  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
$x=\sum_{n=1}^{\infty}a_n\beta^{-n}.$ For any $k\geq 1$ and any $(b_1b_2\cdots
b_k)\in\{0,1\}^{k}$, if there exists some $k_0$ such that
$a_{k_0+1}a_{k_0+2}\cdots a_{k_0+k}=b_1b_2\cdots b_k$, then we call $(a_n)$ 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 $n\geq 2$, 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}.
