On the sharpness of Denjoy's theorem

Abstract

Let $\ddot{I}‰$ be a concave modulus of continuity that is weaker than Lipschitz, meaning $\ddot{I}‰(t)/t$ diverges as $t$ approaches $0$. We construct a diffeomorphism of the circle with irrational rotation number, in the regularity class $C^{1+\ddot{I}‰}$, 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 $\ddot{I}‰(t)=t\log (1/t)$ settles an open problem dating back to Herman's 1979 work on circle diffeomorphisms, which gave constructions for $\ddot{I}‰(t)=t\log (1/t{)}^{1+\epsilon }$ for every $\epsilon >0$. Our examples arise as limits of periodic circle diffeomorphisms with rapidly converging rotation numbers.

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