortion.