Dimension spectrum of digit frequency sets for beta-expansions

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 $1<β<2$.

💡 Research Summary

The paper studies the symbolic dynamics of the β‑shift (Xβ,σ) for 1 < β ≤ 2, focusing on the set Λα of sequences in which the digit “1’’ appears with asymptotic frequency α. By introducing a temperature‑like parameter t∈ℝ and the potential ϕt = t·χC1 (the indicator of the cylinder where the first symbol is 1), the authors define a transfer operator Lt acting on a Banach space of bounded‑variation functions. They prove that Lt is quasi‑compact: all spectral values with modulus larger than a radius rt are isolated eigenvalues of finite multiplicity. For each isolated eigenvalue λ with |λ| > rt they give explicit formulas for an eigenfunction and an eigenfunctional (Theorems 3.6 and 3.3). In particular the leading eigenvalue λt > 1 is simple, and the associated eigenfunction ht and eigenmeasure νt yield a unique equilibrium state μt = ht·νt for the potential t·χC1.

Using the thermodynamic formalism, they show that the pressure function P(t)=log λt is real‑analytic and that its derivative satisfies P′(t)=∫χC1 dμt. The conditional variational principle then gives an exact expression for the Hausdorff dimension of the frequency set: \