Rigidity for circle diffeomorphisms with breaks satisfying a Zygmund smoothness condition

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 ${\omega }_{\gamma }(\delta )=|\log \delta {|}^{-(\gamma /2-1)}.$