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 $psi:[0,1]mapstomathbb{R}$ be a continuous function. Denote by $textsf{A}(psi,x)$ all the accumulation points of $Big{frac{1}{n}sum_{j=0}^{n-1}psi(T^jx): nge 1Big}$. 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 $psi$.