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Published in:
2021 | OriginalPaper | Chapter
Symmetry Groups of Infinite Words
Authors : Sergey Luchinin, Svetlana Puzynina
Published in: Developments in Language Theory
Publisher: Springer International Publishing
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Abstract
In this paper we introduce a new notion of a symmetry group of an infinite word. Given a subgroup \(G_n\) of the symmetric group \(S_n\), it acts on the set of finite words of length n by permutation. For each n, a symmetry group of an infinite word w is a subgroup \(G_n\) of the symmetric group \(S_n\) such that g(v) is a factor of w for each permutation \(g \in G_n\) and each factor v of w. We study general properties of symmetry groups of infinite words and characterize symmetry groups of several families of infinite words. We show 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.