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

TL;DR

This paper establishes a new method to prove the 2-regularity of the $ ext{l}$-abelian complexity in 2-automatic sequences, applying it to key examples like the Thue–Morse and period-doubling words.

Contribution

It introduces a general approach for analyzing the $ ext{l}$-abelian complexity of 2-automatic sequences based on symmetry properties, demonstrating 2-regularity for notable sequences.

Findings

Proves 2-regularity of the $ ext{l}$-abelian complexity for certain sequences.

Shows 2-regularity of the 2-abelian complexity sequences of the period-doubling and Thue–Morse words.

Establishes 2-regularity of 1-abelian complexity sequences for 2-block codings of these words.

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.