Every circle homeomorphism is the composition of two weldings

TL;DR

This paper proves that every orientation-preserving circle homeomorphism can be written as the composition of two conformal weldings. The approach uses log-singular maps and log-singular sets, built via an address-based, Cantor-like partitioning to produce zero-capacity images. A central ingredient is Bishop's log-singular welding theorem, which guarantees that log-singular maps are weldings, enabling the decomposition φ = (φ ∘ h^{-1}) ∘ h with h log-singular. Consequently, conformal weldings are not closed under composition, and the work provides a constructive framework for expressing arbitrary circle homeomorphisms through weldings, with implications for conformal welding theory and related Teichmüller-type structures.

Abstract

We show that every orientation-preserving circle homeomorphism is a composition of two conformal welding homeomorphisms, which implies that conformal welding homeomorphisms are not closed under composition. Our approach uses the log-singular maps introduced by Bishop. The main tool that we introduce are log-singular sets, which are zero capacity sets that admit a log-singular map that maps their complement to a zero capacity set.

Figures (5)

• Figure 1: Definition of conformal welding. The conformal maps $f$ and $g$ map the interior and exterior of the unit circle to the two complementary components of a Jordan curve $\gamma$. The map $h\colon\mathbb{S}^{1}\to\mathbb{S}^{1}$ is an orientation-preserving homeomorphism of $\mathbb{S}^{1}$.
• Figure 2: Representation of a log-singular circle homeomorphism $h\colon\mathbb{S}^{1}\to\mathbb{S}^{1}$. The arcs $I_{j}$, which have small capacity, are mapped to the arcs $h(I_{j})$, which have larger capacity.
• Figure 3: Addresses associated to a partition of an interval $I$ with $L_{1}=3, L_{2}=5, L_{3}=4, L_{4}=2, L_{5}=3$. The address to $I_{\{1,3,0,1,2\}}$ is highlighted in a thicker color and by non-dashed arrows.
• Figure 4: Sketch of the proof of Lemma \ref{['lemma:logsingular']} where two different partitions are matched up. The figure represents two iterations of the process in the proof of Lemma \ref{['lemma:logsingular']}. The lighter intervals, like $I_{\{1,3\}}$, have small capacity in the domain and are mapped to intervals with larger capacity, like $J_{\{1,3\}}=h(I_{\{1,3\}})$. At each step $n$, the maps $h_{n}$ are linear on each $I_{A_{n}}$.
• Figure 5: Finding the partition in Lemma \ref{['lemma:find_logsingular']}. The lighter thicker sub-arcs have small capacity and they are mapped to sub-arcs with small capacity.

Theorems & Definitions (18)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3: ChrisWeldingAnnals Theorem 3
• Theorem 1.4
• Proposition 2.1
• Lemma 2.2
• Theorem 2.3: Pfluger Pfluger
• Theorem 2.4: ChrisWeldingAnnals Theorem 3
• Theorem 2.5: ChrisWeldingAnnals Theorem 1
• Definition 3.1: Log-singular homeomorphism, log-singular set