Resumen de Generalized Brjuno functions associated to a-continued fractions

Laura Luzzi, Stefano Marmi, Hitoshi Nakada, Rie Natsui

• For 0   1 given, we consider the one-parameter family of -continued fraction maps, which include the Gauss map ( D 1), the nearest integer ( D 1=2) and by-excess ( D 0) continued fraction maps. To each of these expansions and to each choice of a positive function u on the interval I we associate a generalized Brjuno function B. ;u/.x/. When D 1=2 or D 1, and u.x/ D ..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 and we prove that the set of . ; u/-Brjuno numbers does not depend on the choice of provided that 6D 0. We then consider the case D 0, u.x/ D ..log.x/ and we prove that x is a Brjuno number (for 6D 0) if and only if both x and ..x are Brjuno numbers for D 0.