On points without universal expansions
Volume 184 / 2018
Abstract
Let $1 \lt \beta \lt 2$. 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 $(b_1\cdots b_k)\in\{0,1\}^{k}$, if there exists some $k_0$ such that $a_{k_0+1}a_{k_0+2}\cdots a_{k_0+k}=b_1\cdots b_k$, then we call $(a_n)$ a universal $\beta$-expansion of $x$. Sidorov (2003) and Dajani and de Vries (2007) proved that for any $1 \lt \beta \lt 2$, Lebesgue almost every point has uncountably many universal $\beta$-expansions. In this paper we consider the set $V_{\beta}$ of points without universal $\beta$-expansions. For any $n\geq 2$, let $\beta_n$ be the $n$-bonacci number, i.e., $\beta^n=\beta^{n-1}+\beta^{n-2}+\cdots +\beta+1.$ Then $\dim_{H}(V_{\beta_n})=1$, where $\dim_{H}$ denotes the Hausdorff dimension. Similar results are available for some other algebraic numbers. As a corollary, we give some results on the Hausdorff dimension of the survivor set generated by some open dynamical systems. This note is another application of our paper [Nonlinearity 30 (2017)].
