An Introduction to the C ∗-Algebra of a One-Sided Shift Space

Carlsen, Toke Meier

doi:10.1007/978-3-642-39459-1_4

Cited by 1 publication

(1 citation statement)

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“…The work of Cuntz and Krieger [15] and Cuntz [14] was seminal in attaching such invariants to topological Markov chains (= two-sided shifts of finite type). Carlsen ([11], [12]) attached C *algebras to one-sided subshifts and studied their properties. Shift-stable sets X are studied in the context of partial isometry actions and C ⋆ algebras attached to them by Dokuchaev and Exel [17].…”

Section: 31mentioning

confidence: 99%

Decimation and Interleaving Operations in One-Sided Symbolic Dynamics

Abram¹,

Lagarias²,

Slonim³

2020

Preprint

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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 ≥ 1. Given subsets X0, X1, ..., Xn−1 of the shift space, the n-ary interleaving operator produces a set whose elements combine individual elements xi, one from each Xi, by interleaving their symbol sequences cyclically in arithmetic progressions (mod n). We determine algebraic relations between decimation and interleaving operators and the shift operator. We study set-theoretic n-fold closure operations X → 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 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. 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.

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