On Substitutions Closed Under Derivation: Examples

Košík, Václav; Starosta, Štěpán

doi:10.1007/978-3-030-28796-2_18

Cited by 1 publication

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“…However, there exists an example of a set M closed under derivation which contains substitutions acting on a binary alphabet and substitutions acting on a ternary alphabet such that no proper subset of M is closed under derivation. This example is given in [12] and the set M contains substitutions fixing derived words to non-empty factors of the period doubling sequence. Recall that the period doubling sequence is fixed by the substitution a → ab, b → aa.…”

Section: Commentsmentioning

confidence: 99%

On Sturmian substitutions closed under derivation

Pelantová¹,

Starosta²

2019

Preprint

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Occurrences of a factor w in an infinite uniformly recurrent sequence u can be encoded by an infinite sequence over a finite alphabet. This sequence is usually denoted du(w) and called the derived sequence to w in u. If w is a prefix of a fixed point u of a primitive substitution ϕ, then by Durand's result from 1998, the derived sequence du(w) is fixed by a primitive substitution ψ as well. For a non-prefix factor w, the derived sequence du(w) is fixed by a substitution only exceptionally. To study this phenomenon we introduce a new notion: A finite set M of substitutions is said to be closed under derivation if the derived sequence du(w) to any factor w of any fixed point u of ϕ ∈ M is fixed by a morphism ψ ∈ M . In our article we characterize the Sturmian substitutions which belong to a set M closed under derivation. The characterization uses either the slope and the intercept of its fixed point or its S-adic representation.

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Section: Commentsmentioning

confidence: 99%