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.
Supported by Russian Foundation of Basic Research (grant 20-01-00488).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)
Allouche, J.-P.: The number of factors in a paperfolding sequence. Bull. Austral. Math. Sot. 46, 23–32 (1992)
Cassaigne, J., Karhumäki, J.: Toeplitz words, generalized periodicity and periodically iterated morphisms. Eur. J. Comb. 18, 497–510 (1997)
Charlier, É., Puzynina, S., Zamboni, L.Q.: On a group theoretic generalization of the Morse-Hedlund theorem. Proc. Amer. Math. Soc. 145(8), 3381–3394 (2017)
Glen, A., Justin, J.: Episturmian words: a survey. RAIRO - Theoret. Inf. Appl. 43, 402–433 (2009)
Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)
Madil, B., Rampersad, N.: The abelian complexity of the paperfolding word. Discrete Math. 313(7), 831–838 (2013)
Thue, A.: Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1, 1–67 (1912)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Luchinin, S., Puzynina, S. (2021). Symmetry Groups of Infinite Words. In: Moreira, N., Reis, R. (eds) Developments in Language Theory. DLT 2021. Lecture Notes in Computer Science(), vol 12811. Springer, Cham. https://doi.org/10.1007/978-3-030-81508-0_22
Download citation
DOI: https://doi.org/10.1007/978-3-030-81508-0_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-81507-3
Online ISBN: 978-3-030-81508-0
eBook Packages: Computer ScienceComputer Science (R0)