Quasi-symmetric conjugacy for circle maps with a flat interval

In this paper we study quasi-symmetric conjugations of ${\mathcal{C}}^{2}$ weakly order-preserving circle maps with a flat interval. Under the assumption that the maps have the same rotation number of bounded type and that bounded geometry holds, we construct a quasi-symmetric conjugation between their non-wandering sets. Further, this conjugation is extended to a quasi-symmetric circle homeomorphism. Our proof techniques hinge on real-dynamic methods, allowing us to construct the conjugation under general and natural assumptions.