Dimension spectrum of digit frequency sets for beta-expansions

Abstract

For any beta-shift $(X_{\hat{I}}^2,\ddot{I}ƒ)$ on two symbols, i.e., the symbolic coding of the beta-map for $1<{\hat{I}}^2\le 2$, we give an exact formula for the Hausdorff dimension ${\dim }_H\hat{I}{›}_{\hat{I}\pm (t)}$ as a function of $t\in \mathbb{R}$, where $\hat{I}{›}_{\hat{I}}\pm$ 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 $\hat{I}\pm \in [0,1]$ and $\hat{I}\pm (t)$ 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 $\hat{I}{»}_t$ of the transfer operator ${\mathcal{L}}_t$ with the potential $t\ddot{I}{‡}_{C_1}$ for $t\in \mathbb{R}$, where $\ddot{I}{‡}_{C_1}$ denotes the indicator function of the cylinder set $C_1=\{(x_i{)}_{i=1}^{\infty }\in X_{\hat{I}}^2;x_1=1\}$. 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 $\hat{I}{»}_t$ 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 ${\hat{I}}^2=2$ and satisfies an analogy of the Hata-Yamaguchi formula, which yields a generalization of the Takagi function for beta-expansions with the base $1<{\hat{I}}^2<2$.

33pages, no figure