Cohomological equations and invariant distributions for minimal circle diffeomorphisms

TL;DR

This paper proves that for smooth circle diffeomorphisms with irrational rotation numbers, the invariant measure is the unique invariant distribution, and characterizes when the space of smooth coboundaries is closed based on Diophantine conditions.

Contribution

It establishes the uniqueness of invariant distributions for smooth circle diffeomorphisms with irrational rotation numbers and links the closure of coboundaries to Diophantine conditions.

Findings

Invariant measure is the only invariant distribution for such diffeomorphisms.

The space of smooth coboundaries is closed if and only if the rotation number is Diophantine.

Provides a cohomological characterization related to 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 smooth coboundaries of such a diffeomorphism is closed if and only if its rotation number is Diophantine.