On Sturmian substitutions closed under derivation [article]

Edita Pelantová, Štěpán Starosta
2019 arXiv   pre-print
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 d_ u(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 d_ u(w) is fixed by a primitive substitution ψ as well. For a non-prefix factor w, the derived sequence d_ u(w) is fixed by a substitution only
more » ... 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 d_ u(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.
arXiv:1908.11095v1 fatcat:hnhtzah6fnbvhoxl5gxfowjdr4