Every circle homeomorphism is the composition of two weldings

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 tools 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.