Hausdorff dimension of specification for the $(α,β)$-shifts
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915431836549120 |
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| author | Takahasi, Hiroki |
| author_facet | Takahasi, Hiroki |
| contents | Specification is an important concept in dynamical systems introduced by Bowen. Schmeling proved that the set of $β>1$ such that the corresponding $β$-shift has specification is of Hausdorff dimension $1$. Hu et al. proved that the set of $β>1$ such that the corresponding $(-β)$-shift has specification is of Hausdorff dimension $1$. We show that the set of $(α,β)\in[0,1)\times(1,\infty)$ such that the corresponding $(α,β)$-shift has specification is of Hausdorff dimension $2$. A new difficulty is a simultaneous control of two critical symbol sequences that determine the ambient shift space. We achieve this by taking intersections of two thick Cantor sets in parameter space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_04345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension of specification for the $(α,β)$-shifts Takahasi, Hiroki Dynamical Systems Number Theory Specification is an important concept in dynamical systems introduced by Bowen. Schmeling proved that the set of $β>1$ such that the corresponding $β$-shift has specification is of Hausdorff dimension $1$. Hu et al. proved that the set of $β>1$ such that the corresponding $(-β)$-shift has specification is of Hausdorff dimension $1$. We show that the set of $(α,β)\in[0,1)\times(1,\infty)$ such that the corresponding $(α,β)$-shift has specification is of Hausdorff dimension $2$. A new difficulty is a simultaneous control of two critical symbol sequences that determine the ambient shift space. We achieve this by taking intersections of two thick Cantor sets in parameter space. |
| title | Hausdorff dimension of specification for the $(α,β)$-shifts |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2508.04345 |