Pisot数的周期性β-展开
Periodic intermediate $β$-expansions of Pisot numbers
AI总结：
本文研究了Pisot数的周期性β-展开性质，证明了在特定条件下β-移位具有子移位有限型性质，并推广了Schmidt的经典结果。
摘要：
子移位有限型性质（也称为马尔可夫性质）在动力系统中无处不在，最简单且最广泛研究的动力系统类是β-移位，即形式为T_{β, α}：x ↦ βx + α mod 1，作用在[-α/(β-1), (1-α)/(β-1)]上的变换，其中(β, α)∈Δ固定，其中Δ={ (β, α) ∈ R² : β∈(1,2) 且 0 ≤ α ≤ 2-β}。最近，李等人（Proc. Amer. Math. Soc. 147(5): 2045-2055, 2019）证明了具有子移位有限型性质的(β, α)集合在参数空间Δ中是稠密的。在此，他们提出了以下问题。给定一个固定的β∈(1, 2)，它是第n个Perron数的根，是否存在一个稠密的α集合在纤维{β}×(0, 2-β)中，使得T_{β, α}具有子移位有限型性质？我们对Pisot数的一类情况给出了肯定回答。进一步，我们探讨当将子移位有限型性质替换为开始sofic性质（即子移位有限型的因子）时，这个问题是否成立。在此过程中，我们推广了Schmidt（Bull. London Math. Soc., 12(4): 269-278, 1980）的经典结果，从α=0的情况推广到α∈(0, 2-β)的情况。也就是说，我们研究了当β是Pisot数或第n个Pisot数的根时，T_{β, α}的最终周期点集合的结构。
英文摘要：
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.
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