On Sturmian substitutions closed under derivation

TL;DR

This paper characterizes Sturmian substitutions that form a set closed under derivation, linking their properties to slope, intercept, and S-adic representation, thus extending understanding of derived sequences in symbolic dynamics.

Contribution

It introduces the concept of sets of substitutions closed under derivation and characterizes Sturmian substitutions within this framework using slope, intercept, and S-adic representation.

Findings

Characterization of Sturmian substitutions in closed under derivation sets.

Connection between derived sequences and S-adic representations.

Extension of Durand's 1998 results to non-prefix factors.

Abstract

Occurrences of a factor $w$ in an infinite uniformly recurrent sequence ${\bf u}$ can be encoded by an infinite sequence over a finite alphabet. This sequence is usually denoted ${\bf d_{\bf u}}(w)$ and called the derived sequence to $w$ in ${\bf u}$. If $w$ is a prefix of a fixed point ${\bf u}$ of a primitive substitution $\varphi$, then by Durand's result from 1998, the derived sequence ${\bf d_{\bf u}}(w)$ is fixed by a primitive substitution $\psi$ as well. For a non-prefix factor $w$, the derived sequence ${\bf d_{\bf u}}(w)$ 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 ${\bf d_{\bf u}}(w)$ to any factor $w$ of any fixed point ${\bf u}$ of $\varphi \in M$ is fixed by a morphism $\psi \in M$. 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.