1 research outputs found
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
they are closed under composition. We also study a family of -ary
interleaving operations, one for each . Given subsets of the shift space, the -ary interleaving operator produces a set
whose elements combine individual elements , one from each , by
interleaving their symbol sequences cyclically in arithmetic progressions
. We determine algebraic relations between decimation and
interleaving operators and the shift operator. We study set-theoretic -fold
closure operations , which interleave decimations of of
modulus level . A set is -factorizable if . The -fold
interleaving operators are closed under composition and are idempotent. To each
we assign the set of all values for which . We characterize the possible sets 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 , but can be different in general. We give a formula
for entropy of interleavings of weakly shift-stable sets in terms of individual
entropies.Comment: 41 pages, 2 figure