The Hausdorff measure and uniform fibre conditions for Barański carpet
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916495977611264 |
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| author | Qiu, Hua Wang, Qi |
| author_facet | Qiu, Hua Wang, Qi |
| contents | For a self-affine carpet $K$ of Barański, we establish a dichotomy:
$
\text{either }\quad 0<\mathcal{H}^{\dim_{\text{H}} K}(K)<+\infty \quad\text{ or } \quad\mathcal{H}^{\dim_{\text{H}} K}(K)=+\infty.
$
We introduce four types of uniform fibre condition for $K$: Hausdorff ($\textbf{u.f.H}$), Box ($\textbf{u.f.B}$), Assouad ($\textbf{u.f.A}$), and Lower ($\textbf{u.f.L}$), which are progressively stronger, with
$
\textbf{u.f.L} \Longrightarrow \textbf{u.f.A} \Longrightarrow \textbf{u.f.B} \Longrightarrow \textbf{u.f.H},
$
and each implication is strict. The condition $\textbf{u.f.H}$ serves as a criterion for the dichotomy. The remaining three conditions provide an equivalent characterization for the coincidence of any two distinct dimensions. The condition $\textbf{u.f.L}$ is also equivalent to the Ahlfors regularity of $K$. As a corollary, $\dim_{\text{H}} K=\dim_{\text{B}} K$ is sufficient but not necessary for $0<\mathcal{H}^{\dim_{\text{H}} K}(K)<+\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17018 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Hausdorff measure and uniform fibre conditions for Barański carpet Qiu, Hua Wang, Qi Classical Analysis and ODEs 28A80 For a self-affine carpet $K$ of Barański, we establish a dichotomy: $ \text{either }\quad 0<\mathcal{H}^{\dim_{\text{H}} K}(K)<+\infty \quad\text{ or } \quad\mathcal{H}^{\dim_{\text{H}} K}(K)=+\infty. $ We introduce four types of uniform fibre condition for $K$: Hausdorff ($\textbf{u.f.H}$), Box ($\textbf{u.f.B}$), Assouad ($\textbf{u.f.A}$), and Lower ($\textbf{u.f.L}$), which are progressively stronger, with $ \textbf{u.f.L} \Longrightarrow \textbf{u.f.A} \Longrightarrow \textbf{u.f.B} \Longrightarrow \textbf{u.f.H}, $ and each implication is strict. The condition $\textbf{u.f.H}$ serves as a criterion for the dichotomy. The remaining three conditions provide an equivalent characterization for the coincidence of any two distinct dimensions. The condition $\textbf{u.f.L}$ is also equivalent to the Ahlfors regularity of $K$. As a corollary, $\dim_{\text{H}} K=\dim_{\text{B}} K$ is sufficient but not necessary for $0<\mathcal{H}^{\dim_{\text{H}} K}(K)<+\infty$. |
| title | The Hausdorff measure and uniform fibre conditions for Barański carpet |
| topic | Classical Analysis and ODEs 28A80 |
| url | https://arxiv.org/abs/2411.17018 |