A sharp smoothness of the conjugation of class P-homeomorphisms to diffeomorphisms

TL;DR

This paper establishes precise smoothness conditions for conjugating class P-homeomorphisms of the circle to diffeomorphisms, providing sharp estimates and characterizing cases where conjugacies are not absolutely continuous.

Contribution

It proves the existence of piecewise analytic conjugacies with prescribed break points and offers sharp smoothness estimates for conjugations under the (D)-property.

Findings

Existence of piecewise analytic conjugacy with prescribed break points.

Sharp estimate for smoothness of conjugation under the (D)-property.

Conjugacy is not absolutely continuous if the (D)-property fails and the product of jumps is non-trivial.

Abstract

Let f be a class P -homeomorphism of the circle. We prove that there exists a piecewise analytic homeomorphism that conjugate f to a one-class P with prescribed break points lying on pairwise distinct orbits. As a consequence, we give a sharp estimate for the smoothness of a conjugation of class P -homeomorphism f of the circle satisfying the (D)-property (i.e. the product of f-jumps in the break points contained in a same orbit is trivial), to diffeomorphism. When f does not satisfy the (D)-property the conjugating homeomorphism is never piecewise C^1 and even more it is not absolutely continuous function if the total product of f-jumps in all the break points is non-trivial.