On the Uniqueness of the Norton-Sullivan Quasiconformal extension

Abstract

We show that the extension map \[ \mathcal{E}_{NS}(f)(z)=\frac{f(x+y)+f(x-y)}{2}+i\frac{f(x+y)-f(x-y)}{2}\mbox{ for all }z=x+iy\in\mathbb{H}\,, \] defined by Norton and Sullivan in '96, is the only locally linear extension map taking bi-Lipschitz functions on $\mathbb{R}$ to quasiconformal functions on $\mathbb{H}$, modulo the action of a group isomorphic to the linear group. In fact, we discovered many other extension like this one (lying in the orbit of such group action), such as: $f(x)\mapsto f(x)+i(f(x)-f(x-y))$.