Moussa P., Marmi S.
Diophantine Conditions and Real or Complex Brjuno Functions
(234K, PostScript)
ABSTRACT. The continued fraction
expansion of the real number $x=a_0+x_0$,
$a_0\in {\bbbz}$, is given by $0\le x_n<1$,
$x_{n}^{-1}=a_{n+1}+ x_{n+1}$, $a_{n+1}\in {\bbbn}$,
for $n\ge 0$. The Brjuno function is then
$B(x)=\sum_{n=0}^{\infty}x_0x_1\ldots
x_{n-1}\ln(x_n^{-1})$, and the number $x$ satisfies
the Brjuno diophantine condition whenever $B(x)$ is bounded.
Invariant circles under a complex
rotation persist when the map is analytically
perturbed, if and only if
the rotation number satisfies the Brjuno condition,
and the same holds for invariant circles in the semi-standard and
standard maps cases.
In this lecture, we will review some properties of the Brjuno function,
and give some generalisations related to familiar
diophantine conditions. The Brjuno function is highly singular
and takes value $+\infty$ on a dense set including
rationals. We present a regularisation leading to
a complex function holomorphic in the upper half plane.
Its imaginary part tends to the Brjuno function on
the real axis, the real part remaining bounded, and we also
indicate its transformation under the modular
group.