Generalized Brjuno functions associated to α-continued fractions

For 0 ≤ α ≤ 1 given, we consider the one-parameter family of α-continued fraction maps, which include the Gauss map (α = 1), 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 we associate a generalized Brjuno function B_{(α, u)} (x). 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. 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 ≠ 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.