Continued fractions and Brjuno functions

摘要

For 0⩽α⩽1 given, we consider the modified continued fraction expansion of the real number x defined by x=a0+ε0x0,a0∈Z, and, xn−1−1=an+εnxn,an∈N for n>0, where α−1⩽εnxn<α,εn=±1, for n⩾0, with xn⩾0. The usual (Gaussian) case is α=1, whereas α=12 is the continued fraction to the nearest integer. The Brjuno function B(α)(x) is then defined by B(α)(x)=−ln(x0)−∑n=1∞x0x1⋯xn−1ln(xn). These functions were introduced by Yoccoz in the α=1 and 12 cases, in his work on the holomorphic conjugacy to a rotation, of an analytic map with an indifferent fixed point. We will review some properties of these functions, namely, all these functions are 1-periodic, and belong to LIocp(R), for 1⩽p<∞, 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 finally the invariant density, generalising the classical result of Gauss for the usual case.