For αin the interval [0,1], we consider the one-parameter family of α-continued fraction maps, which include the Gauss map (α=1) and the nearest integer (α=1/2) and by-excess (α=0) continued fraction maps. To each of these expansions, and to each choice of a positive function u on the interval I_α=(0,max(α,1-α)) we associate a generalized Brjuno function B_(α,u)(x). For α=1/2 or α=1, and u(x)=-\log(x), these functions were introduced by Yoccoz in his work on the linearization of holomorphic maps. Their regularity properties, including BMO regularity and their extension to the complex plane, have been thoroughly investigated. We compare the functions obtained with different values of αand we prove that the set of (α,u)-Brjuno numbers does not depend on the choice of αprovided that α>0. We then consider the case α=0, u(x)=-\log(x) and we prove that x is a Brjuno number (for α> 0) if and only if both x and -x are Brjuno numbers for α=0.
The Brjuno functions of the by-excess, odd, even and odd-odd continued fractions and their regularity properties
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 gene…
