论文标题
周期性的中级$β$ - PISOT数字的扩张
Periodic intermediate $β$-expansions of Pisot numbers
论文作者
论文摘要
有限类型属性（也称为Markov属性）的子班次在动态系统中无处不在，最简单，最广泛研究的动态系统是$β$ - 缩换，即形式的$ t_ {β{β，α} \α} \ colon x \ colon x \ colon x \ mapstoβX +βx +α\ bmmodα\ bmmod/1} $ [1} $ [1} $ [1} $ [1} （1-α）/（β-1）] $，其中$（β，α）\在δ$中是固定的，其中$δ= \ {（β，α，α）\ in \ Mathbb {r}^{2} {2} \ colonβ\ in（1,2）\ in（1,2）\; \ text {and} \; 0 \leqα\ leq 2-β\} $。最近，Li等人表明了这一点。 （Proc。Amer。Math。Soc。147（5）：2045-2055，2019），$（β，α）$的集合，使得$ t_ {β，α} $具有有限类型的亚缩影在参数空间$δ$中是密集的。在这里，他们提出了以下问题。给定（1，2）$的固定$β\，这是perron号的$ n $ th root，是否存在纤维$ \ {β\} \ times（0，2-β）$中的密集的$α$，那么$ t_ {β，α} $具有有限类型的subshift opshift operite typle属性？ 我们以一类PISOT数字为正面的问题。此外，我们调查了这个问题在通过sofic的属性替换有限类型属性的子迁移时是否存在（这是有限的子迁移的因素）。为此，我们概括了Schmidt的经典结果（Bull。LondonMath。Soc。，12（4）：269-278，1980），$α= 0 $ to $α\ in（0，2 -β）$。也就是说，当$β$是pisot号时，当$β$是$β$是pisot号的$ n $ th root时，我们检查了$ t_ {β，α} $的最终周期点的结构。
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 $β$-shifts, namely transformations of the form $T_{β, α} \colon x \mapsto βx + α\bmod{1}$ acting on $[-α/(β- 1), (1-α)/(β- 1)]$, where $(β, α) \in Δ$ is fixed and where $Δ= \{ (β, α) \in \mathbb{R}^{2} \colon β\in (1,2) \; \text{and} \; 0 \leq α\leq 2-β\}$. Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045-2055, 2019), that the set of $(β, α)$ such that $T_{β, α}$ has the subshift of finite type property is dense in the parameter space $Δ$. Here, they proposed the following question. Given a fixed $β\in (1, 2)$ which is the $n$-th root of a Perron number, does there exists a dense set of $α$ in the fiber $\{β\} \times (0, 2- β)$, so that $T_{β, α}$ 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 $α= 0$ to the case when $α\in (0, 2 - β)$. That is, we examine the structure of the set of eventually periodic points of $T_{β, α}$ when $β$ is a Pisot number and when $β$ is the $n$-th root of a Pisot number.
