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

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.