TY - JOUR

T1 - Sequences of symmetry groups of infinite words.

AU - Luchinin, Sergey

AU - Puzynina, Svetlana

N1 - DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

PY - 2023

Y1 - 2023

N2 - In this paper we introduce a new notion of a sequence of symmetry groups of an infinite word. Given a subgroup of the symmetric group, it acts on the set of finite words of length n by permutation. We associate to an infinite word w a sequence of its symmetry groups: For each n, a symmetry group of w is a subgroup of the symmetric group such that is a factor of w for each permutation and each factor v of length n of w. We study general properties of the symmetry groups of infinite words and characterize the sequences of symmetry groups of several families of infinite words. We show that for each subgroup G of there exists an infinite word w with . On the other hand, the structure of possible sequences is quite restrictive: we show that they cannot contain for each order n certain cycles, transpositions and some other permutations. The sequences of symmetry groups can also characterize a generalized periodicity property. We prove that symmetry groups of Sturmian words and more generally Arnoux-Rauzy words are of order two for large enough n; on the other hand, symmetry groups of certain Toeplitz words have exponential growth.

AB - In this paper we introduce a new notion of a sequence of symmetry groups of an infinite word. Given a subgroup of the symmetric group, it acts on the set of finite words of length n by permutation. We associate to an infinite word w a sequence of its symmetry groups: For each n, a symmetry group of w is a subgroup of the symmetric group such that is a factor of w for each permutation and each factor v of length n of w. We study general properties of the symmetry groups of infinite words and characterize the sequences of symmetry groups of several families of infinite words. We show that for each subgroup G of there exists an infinite word w with . On the other hand, the structure of possible sequences is quite restrictive: we show that they cannot contain for each order n certain cycles, transpositions and some other permutations. The sequences of symmetry groups can also characterize a generalized periodicity property. We prove that symmetry groups of Sturmian words and more generally Arnoux-Rauzy words are of order two for large enough n; on the other hand, symmetry groups of certain Toeplitz words have exponential growth.

KW - Infinite words

KW - symmetry groups

KW - Arnoux-Rauzy words

KW - Toeplitz words

KW - Symmetry groups

UR - https://www.mendeley.com/catalogue/7c92c9b5-52d2-3c3d-b8f9-b85ae91ae487/

U2 - 10.1016/j.disc.2022.113171

DO - 10.1016/j.disc.2022.113171

M3 - Article

VL - 346

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1

M1 - 113171

ER -