Ultrametric spaces and the logarithmic ratio
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918255389573120 |
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| author | Movahedi-Lankarani, H. |
| author_facet | Movahedi-Lankarani, H. |
| contents | It is shown that if a compact metric space $(X, d)$ is bi-Hölder equivalent to an ultrametric space, then the logarithmic ratio $R(X,d)$ is finite. Conversely, if the logarithmic ratio $R(X,d)$ is finite and ${\A}^*_p (X) \ne \emptyset$ for some $p \in (1, \infty )$, then $(X, d)$ is bi-Hölder equivalent to an ultrametric space.
It is also shown that for any $s \in [0, \infty]$ there exists a compact countable metric space $(X, d)$ with a unique cluster point such that the logarithmic ratio $R(X, d)$ is equal $s$. Moreover, we prove a bi-Hölder embedding result for a certain class of compact totally disconnected metric spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16820 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ultrametric spaces and the logarithmic ratio Movahedi-Lankarani, H. General Topology It is shown that if a compact metric space $(X, d)$ is bi-Hölder equivalent to an ultrametric space, then the logarithmic ratio $R(X,d)$ is finite. Conversely, if the logarithmic ratio $R(X,d)$ is finite and ${\A}^*_p (X) \ne \emptyset$ for some $p \in (1, \infty )$, then $(X, d)$ is bi-Hölder equivalent to an ultrametric space. It is also shown that for any $s \in [0, \infty]$ there exists a compact countable metric space $(X, d)$ with a unique cluster point such that the logarithmic ratio $R(X, d)$ is equal $s$. Moreover, we prove a bi-Hölder embedding result for a certain class of compact totally disconnected metric spaces. |
| title | Ultrametric spaces and the logarithmic ratio |
| topic | General Topology |
| url | https://arxiv.org/abs/2512.16820 |