On the capacity dimensions of the Brjuno and Perez-Marco sets
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912496334405632 |
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| author | Akramov, Nurali Ashirov, Abduvahhob |
| author_facet | Akramov, Nurali Ashirov, Abduvahhob |
| contents | In this work, we prove that the complement of the Brjuno set $\mathcal{B}$ has a zero capacity with respect to the kernel $k^1_σ(z,ξ)=\ln^2{|z-ξ|}\left|\ln{\ln{\left(e+\frac{1}{|z-ξ|}\right)}}\right|^σ$ for any $σ> 2$. Similarly, the complement of the Perez-Marco set $\mathcal{PM}$ has a zero capacity with respect to the kernel $k^2_σ(z,ξ) = \ln^{2}{\ln\left(e+\frac{1} {\left| {z - ξ}\right|}\right)} \cdot\ln^σ{\ln\ln\left(e^3+\frac{1} {\left| {z - ξ}\right|}\right)}$ for any $σ>2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15980 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the capacity dimensions of the Brjuno and Perez-Marco sets Akramov, Nurali Ashirov, Abduvahhob Complex Variables In this work, we prove that the complement of the Brjuno set $\mathcal{B}$ has a zero capacity with respect to the kernel $k^1_σ(z,ξ)=\ln^2{|z-ξ|}\left|\ln{\ln{\left(e+\frac{1}{|z-ξ|}\right)}}\right|^σ$ for any $σ> 2$. Similarly, the complement of the Perez-Marco set $\mathcal{PM}$ has a zero capacity with respect to the kernel $k^2_σ(z,ξ) = \ln^{2}{\ln\left(e+\frac{1} {\left| {z - ξ}\right|}\right)} \cdot\ln^σ{\ln\ln\left(e^3+\frac{1} {\left| {z - ξ}\right|}\right)}$ for any $σ>2$. |
| title | On the capacity dimensions of the Brjuno and Perez-Marco sets |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2507.15980 |