Derived Sequences of Arnoux–Rauzy Sequences

For an Arnoux–Rauzy sequence \(\mathbf {\mathbf u}\) we describe the set \(\text {Der}(\mathbf {\mathbf u})\) of derived sequences corresponding to all nonempty prefixes of \(\mathbf {\mathbf u}\) using the normalized directive sequence of \(\mathbf {\mathbf u}\). As a corollary, we show that all derived sequences of \(\mathbf {\mathbf u}\) are also Arnoux–Rauzy sequences. Moreover, if \(\mathbf {\mathbf u}\) is primitive substitutive, we precisely determine the cardinality of the set \(\text {Der}(\mathbf {\mathbf u})\).