Triangulation extensions of self-homeomorphisms of the real line (Q522104)

Sense-preserving self-homeomorphism of the real axis can be extended to a self-homeomorphism of the upper half-plane. One such extension was constructed by Hubbard with triangulation. For quasisymmetric homeomorphism of the real line, the extension problem has been developed for a long time, which is very interesting and rich in content. It is well known that the Beurling-Ahlfors extension, the Douady-Earle extension and the Carleson box extension are three important quasiconformal self-homeomorphisms of the upper half-plane. So it is natural to ask how about the triangulation extension. This paper concerns about the following two problems: one is the question whether the triangulation extension of a quasisymmetric homeomorphism is quasiconformal, the other one is to finf when the triangulation extension of homeomorphism is a David mapping. In the construction of the triangulation extension, the points \(f(k+\frac{2j}{2^i})\) play an important role, where \(k\in\mathbb{Z}\), \(i\in\mathbb{N}\), and \(j=0,1,2,\dots,2^{i-1}-2\), and \(f:\mathbb{R}\rightarrow\mathbb{R}\) is a homeomorphism. Lines from these points with slope \(1\) and \(-1\) form a triangulation of the upper half-plane. An extension mapping is defined to be piecewise linear and isometric on vertical line onto these triangles. So the maximal dilatation of the triangulation extension is bounded by the scalewise distortion \(\rho_f(t)\) of \(f(x)\), which is Lemma 2 in the paper and plays a key role in the proof of the main results.