On the sharpness of Denjoy's theorem

Abstract

Let $\omega$ be a concave modulus of continuity that is weaker than Lipschitz, meaning $\omega(t)/t$ diverges as $t$ approaches $0$. We construct a diffeomorphism of the circle with irrational rotation number, in the regularity class $C^{1+\omega}$, with a wandering interval. This construction implies that Denjoy's 1932 theorem is sharp in regularity, unless additional restrictions are imposed on the rotation number. The construction in the special case $\omega(t) = t\log(1/t)$ settles an open problem dating back to Herman's 1979 work on circle diffeomorphisms, which gave constructions for $\omega(t) = t\log(1/t)^{1+\varepsilon}$ for every $\varepsilon>0$. Our examples arise as limits of periodic circle diffeomorphisms with rapidly converging rotation numbers.