$C^1$-Diffeomorphism Class of some Circle Maps with a Flat Interval

TL;DR

This paper investigates a class of circle maps with flat intervals and boundary singularities, proving that bi-Lipschitz conjugacies imply $C^1$ smoothness, thus linking topological and smooth classifications.

Contribution

It establishes that for this class of circle maps, bi-Lipschitz conjugacies are actually $C^1$ diffeomorphisms, connecting topological and smooth conjugacy classes.

Findings

Bi-Lipschitz conjugacy implies $C^1$ conjugacy.

Topological and $C^1$ classes coincide for these maps.

The class includes maps with boundary singularities of different degrees.

Abstract

We study a certain class circle maps which are constant on one interval (called flat piece), and such that the degrees of the singularities at the boundary of the flat piece are different. In this paper, we show that if the topological conjugacy between two maps of my class is a bi-Lipschitz homeomorphism, then it is a $C^1$ diffeomorphism; that is, the bi-Lipschitz homeomorphism class and $C^1$ diffeomorphism class of a map in our class are equivalent.