Discoveries / analysis

Sharpness of Denjoy's Theorem

Denjoy's 1932 theorem says a $C^{1+\mathrm{bv}}$ circle diffeomorphism with irrational rotation number has no wandering interval. Whether it is sharp in regularity: for every concave modulus of continuity $\omega$ weaker than Lipschitz, there is a $C^{1+\omega}$ circle diffeomorphism with irrational rotation number and a wandering interval. The case $\omega(t) = t\log(1/t)$ settles an open problem going back to Herman's 1979 work, which had constructions only for $\omega(t) = t\log(1/t)^{1+\varepsilon}$.