Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T04:25:15.113Z Has data issue: false hasContentIssue false
  • English
  • Français

Least Periods of Factors of Infinite Words

Published online by Cambridge University Press:  12 March 2008

James D. Currie  and
Kalle Saari
James D. Currie
Affiliation:
Department of Mathematics & Statistics, University of Winnipeg, Winnipeg, R3B2E9, Canada
Kalle Saari
Affiliation:
Department of Mathematics and Turku Centre for Computer Science, University of Turku, Turku, Finland; [email protected]
Get access

Abstract

We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a Sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a by-product of our results, we give several new proofs and tightenings of well-known properties of Sturmian words.

Keywords

PeriodicityFibonacci wordThue-Morse wordSturmian word.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 43 , Issue 1 , January 2009 , pp. 165 - 178
Copyright
© EDP Sciences, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.-P. Allouche and J. Shallit, The ubiquitous Prouhet-Thue-Morse sequence, in Sequences and Their Applications: Proceedings of SETA'98. Springer Series in Discrete Mathematics and Theoretical Computer Science, C. Ding, T. Helleseth and H. Niederreiter, Eds., Springer-Verlag, London (1999) 1–16.
J. Berstel, On the index of Sturmian words. In Jewels are forever. Springer, Berlin (1999) 287–294.
Cao, W.-T. and Wen, Z.-Y., Some properties of the factors of Sturmian sequences. Theor. Comput. Sci. 304 (2003) 365–385. CrossRef
C. Choffrut and J. Karhumäki, Combinatorics on words. In A. Salomaa and G. Rozenberg, Eds., Handbook of Formal Languages, volume 1. Springer, Berlin (1997) 329–438.
Cummings, L.J., Moore, D.W. and Karhumäki, J., Borders of Fibonacci strings. J. Comb. Math. Comb. Comput. 20 (1996) 81–87.
Damanik, D. and Lenz, D., Powers in Sturmian sequences. Eur. J. Combin. 24 (2003) 377–390. CrossRef
de Luca, A. and De Luca, A., Some characterizations of finite Sturmian words. Theor. Comput. Sci. 356 (2006) 118–125. CrossRef
Fine, N.J. and Wilf, H.S., Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16 (1965) 109–114. CrossRef
Harju, T. and Nowotka, D., Minimal Duval extensions. Int. J. Found. Comput. Sci. 15 (2004) 349–354. CrossRef
M. Lothaire, Combinatorics on Words. Cambridge University Press, Cambridge (1997).
M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, Vol. 90. Cambridge University Press, Cambridge (2002).
Mignosi, F. and Zamboni, L.Q., A note on a conjecture of Duval and Sturmian words. RAIRO-Theor. Inf. Appl. 36 (2002) 1–3. CrossRef
Mohammad-Noori, M. and Currie, J.D., Dejean's conjecture and Sturmian words. Eur. J. Combin. 28 (2007) 876–890. CrossRef
K. Saari, Periods of factors of the Fibonacci word. in Proceedings of the Sixth International Conference on Words (WORDS'07). Institut de Mathématiques de Luminy (2007) 273–279.