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

TL;DR

This paper proves that neither the Arshon sequence nor the $\sigma$-sequence can be generated through iterative morphisms, highlighting their unique non-morphic nature and connections to geometric structures.

Contribution

It provides an alternative proof that the Arshon sequence is non-morphic and establishes that the $\sigma$-sequence also cannot be generated by morphic iteration.

Findings

Both sequences cannot be obtained by morphic iteration.

The $\sigma$-sequence is connected to the Dragon curve.

The results clarify the non-morphic nature of these sequences.

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 $\sigma$-sequence was constructed by Evdokimov in order to construct chains of maximal length in the n-dimensional unit cube. It turns out that the $\sigma$-sequence has a close connection to the Dragon curve. We prove that the $\sigma$-sequence can not be defined by iteration of a morphism.