On the points without universal expansions

Abstract

Let $1<β<2$. Given any $x\in[0, (β-1)^{-1}]$, a sequence $(a_n)\in\{0,1\}^{\mathbb{N}}$ is called a $β$-expansion of $x$ if $x=\sum_{n=1}^{\infty}a_nβ^{-n}.$ 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 $β$-expansion of $x$. Sidorov \cite{Sidorov2003}, Dajani and de Vries \cite{DajaniDeVrie} proved that given any $1<β<2$, then Lebesgue almost every point has uncountably many universal expansions. In this paper we consider the set $V_β$ of points without universal expansions. For any $n\geq 2$, let $β_n$ be the $n$-bonacci number satisfying the following equation: $β^n=β^{n-1}+β^{n-2}+\cdots +β+1.$ Then we have $\dim_{H}(V_{β_n})=1$, 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}.