Diophantine conditions and real or complex Brjuno functions

The continued fraction expansion of the real number $x=a_0+x_0, a_0\in {\ZZ},$ is given by $0\leq x_n<1, x_{n}^{-1}=a_{n+1}+ x_{n+1}, a_{n+1}\in {\NN},$ for $n\geq 0.$ The Brjuno function is then $B(x)=\sum_{n=0}^{\infty}x_0x_1... x_{n-1}\ln(x_n^{-1}),$ 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.