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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2603.14862 |
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| _version_ | 1866915865457328128 |
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| author | Huang, Yan Sun, Yun |
| author_facet | Huang, Yan Sun, Yun |
| contents | We study the negative beta transformations $T_{-β}:=-βx +\lfloorβx\rfloor+1$ for $x\in(0,1]$ and $β>1$. We present a complete characterization of pairs of dstinct non-integers with the same $T_{-β}$-invariant measure: for two non-integers $β_1 ,β_2 >1$, the invariant measures of negative $β$-transformation 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 $β_2 = β_1 + 1$. Furthermore, we show that $T_{-β}$ has matching property for all $β$ being generalized multinacci numbers. We also prove that the set of simple $-β$ numbers, whose $-β$-shifts are subshifts of finite type, is dense in the parameter interval $(1,\infty)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14862 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Negative $β$-transformations: invariant measures, subshifts of finite type and matching property Huang, Yan Sun, Yun Dynamical Systems Number Theory 37E05, 37B10 We study the negative beta transformations $T_{-β}:=-βx +\lfloorβx\rfloor+1$ for $x\in(0,1]$ and $β>1$. We present a complete characterization of pairs of dstinct non-integers with the same $T_{-β}$-invariant measure: for two non-integers $β_1 ,β_2 >1$, the invariant measures of negative $β$-transformation 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 $β_2 = β_1 + 1$. Furthermore, we show that $T_{-β}$ has matching property for all $β$ being generalized multinacci numbers. We also prove that the set of simple $-β$ numbers, whose $-β$-shifts are subshifts of finite type, is dense in the parameter interval $(1,\infty)$. |
| title | Negative $β$-transformations: invariant measures, subshifts of finite type and matching property |
| topic | Dynamical Systems Number Theory 37E05, 37B10 |
| url | https://arxiv.org/abs/2603.14862 |