On the uniqueness of the Norton–Sullivan quasiconformal extension
Abstract
We show that the extension map \(\mathcal{E}_{NS}(f)(z)=\frac{f(x+y)+f(x-y)}{2}+i\frac{f(x+y)-f(x-y)}{2}\) for all \(z=x+iy\in\mathbb{H}\), defined by Norton and Sullivan in 1996, 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 extensions like this one (lying in the orbit of such group action), such as \(f(x)\mapsto f(x)+i(f(x)-f(x-y))\).
Fields of Study & Subjects
Analytic and geometric function theory
Mathematical Dynamics and Fractals
Geometric and Algebraic Topology
Publication details
- Journal / source
- Annales Fennici Mathematici
- Volume
- 51
- Issue
- 1
- Pages
- 113–126
- Publication date
- 9 February 2026
- Language
- EN