Dimension spectrum of digit frequency sets for beta-expansions

Abstract: For any beta-shift $(X_β,σ)$ on two symbols, i.e., the symbolic coding of the beta-map for $1&lt;β\leq2$, we give an exact formula for the Hausdorff dimension $\dim_{H} Λ<em>{α(t)}$ as a function of $t\in\mathbb{R}$, where $Λ</em>α$ denotes the frequency set of the digit $1$ defined by [Λα=\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 $λ<em>t$ of the transfer operator $\mathcal{L}_t$ with the potential $tχ</em>{C_1}$ for $t\in\mathbb{R}$, where $χ<em>{C</em>{1}}$ denotes the indicator function of the cylinder set $$C_1={(x_i)<em>{i=1}^\infty\in X</em>β; x_1=1}$$. 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 $1<β<2$.