On univoque Pisot numbers

We study Pisot numbers $\beta \in (1, 2)$ which are univoque, i.e., such that there exists only one representation of 1 as $1 = \sum_{n \geq 1} s_n\beta^{-n}$, with $s_n \in \{0, 1\}$. We prove in particular that there exists a smallest univoque Pisot number, which has degree 14. Furthermore we give the smallest limit point of the set of univoque Pisot numbers.Comment: Accepted by Mathematics of COmputatio

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 $x=\sum_{n=1}^{\infty}a_n\beta^{-n}.$ 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}.Comment: 15page

Periodic intermediate ββ-expansions of Pisot numbers

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are $\beta$-shifts, namely transformations of the form $T_{\beta, \alpha} \colon x \mapsto \beta x + \alpha \bmod{1}$ acting on $[-\alpha/(\beta - 1), (1-\alpha)/(\beta - 1)]$, where $(\beta, \alpha) \in \Delta$ is fixed and where $\Delta = \{ (\beta, \alpha) \in \mathbb{R}^{2} \colon \beta \in (1,2) \; \text{and} \; 0 \leq \alpha \leq 2-\beta \}$. Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045-2055, 2019), that the set of $(\beta, \alpha)$ such that $T_{\beta, \alpha}$ has the subshift of finite type property is dense in the parameter space $\Delta$. Here, they proposed the following question. Given a fixed $\beta \in (1, 2)$ which is the $n$-th root of a Perron number, does there exists a dense set of $\alpha$ in the fiber $\{\beta\} \times (0, 2- \beta)$, so that $T_{\beta, \alpha}$ has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the property of beginning sofic (that is a factor of a subshift of finite). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269-278, 1980) from the case when $\alpha = 0$ to the case when $\alpha \in (0, 2 - \beta)$. That is, we examine the structure of the set of eventually periodic points of $T_{\beta, \alpha}$ when $\beta$ is a Pisot number and when $\beta$ is the $n$-th root of a Pisot number.Comment: 13 pages, 1 figur