The orbit of a $β$-transformation cannot lie in a small interval
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910827639996416 |
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| author | Kwon, DoYong |
| author_facet | Kwon, DoYong |
| contents | For $β>1$, let $T_β:[0,1]\rightarrow [0,1)$ be the $β$-transformation. We consider an invariant $T_β$-orbit closure contained in a closed interval with diameter $1/β$, then define a function $Ξ(α,β)$ by the supremum of such $T_β$-orbit with frequency $α$ in base $β$, i.e., the maximum value in the $T_β$-orbit closure. This paper effectively determines the maximal domain of $Ξ$, and explicitly specifies all possible minimal intervals containing $T_β$-orbits.
For Addendum: The paper mentioned in the title is completed by this Addendum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10132 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The orbit of a $β$-transformation cannot lie in a small interval Kwon, DoYong Number Theory 11A63, 37B10, 68R15 For $β>1$, let $T_β:[0,1]\rightarrow [0,1)$ be the $β$-transformation. We consider an invariant $T_β$-orbit closure contained in a closed interval with diameter $1/β$, then define a function $Ξ(α,β)$ by the supremum of such $T_β$-orbit with frequency $α$ in base $β$, i.e., the maximum value in the $T_β$-orbit closure. This paper effectively determines the maximal domain of $Ξ$, and explicitly specifies all possible minimal intervals containing $T_β$-orbits. For Addendum: The paper mentioned in the title is completed by this Addendum. |
| title | The orbit of a $β$-transformation cannot lie in a small interval |
| topic | Number Theory 11A63, 37B10, 68R15 |
| url | https://arxiv.org/abs/2502.10132 |