Multifractal analysis of the Brjuno function

TL;DR

This paper analyzes the pointwise regularity of the Brjuno function, establishing its multifractal nature by determining its 1-exponent everywhere, and discusses different notions of regularity.

Contribution

It provides the first comprehensive determination of the 1-exponent of the Brjuno function, demonstrating its multifractal properties and exploring various regularity notions.

Findings

The Brjuno function is multifractal with a well-defined 1-exponent everywhere.

The paper characterizes the pointwise regularity of the Brjuno function.

It introduces new insights into the regularity notions applicable to the Brjuno function.

Abstract

We determine the $1$-exponent (according to the Calder\'on-Zygmund definition) of the Brjuno function $B$ everywhere, thus showing that it is a new example of multifractal function. We also discuss various notions of pointwise regularity of the function $B$.