Let $\alpha\in\mathbb R\setminus \mathbb Q$ and let $(p_n/q_n )_{n\ge 0}$ be the sequence of the convergents of its continued fraction expansion. A Brjuno number is an irrational number $\alpha$ such that $\sum_{n=0}^\infty q_n log q_{n+1} < +\infty$. The importance of Brjuno numbers comes from the study of analytic small divisors problems in dimension one: Yoccoz proved that they give the optimal arithmetical condition for the existence of linearizations. In a series of papers Moussa, Yoccoz and myself studied the associated Brjuno function from the point of view of its real ($L^p$ and BMO) regularity properties and we constructed its complex analytic extension to the upper half–plane. Both the real and the complex analysis systematically exploit its relationship with continued fractions and the cocycle relation. The harmonic conjugate of the Brjuno function is pretty remarkable: it is continuous at all irrationals and has a decreasing jump of $\pi/q$ at each rational number $p/q$.

In a more recent unifinished work with Izabela Petrykiewicz, motivated by her work on regularity properties of modular integrals, we also studied $k$-Brjuno functions. (Hopefully) I will also briefly report on this work in progress.