丢番图条件与实或复Brjuno函数
Diophantine conditions and real or complex Brjuno functions
摘要：
实数$x=a_0+x_0, a_0\in {\ZZ},$的连分数展开式为：$0\leq x_n<1, x_{n}^{-1}=a_{n+1}+ x_{n+1}, a_{n+1}\in {\NN},$ for $n\geq 0.$则Brjuno函数为$B(x)=\sum_{n=0}^{\infty}x_0x_1... x_{n-1}\ln(x_n^{-1}),$，且只要$B(x)$有界，数$x$满足Brjuno丢番图条件。当映射解析扰动时，复旋转下的不变圆持续存在，当且仅当旋转数满足Brjuno条件，并且在半标准和标准映射的情况下，不变圆也是如此。在这个讲座中，我们将回顾Brjuno函数的一些性质，并给出一些与熟悉的丢番图条件相关的概括。Brjuno函数是高度奇异的，并且在包含有理数的稠密集上取值$+\infty$。我们提出了一个正则化导致一个复杂的功能全纯的上半平面。它的虚部趋向于实轴上的Brjuno函数，实部保持有界，我们还指出了它在模群下的变换。
英文摘要：
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.
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