Multifractal analysis of the divergence points of Birkhoff averages in $beta$-dynamical systems

This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of $\beta$-expansions. More precisely, let $([0,1),T_{\beta})$ be the $\beta$-dynamical system for a general $\beta>1$ and $\psi:[0,1]\mapsto\mathbb{R}$ be a continuous function. Denote by $\textsf{A}(\psi,x)$ all the accumulation points of $\Big\{\frac{1}{n}\sum_{j=0}^{n-1}\psi(T^jx): n\ge 1\Big\}$. The Hausdorff dimensions of the sets $$\Big\{x:\textsf{A}(\psi,x)\supset[a,b]\Big\},\ \ \Big\{x:\textsf{A}(\psi,x)=[a,b]\Big\}, \ \Big\{x:\textsf{A}(\psi,x)\subset[a,b]\Big\}$$ i.e., the points for which the Birkhoff averages of $\psi$ do not exist but behave in a certain prescribed way, are determined completely for any continuous function $\psi$.