Decimation and Interleaving Operations in One-Sided Symbolic Dynamics

Abstract

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.