Capacity dimension of the Brjuno set in ${\mathbb{C}}^n$

Abstract

In this work, we prove that the complement of the Brjuno set in ${\mathbb{C}}^n$ has zero $C_{\sigma }$-capacity with respect to the kernel $k_{\sigma }(z,\xi )=\Vert z-\xi {\Vert }^{-2n+2}|\log \Vert z-\xi \Vert {|}^{\sigma }$ for any $\sigma >n$. In particular, it follows that it has zero $h_{\delta }$-Hausdorff measure with respect to the $h_{\delta }(t)=t^{2n-2}|\log t{|}^{-\delta }$, for any $\delta >n+1$. This generalizes a previous result of Sadullaev and the second author in dimension one to higher dimensions.