Quasi-Symmetric Conjugacy for Circle Maps with a Flat Interval

TL;DR

This paper establishes the existence of quasi-symmetric conjugations between certain circle maps with flat intervals, based on their rotation number and geometric properties, extending to circle homeomorphisms.

Contribution

It introduces a method to construct quasi-symmetric conjugations for $C^2$ circle maps with flat intervals, under bounded type rotation numbers and bounded geometry.

Findings

Constructed quasi-symmetric conjugation between non-wandering sets.

Extended conjugation to a quasi-symmetric circle homeomorphism.

Applied real-dynamic methods for the construction.

Abstract

In this paper we study quasi-symmetric conjugations of $C^2$ 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.