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arXiv 2602.06368math.DS
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β展开中数字频率集的维度谱

Dimension spectrum of digit frequency sets for beta-expansions

Tokyo Gakugei University（东京学艺大学）

Shintaro Suzuki

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35pages, no figures, revised version
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AI总结：

本文研究了β展开中数字频率集的维度谱，通过转移算子的特征函数和特征函数推导出Hausdorff维度的精确公式，并推广了Takagi函数。

摘要：

对于任何β移位$(X_β,σ)$，即$1<β\leq2$的β映射的符号编码，我们给出了Hausdorff维度$\dim_{H} Λ_{α(t)}$作为$t\in\mathbb{R}$函数的精确公式，其中$Λ_α$定义为$\Lambda_α=\Biggl\{(x_i)_{i=1}^\infty\in X_β;\\ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}x_i=α\Biggr\}$对于$α\in[0,1]$，而$α(t)$是与$1$的准贪心展开相关的显式函数。该公式来源于对应于转移算子$\mathcal{L}_t$的主导特征值$λ_t$的特征函数和特征函数的显式公式，其中$χ_{C_1}$表示圆柱集$C_1=\{(x_i)_{i=1}^\infty\in X_β; x_1=1\}$的指示函数。这些公式不仅适用于主导特征值，还适用于$\mathcal{L}_t$的其他孤立特征值，从而得到$\mathcal{L}_t$的精确谱分解。作为进一步应用，我们研究了由编码映射的逆映射推动的特征测度的分布函数。我们证明，经过变量替换后，对于$β=2$，分布函数等于勒贝格奇异函数，并满足Hata-Yamaguchi公式的类比，从而为$1<β<2$的β展开中的Takagi函数提供了推广。

英文摘要：

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 $1<β<2$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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