Dimension spectrum of digit frequency sets for beta-expansions

TL;DR

This paper derives an explicit formula for the Hausdorff dimension of digit frequency sets in beta-expansions, using spectral analysis of transfer operators, and explores related distribution functions generalizing classical singular functions.

Contribution

It provides an exact formula for the Hausdorff dimension of digit frequency sets in beta-shifts, connecting spectral properties of transfer operators with fractal dimensions.

Findings

Explicit Hausdorff dimension formula for digit frequency sets.

Spectral decomposition of transfer operators for beta-shifts.

Distribution functions related to eigenmeasures generalize classical singular functions.

Abstract

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 $1<\beta<2$.