On the points without universal expansions

Let $1<\beta <2$. Given any $x\in [0,(\beta -1{)}^{-1}]$, a sequence $(a_n)\in \{0,1{\}}^{\mathbb{N}}$ is called a $\beta$-expansion of $x$ if $x={\sum }_{n=1}^{\mathrm{\infty }}{a}_n{\beta }^{-n}\mathrm{.}$ For any $k\ge 1$ and any $(b_1b_2\cdots b_k)\in{0,1}^{k}$,ifthereexistssome$k_0$suchthat$a_{k_0+1}a_{k_0+2}\cdots a_{k_0+k}=b_1b_2\cdots b_k$,thenwecall$(a_n)$ a universal $\beta$-expansion of $x$. Sidorov \cite{Sidorov2003}, Dajani and de Vries \cite{DajaniDeVrie} proved that given any $1<\beta <2$, then Lebesgue almost every point has uncountably many universal expansions. In this paper we consider the set $V_{\beta }$ of points without universal expansions. For any $n\ge 2$, let ${\beta }_n$ be the $n$-bonacci number satisfying the following equation: ${\beta }^n={\beta }^{n-1}+{\beta }^{n-2}+\cdots +\beta +1.$ Then we have ${\dim }_H(V_{{\beta }_n})=1$, where ${\dim }_H$ denotes the Hausdorff dimension. Similar results are still available for some other algebraic numbers. As a corollary, we give some results of the Hausdorff dimension of the survivor set generated by some open dynamical systems. This note is another application of our paper \cite{KarmaKan}.