On Sturmian substitutions closed under derivation

Abstract

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.