Capacity dimension of the Brjuno set (Q2788640)

The authors study the capacity dimension of the Brjuno set \(\mathcal{B}\). The set \(\mathcal{B}\) consists of the irrational numbers whose partial fraction expansion satisfies the condition \(\sum\limits_{n=1}^{\infty}\frac{\ln Q_{n+1}}{Q_{n}}<\infty\), where \(Q_{n}\) is associated to the \(n^{th}\) stage partial fraction \(\frac{P_{n}}{Q_{n}}\) of the Brjuno number \(\alpha\). The set \(\mathcal{B}\) of Brjuno numbers arises in connection with the problem of linearization of holomorphic germs. This problem is important in the dynamics of functions in the complex plane in a neighborhood of a fixed point. Mainly, it shows that the complement \(\mathbb{R}\backslash \mathcal{B}\) of the Brjuno set has zero \(C_{\sigma}\)-capacity with respect to the kernel \(k_{\sigma}(z,\xi)=|\ln |z-\xi | |^{\sigma}\), \(\sigma >2\).