Periodic points in the $β$-transformation with a hole at 0
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913733688688640 |
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| author | Bi, Yuzheng |
| author_facet | Bi, Yuzheng |
| contents | For $β\in(1,2]$ let $T_β: [0,1)\to[0,1); x\mapsto βx\pmod 1$. In this paper we study the periodic points in the open dynamical system $([0,1), T_β)$ with a hole $[0,t)$. For $p\in\mathbb{N}$ we characterize the largest $t$, denoted by $S_β(p)$, in which the survivor set $K_β(t)$ has a periodic point of smallest period $p$. More precisely, we give precise formulae for this critical value $S_β(p)$ when $β=2$, $β=\frac{1+\sqrt{5}}{2}$ and $β$ being the tribonacci number. We show that for $β=2$ the critical value $S_2(p)$ converges to $1/2$ as $p\to \infty$. When $β=\frac{1+\sqrt{5}}{2}$, the critical value $S_β(p)\to \frac{1}{β^3-β}$. While $β$ is the tribinacci number, the critical value $S_β(p)\to \frac{β^{2}+1}{β^4-β}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10039 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Periodic points in the $β$-transformation with a hole at 0 Bi, Yuzheng Dynamical Systems For $β\in(1,2]$ let $T_β: [0,1)\to[0,1); x\mapsto βx\pmod 1$. In this paper we study the periodic points in the open dynamical system $([0,1), T_β)$ with a hole $[0,t)$. For $p\in\mathbb{N}$ we characterize the largest $t$, denoted by $S_β(p)$, in which the survivor set $K_β(t)$ has a periodic point of smallest period $p$. More precisely, we give precise formulae for this critical value $S_β(p)$ when $β=2$, $β=\frac{1+\sqrt{5}}{2}$ and $β$ being the tribonacci number. We show that for $β=2$ the critical value $S_2(p)$ converges to $1/2$ as $p\to \infty$. When $β=\frac{1+\sqrt{5}}{2}$, the critical value $S_β(p)\to \frac{1}{β^3-β}$. While $β$ is the tribinacci number, the critical value $S_β(p)\to \frac{β^{2}+1}{β^4-β}$. |
| title | Periodic points in the $β$-transformation with a hole at 0 |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2503.10039 |