独特的双基地扩展

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引用次数: 0

摘要

Abstract 对于两个实基 \(q_0, q_1 > 1\), 我们考虑实数的展开形式为 \(\sum _{k=1}^\{infty } i_k/(q_{i_1}q_{i_2}\ldots q_{i_k})\) with \(i_k \in \{0,1\}\), 我们称之为 \((q_0,q_1)\)-展开。一个序列 \((i_k)\) 被称为一个唯一的 \((q_0,q_1)\)-扩展,如果所有其他序列都有不同的值((q_0,q_1)\)-展开式的集合表示为-展开的集合用 \(U_{q_0,q_1}\) 表示。在特殊情况下 \(q_0 = q_1 = q\) ,如果 q 低于黄金分割率,那么集合 \(U_{q,q}\) 是微不足道的;如果 q 高于科莫尼克-洛雷蒂常数,那么集合 \(U_{q,q}\) 是不可数的。将具有琐碎的\(U_{q_0,q_1}\)和具有非琐碎的\(U_{q_0,q_1}\)的基对分开的曲线是函数 \(\mathcal {G}(q_0)\) 的图形,我们称之为广义黄金比率。同样,将可数\(U_{q_0,q_1}\)和不可数\(U_{q_0,q_1}\)的数对\((q_0, q_1)\)分开的曲线是函数\(\mathcal {K}(q_0)\) 的图,我们称之为广义科莫尼克-洛雷蒂常数。我们证明这两条曲线在 \(q_0\) 和 \(q_1\) 上是对称的,\(\mathcal {G}\) 和 \(\mathcal {K}\) 是连续的、严格递减的,因此在 \((1,\infty )\) 上几乎无处不可变。并且满足 \(\mathcal {G}(q_0)=\mathcal {K}(q_0)\) 的 \(q_0\) 的集合的豪斯多夫维度为零。对于所有的 \(q_0 > 1\) ,我们给出了 \(\mathcal {G}(q_0)\) 和 \(\mathcal {K}(q_0)\) 的公式,分别使用了避开词典区间的二元子移位是微不足道的、可数的、熵为零的不可数的和熵为正的不可数的时的特征。我们用 S-adic 序列(包括 Sturmian 序列和 Thue-Morse 序列)进行的描述比 Labarca 和 Moreira (Ann Inst Henri Poincaré Anal Non Linéaire 23, 683-694, 2006) 以及 Glendinning 和 Sidorov (Ergod Theory Dyn Syst 35, 1208-1228, 2015) 的描述更简单,而且也适用于其他开放动力学系统。
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Unique double base expansions

Abstract

For two real bases \(q_0, q_1 > 1\) , we consider expansions of real numbers of the form \(\sum _{k=1}^{\infty } i_k/(q_{i_1}q_{i_2}\ldots q_{i_k})\) 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 (Ann Inst Henri Poincaré Anal Non Linéaire 23, 683–694, 2006) and Glendinning and Sidorov (Ergod Theory Dyn Syst 35, 1208–1228, 2015), and are relevant also for other open dynamical systems.

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