On Sturmian Substitutions Closed Under Derivation

Occurrences of a factor win 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 win u. If wis a prefix of a fixed point uof a primitive substitution phi, then by Durand's result from 1998, the derived sequence d(u)(w) is fixed by a primitive substitution psi as well. For a non-prefix factor w, the derived sequence d(u)(w) is fixed by a substitution only exceptionally. To study this phenomenon we introduce a new notion: A finite set Mof substitutions is said to be closed under derivation if the derived sequence d(u)(w) to any factor wof any fixed point u of phi is an element of M is fixed by a morphism psi is an element of M. In our article we characterize the Sturmian substitutions which belong to a set Mclosed under derivation. The characterization uses either the slope and the intercept of its fixed point or its S-adic representation. (c) 2021 Elsevier B.V. All rights reserved.