For every sense-preserving self-homeomorphism of the real axis, Hubbard constructed an extension that is a self-homeomorphism of the upper half-plane by triangulation. It is natural to ask if such extensions of quasisymmetric homeomorphisms of the real axis are all quasiconformal. Furthermore, for what sense-preserving self- homeomorphisms are such extensions David mappings? In this article, a sufficient and necessary condition for such extensions to be quasiconformal and a sufficient condition for such extensions to be David mappings are given.
"Triangulation extensions of self-homeomorphisms of the real line." Kyoto J. Math. 57 (1) 1 - 15, April 2017. https://doi.org/10.1215/21562261-3664950