论文标题
独特的双基础扩展
Unique double base expansions
论文作者
论文摘要
对于两个真实基础$ q_0，q_1> 1 $，我们考虑$ \ sum_ {k = 1}^{\ infty} i_k/（q_ {i_1} q_ {i_2} {i_2} \ cdots q_ {i_k}）$ with $ i_k \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in， $（q_0，q_1）$ - 扩展。序列$（i_k）$称为唯一$（q_0，q_1）$ - 扩展如果所有其他序列都具有不同的值，则为$（q_0，q_1）$ - 扩展，以及一组唯一的$（q_0，q_1，q_1）$ - 扩展由$ u_ {q_0，q_1} $表示。在特殊情况下，$ q_0 = q_1 = q $，如果$ q $低于黄金比率，则设置$ u_ {q，q} $是微不足道的，如果$ q $高于komornik-loreti常数，则无数。与非客气$ u_ {q_1} $的曲线分隔基本对$（q_0，q_1）$，带有微不足道的$ u_ {q_1} $，是函数$ \ nathcal {g}（q_0）$的曲线。同样，曲线分离对$（q_0，q_1）$，可计数$ u_ {q_1，q_1} $与那些具有无数$ u_ {q_1} $的人的图形是函数$ \ nathcal {k}（k}（q_0）$的图形，我们称之为我们称之为generalized komornik-loreti nonstant。我们表明，这两条曲线在$ q_0 $和$ q_1 $中是对称的，$ \ mathcal {g} $和$ \ Mathcal {k} $是连续的，严格降低，因此几乎在$（1，\ infty）$的$（1，\ infty）$上差异，以及$ q_0 $ q_0 $ q_0 $ q_0 $ q_0 $ q_0 $ q_0 $ q_0 $ q_0 $ $ \ MATHCAL {G}（Q_0）= \ Mathcal {K}（Q_0）$为零。我们使用$ \ Mathcal {g}（q_0）$和$ \ Mathcal {k}（q_0）$的公式，用于所有$ q_0> 1 $，使用二进制二进制乘坐词法间隔的特征是微不足道的，可计数的，可计数，无可计算，无需零词，并且与阳性呈阳性无关。我们的特征在包括Sturmian和Thue-Morse序列在内的$ s $ - 多序列方面比Labarca和Moreira（2006）以及Glendinning和Sidorov（2015）更简单，并且与其他开放的动态系统也相关。
For two real bases $q_0, q_1 > 1$, we consider expansions of real numbers of the form $\sum_{k=1}^{\infty} i_k/(q_{i_1}q_{i_2}\cdots q_{i_k})$ with $i_k \in \{0,1\}$, which we call $(q_0,q_1)$-expansions. A sequence $(i_k)$ is called a unique $(q_0,q_1)$-expansion if all other sequences have different values as $(q_0,q_1)$-expansions, and the set of unique $(q_0,q_1)$-expansions is denoted by $U_{q_0,q_1}$. In the special case $q_0 = q_1 = q$, the set $U_{q,q}$ is trivial if $q$ is below the golden ratio and uncountable if $q$ is above the Komornik--Loreti constant. The curve separating pairs of bases $(q_0, q_1)$ with trivial $U_{q_0,q_1}$ from those with non-trivial $U_{q_0,q_1}$ is the graph of a function $\mathcal{G}(q_0)$ that we call generalized golden ratio. Similarly, the curve separating pairs $(q_0, q_1)$ with countable $U_{q_0,q_1}$ from those with uncountable $U_{q_0,q_1}$ is the graph of a function $\mathcal{K}(q_0)$ that we call generalized Komornik--Loreti constant. We show that the two curves are symmetric in $q_0$ and $q_1$, that $\mathcal{G}$ and $\mathcal{K}$ are continuous, strictly decreasing, hence almost everywhere differentiable on $(1,\infty)$, and that the Hausdorff dimension of the set of $q_0$ satisfying $\mathcal{G}(q_0)=\mathcal{K}(q_0)$ is zero. We give formulas for $\mathcal{G}(q_0)$ and $\mathcal{K}(q_0)$ for all $q_0 > 1$, using characterizations of when a binary subshift avoiding a lexicographic interval is trivial, countable, uncountable with zero entropy and uncountable with positive entropy respectively. Our characterizations in terms of $S$-adic sequences including Sturmian and the Thue--Morse sequences are simpler than those of Labarca and Moreira (2006) and Glendinning and Sidorov (2015), and are relevant also for other open dynamical systems.
