Marmi S., Moussa P., Yoccoz J.-C.
Complex Brjuno Functions
(997K, PostScript)
ABSTRACT. 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
$p/q\in {\Bbb Q}$.\par