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}.
Categories: math.DS math.MG math.NT