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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.06368 |
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| _version_ | 1866915779670179840 |
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| author | Suzuki, Shintaro |
| author_facet | Suzuki, Shintaro |
| contents | For any beta-shift $(X_β,σ)$ on two symbols, i.e., the symbolic coding of the beta-map for $1<β\leq2$, we give an exact formula for the Hausdorff dimension $\dim_{H} Λ_{α(t)}$ as a function of $t\in\mathbb{R}$, where $Λ_α$ denotes the frequency set of the digit $1$ defined by \[Λ_α=\Biggl\{(x_i)_{i=1}^\infty\in X_β;\ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}x_i=α\Biggr\}\] for $α\in[0,1]$ and $α(t)$ is an explicit function related to the quasi-greedy expansion of $1$. The formula is derived from explicit formulae for eigenfunctions and eigenfunctionals corresponding to the leading eigenvalue $λ_t$ of the transfer operator $\mathcal{L}_t$ with the potential $tχ_{C_1}$ for $t\in\mathbb{R}$, where $χ_{C_{1}}$ denotes the indicator function of the cylinder set $C_1=\{(x_i)_{i=1}^\infty\in X_β; x_1=1\}$. These formulae can be applied not only to the leading eigenvalue but also to the other isolated eigenvalues of $\mathcal{L}_t$, which yields a precise spectral decomposition of $\mathcal{L}_t$. As a further application, we investigate the distribution function of the push-forward of the eigenmeasure corresponding to $λ_t$ by the inverse map of the coding map. We show that the distribution function after a change of variables for $t$ is equal to the Lebesgue singular function if $β=2$ and satisfies an analogy of the Hata-Yamaguchi formula, which yields a generalization of the Takagi function for beta-expansions with the base $1<β<2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06368 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dimension spectrum of digit frequency sets for beta-expansions Suzuki, Shintaro Dynamical Systems For any beta-shift $(X_β,σ)$ on two symbols, i.e., the symbolic coding of the beta-map for $1<β\leq2$, we give an exact formula for the Hausdorff dimension $\dim_{H} Λ_{α(t)}$ as a function of $t\in\mathbb{R}$, where $Λ_α$ denotes the frequency set of the digit $1$ defined by \[Λ_α=\Biggl\{(x_i)_{i=1}^\infty\in X_β;\ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}x_i=α\Biggr\}\] for $α\in[0,1]$ and $α(t)$ is an explicit function related to the quasi-greedy expansion of $1$. The formula is derived from explicit formulae for eigenfunctions and eigenfunctionals corresponding to the leading eigenvalue $λ_t$ of the transfer operator $\mathcal{L}_t$ with the potential $tχ_{C_1}$ for $t\in\mathbb{R}$, where $χ_{C_{1}}$ denotes the indicator function of the cylinder set $C_1=\{(x_i)_{i=1}^\infty\in X_β; x_1=1\}$. These formulae can be applied not only to the leading eigenvalue but also to the other isolated eigenvalues of $\mathcal{L}_t$, which yields a precise spectral decomposition of $\mathcal{L}_t$. As a further application, we investigate the distribution function of the push-forward of the eigenmeasure corresponding to $λ_t$ by the inverse map of the coding map. We show that the distribution function after a change of variables for $t$ is equal to the Lebesgue singular function if $β=2$ and satisfies an analogy of the Hata-Yamaguchi formula, which yields a generalization of the Takagi function for beta-expansions with the base $1<β<2$. |
| title | Dimension spectrum of digit frequency sets for beta-expansions |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2602.06368 |