On $C^2$ circle diffeomorphisms with rotation numbers of constant type

Abstract

For every irrational rotation number $α$ of constant type and every $q>1$, we construct a $C^2$ circle diffeomorphism $F$ arbitrarily close to the rigid rotation $R_α$ in the combined $C^2$ and $B^2_{2,q}$ topology whose normalized conjugacy is not Lipschitz. We also prove that $F''\in B^0_{2,1}$ implies global $C^1$ conjugacy. Thus the thresholds in the Besov summability and weak Dini scales are $q=1$ and $γ=1/2$, respectively. These results complement Herman's Baire category theorem on the residual existence of $C^2$ diffeomorphisms without $C^1$ conjugacy by providing quantitative counterexamples and sufficient conditions for differentiable conjugacy.