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 $\mathsf{\text{A}}(\psi ,x)$ all the accumulation points of $\{\frac{1}{n}{\sum }_{j=0}^{n-1}\psi (T^{j}x):n\ge 1\}$. The Hausdorff dimensions of the sets $$\{x:\mathsf{\text{A}}(\psi ,x)\supset [a,b]\},\{x:\mathsf{\text{A}}(\psi ,x)=[a,b]\},\{x:\mathsf{\text{A}}(\psi ,x)\subset [a,b]\}$$ 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 $\psi$.