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arXiv · 2505.18801

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

Nurali Akramov·Karim Rakhimov
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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.

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Nurali Akramov, Karim Rakhimov. 2025-05-24. Capacity dimension of the Brjuno set in $\mathbb{C}^n$. https://arxiv.org/abs/2505.18801

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