Dimension spectrum of digit frequency sets for beta-expansions

For any beta-shift $(X_\beta,\sigma)$ on two symbols, i.e., the symbolic coding of the beta-map for $1<\beta\leq2$, we give an exact formula for the Hausdorff dimension $\dim_{H} \Lambda_{\alpha(t)}$ as a function of $t\in\mathbb{R}$, where $\Lambda_\alpha$ denotes the frequency set of the digit $1$ defined by $$\Lambda_\alpha=\Biggl\{(x_i)_{i=1}^\infty\in X_\beta;\ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}x_i=\alpha \Biggr\}$$ for $\alpha\in[0,1]$ and $\alpha(t)$ 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 $\lambda_t$ of the transfer operator $\mathcal{L}_t$ with the potential $t\chi_{C_1}$ for $t\in\mathbb{R}$, where $\chi_{C_{1}}$ denotes the indicator function of the cylinder set $C_1=\{(x_i)_{i=1}^\infty\in X_\beta; x_1=1\}$. 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 $\lambda_t$ 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 $\beta=2$ and satisfies an analogy of the Hata-Yamaguchi formula, which yields a generalization of the Takagi function for beta-expansions with the base $1<\beta<2$.