Decimation and Interleaving Operations in One-Sided Symbolic Dynamics

TL;DR

This paper explores decimation and interleaving operations in one-sided symbolic dynamics, revealing their algebraic properties, closure characteristics, and effects on entropy in shift spaces.

Contribution

It introduces and characterizes the algebraic structure of decimation and interleaving operators, and studies their impact on shift space properties and entropy.

Findings

Decimation operations are closed under composition.

Interleaving operators are idempotent and closed under composition.

Sets of interleaving levels form a distributive lattice.

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.