Generalized Brjuno functions associated to $\alpha$-continued fractions

Abstract

For \alpha in the interval [0,1], we consider the one-parameter family of \alpha-continued fraction maps, which include the Gauss map (\alpha=1) and the nearest integer (\alpha=1/2) and by-excess (\alpha=0) continued fraction maps. To each of these expansions, and to each choice of a positive function u on the interval I_\alpha=(0,max(\alpha,1-\alpha)) we associate a generalized Brjuno function B_(\alpha,u)(x). For \alpha=1/2 or \alpha=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 \alpha and we prove that the set of (\alpha,u)-Brjuno numbers does not depend on the choice of \alpha provided that \alpha>0. We then consider the case \alpha=0, u(x)=-\log(x) and we prove that x is a Brjuno number (for \alpha> 0) if and only if both x and -x are Brjuno numbers for \alpha=0.