Abstract
We provide a formula for the critical exponent and the asymptotic critical exponent of regular Arnoux-Rauzy sequences. Over the binary alphabet it coincides with a well-known formula for Sturmian sequences based on their S-adic representation. We show that among regular d-ary Arnoux-Rauzy sequences, the minimal (asymptotic) critical exponent is reached by the d-bonacci sequence.
We would like to thank Edita Pelantová for fruitful discussions and helpful advice. We thank Sébastien Labbé for his advice in programming. Our thanks belong also to the referees for their useful comments. The first author was supported by the Ministry of Education, Youth and Sports of the Czech Republic through the project CZ.02.1.01/0.0/0.0/16_019/0000778. The second author acknowledges financial support by The French Institute in Prague and the Czech Ministry of Education, Youth and Sports through the Barrande fellowship programme, Agence Nationale de la Recherche through the project Codys (ANR-18-CE40-0007), and the support by Grant Agency of Czech Technical University in Prague, through the project SGS23/187/OHK4/3T/14.
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Notes
- 1.
Note that sequences are denoted in bold characters.
- 2.
In the case of a Sturmian sequence \(\textbf{u}\), we can proceed similarly and associate to the directive sequence the number \(\theta \) with the continued fraction \([0;a_1, a_2, a_3,\dots ]\). It holds that \(\theta \) is equal to the ratio of the less frequent letter to the more frequent letter. The geometrical interpretation of \(\theta \) is the slope of the straight line producing \(\textbf{u}\) as a cutting sequence [24].
- 3.
The most frequent return words to primary bispecial factors coincide with the generalized standard words defined in [17].
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Dvořáková, L., Lepšová, J. (2023). Critical Exponents of Regular Arnoux-Rauzy Sequences. In: Frid, A., Mercaş, R. (eds) Combinatorics on Words. WORDS 2023. Lecture Notes in Computer Science, vol 13899. Springer, Cham. https://doi.org/10.1007/978-3-031-33180-0_10
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