The Brjuno functions of the by-excess, odd, even and odd-odd continued fractions and their regularity properties

TL;DR

This paper proves conjectures about the regularity of Brjuno functions linked to various continued fractions, showing their differences with classical Brjuno functions are H"older continuous and exploring their integrability properties.

Contribution

It establishes the H"older continuity of differences between classical and by-excess Brjuno functions and extends these results to functions with positive exponents, also analyzing odd and even continued fractions.

Findings

The difference between classical and by-excess Brjuno functions extends to a H"older continuous function.

Brjuno functions for odd and even continued fractions belong to all L^p spaces, p ≥ 1.

The Brjuno function for odd continued fractions differs from the classical one by a H"older continuous function.

Abstract

The Brjuno function was introduced by Yoccoz to study the linearizability of holomorphic germs and other one-dimensional small divisor problems. The Brjuno functions associated with various continued fractions including the by-excess continued fraction were subsequently investigated: it was conjectured that the difference between the classical Brjuno function and the even part of the Brjuno function associated with the by-excess continued fraction extends to a H\"older continuous function of the whole real line. In this paper, we prove this conjecture and we extend its validity to the more general case of Brjuno functions with positive exponents. Moreover, we study the Brjuno functions associated to the odd and even continued fractions introduced by Schweiger. We show that they belong to all $L^p$ spaces, $p\ge1$. We prove that the Brjuno function associated to the odd continued fraction differs from the classical Brjuno function by a H\"older continuous function. On the other hand, the Brjuno function associated to the even continued fraction differs from the classical Brjuno function by a sum of a H\"older continuous function and a Brjuno-type function associated to the odd-odd continued fraction, introduced in the study of the best approximations of the form odd/odd.