Every circle homeomorphism is the composition of two weldings

TL;DR

This paper proves that any orientation-preserving circle homeomorphism can be expressed as a composition of two conformal welding homeomorphisms, revealing new insights into their compositional structure and properties.

Contribution

It introduces the concept of log-singular sets and demonstrates that conformal welding homeomorphisms are not closed under composition, advancing understanding of their algebraic structure.

Findings

Every circle homeomorphism is a composition of two weldings

Conformal welding homeomorphisms are not closed under composition

Introduction of log-singular sets as a new tool

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.