There are no iterated morphisms that define the Arshon sequence and the $σ$-sequence

Abstract

Berstel proved that the Arshon sequence cannot be obtained by iteration of a morphism. An alternative proof of this fact is given here. The $σ$-sequence was constructed by Evdokimov in order to construct chains of maximal length in the n-dimensional unit cube. It turns out that the $σ$-sequence has a close connection to the Dragon curve. We prove that the $σ$-sequence can not be defined by iteration of a morphism.