论文标题
单方面的符号动力学的拆卸和交织操作
Decimation and Interleaving Operations in One-Sided Symbolic Dynamics
论文作者
论文摘要
本文研究有限字母上的单侧移位空间的子集。这种子集出现在符号动力学，分形构造和数理论中。我们研究了一个分解操作的家族，这些家族在无限算术渐进术中提取符号序列的子序列，并表明它们在组成下是封闭的。我们还研究了一个由$ n $ ar的交织行动的家庭，每个$ n \ ge 1 $。给定的子集$ x_0，x_1，...，x_ {n-1} $的移位空间，$ n $ - ary交织的运算符会产生一个集合的集合，其元素组合单个元素$ {\ bf x} _i $，一个来自每个$ x_i $的元素，通过循环地将其符号序列循环到artmety $ x_i $中。我们确定拆卸与交织运算符与转移操作员之间的代数关系。我们研究设置理论$ n $ - 折叠式操作$ x \ mapsto x^{[n]} $，这交通划分了$ x $的模量级别$ n $。如果$ x = x^{[n]} $，一组为$ n $ -factorizable。 $ n $折的交织运算符在组成下关闭，并且是势力。对于每个$ x $，我们分配了所有值$ n \ ge 1 $的设置$ \ natercal {n}（x）$，其中$ x = x^{[n]} $。我们将可能的集合$ \ Mathcal {n}（x）$表征为非空整数集，它们在划分部分下形成分布晶格，并在划分性下向下关闭。我们证明了所有此类类型的集合。我们介绍了一类弱移动稳定的集合，并表明该类别在所有拆卸，交织和转移操作下都关闭。此类包括所有Shift-Invariant集合。我们研究了整个单方面偏移的子集的两个熵概念，并表明它们对于弱转移稳定的$ x $重合，但总体上可能是不同的。我们给出了一个在单个熵方面的弱移动稳定集相互作用的熵的公式。
This paper studies subsets of one-sided shift spaces on a finite alphabet. Such subsets arise in symbolic dynamics, in fractal constructions, and in number theory. We study a family of decimation operations, which extract subsequences of symbol sequences in infinite arithmetic progressions, and show they are closed under composition. We also study a family of $n$-ary interleaving operations, one for each $n \ge 1$. Given subsets $X_0, X_1, ..., X_{n-1}$ of the shift space, the $n$-ary interleaving operator produces a set whose elements combine individual elements ${\bf x}_i$, one from each $X_i$, by interleaving their symbol sequences cyclically in arithmetic progressions $(\bmod\,n)$. We determine algebraic relations between decimation and interleaving operators and the shift operator. We study set-theoretic $n$-fold closure operations $X \mapsto X^{[n]}$, which interleave decimations of $X$ of modulus level $n$. A set is $n$-factorizable if $X=X^{[n]}$. The $n$-fold interleaving operators are closed under composition and are idempotent. To each $X$ we assign the set $\mathcal{N}(X)$ of all values $n \ge 1$ for which $X= X^{[n]}$. We characterize the possible sets $\mathcal{N}(X)$ as nonempty sets of positive integers that form a distributive lattice under the divisibility partial order and are downward closed under divisibility. We show that all sets of this type occur. We introduce a class of weakly shift-stable sets and show that this class is closed under all decimation, interleaving, and shift operations. This class includes all shift-invariant sets. We study two notions of entropy for subsets of the full one-sided shift and show that they coincide for weakly shift-stable $X$, but can be different in general. We give a formula for entropy of interleavings of weakly shift-stable sets in terms of individual entropies.
