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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.08116 |
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| _version_ | 1866916566273097728 |
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| author | Huang, Yan Wang, Zhiqiang |
| author_facet | Huang, Yan Wang, Zhiqiang |
| contents | We present a complete characterization of two different non-integers with the same Rényi-Parry measure. We prove that for two non-integers $β_1 ,β_2 >1$, the Rényi-Parry measures coincide if and only if $β_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in\mathbb{N}$ with $p\leq q$, and $β_2 = β_1 + 1$, which confirms a conjecture of Bertrand-Mathis in \cite[Section III]{Bertrand-1998}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The coincidence of Rényi-Parry measures for $β$-transformation Huang, Yan Wang, Zhiqiang Dynamical Systems Classical Analysis and ODEs 28D05 We present a complete characterization of two different non-integers with the same Rényi-Parry measure. We prove that for two non-integers $β_1 ,β_2 >1$, the Rényi-Parry measures coincide if and only if $β_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in\mathbb{N}$ with $p\leq q$, and $β_2 = β_1 + 1$, which confirms a conjecture of Bertrand-Mathis in \cite[Section III]{Bertrand-1998}. |
| title | The coincidence of Rényi-Parry measures for $β$-transformation |
| topic | Dynamical Systems Classical Analysis and ODEs 28D05 |
| url | https://arxiv.org/abs/2501.08116 |