Dimension spectrum of digit frequency sets for beta-expansions
Shintaro Suzuki
TL;DR
The paper addresses the dimension spectrum of digit-frequency sets in beta-expansions by leveraging thermodynamic formalism for symbolic beta-shifts. It develops an explicit spectral analysis of the transfer operator $\mathcal{L}_t$ with potential $tχ_{C_1}$, including the leading eigenvalue $λ_t$, eigenfunction $h_t$, and eigenmeasure $ν_t$, and constructs a determinant-like object $Φ_t$ to obtain a full spectral decomposition. A conditional variational principle then yields exact formulas for the Hausdorff dimension $\dim_H Λ_{α(t)}$ in terms of the equilibrium state $μ_t$, with $α(t)=∫ χ_{C_1}dμ_t$ and $P(t)=\log λ_t$. The work further explores the analytic structure of the pressure, the distribution function of the eigenmeasure, and a Hata–Yamaguchi-type generalization (Taiagi-like) for $1<β<2$, including explicit examples in multinacci/Parry cases. Overall, the results provide sharp, computable descriptions of fractal dimension spectra for beta-expansions and connect fractal geometry with transfer-operator dynamics in a precise way.
Abstract
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 $1<β<2$.