Cohomological equations and invariant distributions for minimal circle diffeomorphisms

Abstract

Given any smooth circle diffeomorphism with irrational rotation number, we show that its invariant probability measure is the only invariant distribution (up to multiplication by a real constant). As a consequence of this, we show that the space of real $C^{\infty }$-coboundaries of such a diffeomorphism is closed in $C^{\infty }(\mathbb{T})$ if and only if its rotation number is Diophantine.