The one-frequency cohomological equation, Brjuno-like functions and Khintchine-L\'evy numbers

TL;DR

This paper characterizes the frequencies for which the one-frequency cohomological equation on the 2-torus has an analytic solution, using Brjuno-like functions, and explores the solvability conditions for Diophantine and Khintchine-Lévy numbers.

Contribution

It introduces explicit Brjuno-like functions to determine analytic solvability of the cohomological equation and provides estimates for these functions for specific number classes.

Findings

Identifies all frequencies with analytic solutions in terms of Brjuno-like functions.

Provides explicit estimates of Brjuno-like functions for Diophantine and Khintchine-Lévy numbers.

Constructs examples where small functions lack solutions due to infinite Brjuno-like function values.

Abstract

In the paper we consider the one-frequency cohomological equation \begin{equation*} (\partial_x + \omega \partial_y) g(x,y) = a(x,y) \end{equation*} on the 2-torus with unknown $g$ and analytic initial data $a$. We identify all the frequencies $\omega$ for which the equation has an analytic solution and express the analytic solvability condition in terms of two Brjuno-like functions, providing explicit estimates on the sup-norm of $g$. As an example we estimate the Brjuno-like functions for Diophantine and Khintchine-L\'evy numbers. We also construct an example of an arbitrarily small function $a$ for which an analytic $g$ does not exist when one of the Brjuno-like functions has infinite value.