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

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 Hölder 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 Hölder 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 Hölder 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.