Abstract: An irrational number is called a Brjuno number if the sum of the series of log(q_{n+1})/q_n converges, where q_n is the denominator of the n-th principal convergent of the regular continued fraction. The importance of Brjuno numbers comes from the study of analytic small divisors problems in dimension one. In 1988, J.-C. Yoccoz introduced the Brjuno function which characterizes the Brjuno numbers to estimate the size of Siegel disks.

In this talk, at first, we consider the k-Brjuno functions and the Wilton function which are related to the classical Brjuno function. We study their BMO (Bounded Mean Oscillation) regularity properties. Then we complexify the functional equations which they fulfill and we construct analytic extensions of the k-Brjuno and of the Wilton function to the upper half-plane. This is joint work with Stefano Marmi, Izabela Petrykiewicz, and Tanja I. Schindler.

Secondly, we introduce Brjuno-type functions associated with by-excess (backward), odd, even and odd-odd continued fractions. We see that the Brjuno numbers are characterized by the Brjuno-type functions. Then we deal with their Holder continuity properties. This is joint work with Stefano Marmi.