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 these operations 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 such a shift space, the n-ary interleaving operation produces a set whose elements combine individual elements ${\mathbf{x}}_i$, one from each $X_i$, by interleaving their symbol sequences in arithmetic progressions $(\mathrm{mod}n)$. We determine algebraic relations between decimation and interleaving operations and the shift map. 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 operations 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. 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.

本文研究有限字母上单侧移位空间的子集。这些子集出现在符号动力学，分形构造和数论中。我们研究了一个抽取操作族，该抽取操作以无限的算术级数提取符号序列的子序列，并显示这些操作在组成下是封闭的。我们还研究了n元交错操作家族，每个交错操作$ñ\ge \mathrm{1个}$。给定子集$X_0，X_{\mathrm{1个}}，。。。，X_{ñ-\mathrm{1个}}$在这种移位空间中，n元交织操作生成了一个集合，其元素组合了各个元素${\mathbf{X}}_{\mathrm{一世}}$，每个一个 $X_{\mathrm{一世}}$，通过以算术级数交错它们的符号序列 $（\mathrm{模}ñ）$。我们确定抽取和交织运算与移位图之间的代数关系。我们研究集合理论的n折闭合运算$X\mapsto X^{[ñ]}$，交错插入模数级别n的X的抽取。一个集合是n可分解的，如果$X=X^{[ñ]}$。所述Ñ倍交织操作是在组合物封闭，是幂等的。我们为每个X分配集合$\mathcal{ñ}（X）$ 所有价值 $ñ\ge \mathrm{1个}$ 为此 $X=X^{[ñ]}$。我们描述可能的集合$\mathcal{ñ}（X）$作为正整数的非空集，它们在可分解部分顺序下形成一个分布格，并在可分解性下向下封闭。我们显示了所有这种类型的集合都发生了。我们引入了一类弱移位稳定集，并表明该类在所有抽取，交织和移位操作下都是封闭的。我们研究了单侧移位的子集的两个熵概念，并表明它们对于弱移位稳定X一致，但通常可以不同。我们给出了一个弱移位稳定集的交织熵的单个熵公式。