Generalized Brjuno functions associated to α-continued fractions

For 0 ≤ α ≤ 1 given, we consider the α-continued fraction expansion of a real number obtained by iterating the map Aα(x) = ̨̨ x − ˆ x + 1− α ̃ ̨̨ defined on the interval Iα = (0, ᾱ), with ᾱ = max(α, 1− α). These maps generalize the classical (Gauss) continued fraction map which corresponds to the choice α = 1, and include the nearest integer (α = 1/2) and byexcess (α = 0) continued fraction expansion. To each of these expansions and to each choice of a positive function u on the interval Iα we associate a generalized Brjuno function B(α,u)(x) = P∞ n=0 βn−1u(xα,n), where xα,n = Aα(xα,n−1) for all n ≥ 1, xα,0 = |x− [x+ 1− α]|, βα,n = xα,0 · · ·xα,n, βα,−1 = 1. When α = 1/2 or α = 1, and u(x) = − log(x), these functions were introduced by Yoccoz in his work on 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 α and we prove that the set of (α, u)-Brjuno numbers does not depend on the choice of α provided that α 6= 0. We then consider the case α = 0, u(x) = − log(x) and we prove that x is a Brjuno number (for α 6= 0) if and only if both x and −x are Brjuno numbers for α = 0. Subject Classification: Primary: 11J70; Secondary: 37F50