For 06 61 given, we consider the modiΓΏed continued fraction expansion of the real number x deΓΏned by x = a0 + 0x0; a0 β Z, and, x -1 n-1 = an + nx n; an β N for nΒΏ0, where -16 nx nΒ‘ ; n = Β±1, for nΒΏ0, with xnΒΏ0. The usual (Gaussian) case is = 1, whereas = 1 2 is the continued fraction to the nearest integer. The Brjuno function B
. These functions were introduced by Yoccoz in the = 1 and 1 2 cases, in his work on the holomorphic conjugacy to a rotation, of an analytic map with an indi erent ΓΏxed point. We will review some properties of these functions, namely, all these functions are 1-periodic, and belong to L p Ioc (R), for 16pΒ‘β, and also to the space BMO(T). In this communication, we will mainly report on some of the technical tools related to the continued fraction expansions required by the above mentioned results. These results deal with the growth of the denominator of the reduced fractions pn=qn of the above continued fraction expansion, which gives the maximal error rate of approximation, the relation between these continued fraction and the usual Gaussian case, and ΓΏnally the invariant density, generalising the classical result of Gauss for the usual case.