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

Kitaev, Sergey (2003) There are no iterated morphisms that define the Arshon sequence and the sigma-sequence. Journal of Automata, Languages and Combinatorics, 8 (1). pp. 43-50.

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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.