Generalized Brjuno functions associated to image-continued fractions Original Research Article

For 0≤α≤10≤α≤1 given, we consider the one-parameter family of αα-continued fraction maps, which include the Gauss map (α=1α=1), the nearest integer (α=1/2α=1/2) and by-excess (α=0α=0) continued fraction maps. To each of these expansions and to each choice of a positive function uu on the interval IαIα we associate a generalized Brjuno function B(α,u)(x)B(α,u)(x). When α=1/2α=1/2 or α=1α=1, and u(x)=−log(x)u(x)=−log(x), these functions were introduced by Yoccoz in his work on linearization of holomorphic maps. We compare the functions obtained with different values of αα and we prove that the set of (α,u)(α,u)-Brjuno numbers does not depend on the choice of αα provided that α≠0α≠0. We then consider the case α=0α=0, u(x)=−log(x)u(x)=−log(x) and we prove that xx is a Brjuno number (for α≠0α≠0) if and only if both xx and −x−x are Brjuno numbers for α=0α=0.