Rigidity for circle diffeomorphisms with breaks satisfying a Zygmund smoothness condition

TL;DR

This paper proves that circle diffeomorphisms with breaks satisfying a Zygmund smoothness condition are smoothly conjugate under certain conditions, extending rigidity results to less smooth settings.

Contribution

It establishes $C^{1+ ext{Zygmund}}$-smooth conjugacy for circle diffeomorphisms with breaks under Zygmund smoothness, generalizing previous rigidity results.

Findings

Diffeomorphisms with breaks and Zygmund smoothness are conjugate under certain conditions.

The conjugacy is $C^{1+ ext{Zygmund}}$-smooth with a specific modulus.

Results apply to diffeomorphisms with bounded type irrational rotation numbers.

Abstract

Let $f$ and $\tilde{f}$ be two circle diffeomorphisms with a break point, with the same irrational rotation number of bounded type, the same size of the break $c$ and satisfying a certain Zygmund type smoothness condition depending on a parameter $\gamma>2.$ We prove that under a certain condition imposed on the break size $c$, the diffeomorphisms $f$ and $\tilde{f}$ are $C^{1+\omega_{\gamma}}$-smoothly conjugate to each other, where $\omega_{\gamma}(\delta)=|\log \delta|^{-(\gamma/2-1)}.$