Multifractal aspects of $\alpha$-expansions

In \cite{[NE]} we introduce $\alpha$-expansions a real numbers in $(0,1]$, given by $$\sum _{i=1}^{\mathrm{\infty }}(\alpha -1{)}^{i-1}{\alpha }^{-(d_1+\cdots +d_i)}$$ with $\alpha >1$ and $d_i\in \mathbb{N}$ and discuss ergodic theoretical and dimension theoretical aspects of this expansions. In this sequel we study mutifractal aspects of this expansions.