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

TL;DR

This paper demonstrates that the complement of the Brjuno set in complex n-dimensional space has zero capacity and Hausdorff measure under specific kernels, extending previous one-dimensional results to higher dimensions.

Contribution

It generalizes the known one-dimensional capacity and measure results of the Brjuno set to higher-dimensional complex spaces.

Findings

Complement of the Brjuno set has zero $C_\sigma$-capacity for $\sigma>n$

It has zero $h_\delta$-Hausdorff measure for $\delta>n+1$

Extends previous dimension-one results to $\mathbb{C}^n$

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 $k_\sigma(z,\xi)=\|z-\xi\|^{-2n+2}|\log{\|z-\xi\||^{\sigma}}$ 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.