Decimation and interleaving operations in one-sided symbolic dynamics

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 >= 1. Given subsets X-0, X-1, ..., Xn-1 of such a shift space, the n-ary interleaving operation produces a set whose elements combine individual elements x(i), one from each X-i, by interleaving their symbol sequences in arithmetic progressions (modn). We determine algebraic relations between decimation and interleaving operations and the shift map. We study set-theoretic n-fold closure operations X bar right arrow 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 N(X) of all values n >= 1 for which X = X-[n]. We characterize the possible sets 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. (C) 2021 The Authors. Published by Elsevier Inc.

Automated summary

This paper investigates subsets of one-sided shift spaces on a finite alphabet, examining decimation and interleaving operations and their algebraic relations. Additionally, n-fold closure operations and weakly shift-stable sets are studied, along with their impact on entropy.