On the invariant distributions of $$C^2$$ circle diffeomorphisms of irrational rotation number

Abstract

Although invariant measures are a fundamental tool in Dynamical Systems, very little is known about distributions (i.e. linear functionals defined on some space of smooth functions on the underlying space) that remain invariant under a dynamics. Perhaps the most general definite result in this direction is the remarkable theorem of Avila and Kocsard [1] according to which no C∞ circle diffeomorphism of irrational rotation number has an invariant distribution different from (a scalar multiple of integration with respect to) the (unique) invariant (probability) measure. The main result of this Note is an analogous result in low regularity. Unlike [1] which involves very hard computations, our approach is more conceptual. It relies on the work of Douady and Yoccoz [3] concerning automorphic measures for circle diffeomorphisms.