Complex Brjuno functions

The Brjuno function arises naturally in the study of one--dimensional analytic small divisors problems. It belongs to $\hbox{BMO}({\Bbb T}^{1})$ and it is stable under H\"older perturbations. It is related to the size of Siegel disks by various rigorous and conjectural results. In this work we show how to extend the Brjuno function to a holomorphic function on ${\Bbb H}/{\Bbb Z}$, the complex Brjuno function. This has an explicit expression in terms of a series of transformed dilogarithms under the action of the modular group. The extension is obtained using a complex analogue of the continued fraction expansion of a real number. Since our method is based on the use of hyperfunctions it applies to less regular functions than the Brjuno function and it is quite general. We prove that the harmonic conjugate of the Brjuno function is bounded. Its trace on ${\Bbb R}/{\Bbb Z}$ is continuous at all irrational points and has a jump of $\pi/q$ at each rational point $p/q\in {\Bbb Q}$.