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arXiv · 1405.3532

A new approach to the $2$-regularity of the $\ell$-abelian complexity of $2$-automatic sequences

Aline Parreau·Michel Rigo·Eric Rowland·Elise Vandomme
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Abstract

We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general approach for studying the $\ell$-abelian complexity of $2$-automatic sequences. In particular, we prove that the period-doubling word and the Thue--Morse word have $2$-abelian complexity sequences that are $2$-regular. Along the way, we also prove that the $2$-block codings of these two words have $1$-abelian complexity sequences that are $2$-regular.

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Aline Parreau, Michel Rigo, Eric Rowland, Elise Vandomme. 2014-05-14. A new approach to the $2$-regularity of the $\ell$-abelian complexity of $2$-automatic sequences. https://arxiv.org/abs/1405.3532

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