Capacity dimension of the Brjuno set in $\mathbb{C}^n$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916756820328448 |
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| author | Akramov, Nurali Rakhimov, Karim |
| author_facet | Akramov, Nurali Rakhimov, Karim |
| contents | In this work, we prove that the complement of the Brjuno set in $\mathbb{C}^n$ has zero $C_σ$-capacity with respect to the kernel $k_σ(z,ξ)=\|z-ξ\|^{-2n+2}|\log{\|z-ξ\||^σ}$ for any $σ>n$. In particular, it follows that it has zero $h_δ$-Hausdorff measure with respect to the $h_δ(t)=t^{2n-2}|\log{t}|^{-δ}$, for any $δ>n+1$. This generalizes a previous result of Sadullaev and the second author in dimension one to higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_18801 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Capacity dimension of the Brjuno set in $\mathbb{C}^n$ Akramov, Nurali Rakhimov, Karim Complex Variables In this work, we prove that the complement of the Brjuno set in $\mathbb{C}^n$ has zero $C_σ$-capacity with respect to the kernel $k_σ(z,ξ)=\|z-ξ\|^{-2n+2}|\log{\|z-ξ\||^σ}$ for any $σ>n$. In particular, it follows that it has zero $h_δ$-Hausdorff measure with respect to the $h_δ(t)=t^{2n-2}|\log{t}|^{-δ}$, for any $δ>n+1$. This generalizes a previous result of Sadullaev and the second author in dimension one to higher dimensions. |
| title | Capacity dimension of the Brjuno set in $\mathbb{C}^n$ |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2505.18801 |