no. 1
On critical exponents in fixed points of k-uniform binary morphisms
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 41-68.

Let 𝐰 be an infinite fixed point of a binary k-uniform morphism f, and let E(𝐰) be the critical exponent of 𝐰. We give necessary and sufficient conditions for E(𝐰) to be bounded, and an explicit formula to compute it when it is. In particular, we show that E(𝐰) is always rational. We also sketch an extension of our method to non-uniform morphisms over general alphabets.

DOI : 10.1051/ita:2007042
Classification : 68R15
Mots clés : critical exponent, binary $k$-uniform morphism 
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     author = {Krieger, Dalia},
     title = {On critical exponents in fixed points of $k$-uniform binary morphisms},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {41--68},
     publisher = {EDP-Sciences},
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     year = {2009},
     doi = {10.1051/ita:2007042},
     mrnumber = {2483444},
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     url = {http://www.numdam.org/articles/10.1051/ita:2007042/}
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Krieger, Dalia. On critical exponents in fixed points of $k$-uniform binary morphisms. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 41-68. doi : 10.1051/ita:2007042. http://www.numdam.org/articles/10.1051/ita:2007042/

[1] J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press (2003). | MR | Zbl

[2] J. Berstel, Axel Thue's Papers on Repetitions in Words: A Translation. Publications du Laboratoire de Combinatoire et d'Informatique Mathématique 20, Université du Québec à Montréal (1995).

[3] J. Berstel, On the Index of Sturmian Words, in Jewels are forever. Springer, Berlin (1999) 287-294. | MR | Zbl

[4] W.-T. Cao and Z.-Y. Wen, Some properties of the factors of Sturmian sequences. Theoret. Comput. Sci. 304 (2003) 365-385. | MR | Zbl

[5] A. Carpi and A. De Luca, Special factors, periodicity, and an application to Sturmian words. Acta Informatica 36 (2000) 983-1006. | MR | Zbl

[6] J. Cassaigne, An algorithm to test if a given circular HD0L-language avoids a pattern, in IFIP World Computer Congress'94 1 (1994) 459-464. | MR

[7] D. Damanik and D. Lenz, The index of Sturmian sequences. Eur. J. Combin. 23 (2002) 23-29. | MR | Zbl

[8] A. Ehrenfeucht and G. Rozenberg, Repetition of subwords in D0L languages. Inform. Control 59 (1983) 13-35. | MR | Zbl

[9] N.J. Fine and H.S. Wilf, Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16 (1965) 109-114. | MR | Zbl

[10] A.E. Frid, On uniform DOL words. STACS'98 1373 (1998) 544-554. | MR

[11] J. Justin and G. Pirillo, Fractional powers in Sturmian words. Theoret. Comput. Sci. 255 (2001) 363-376. | MR | Zbl

[12] A.V. Klepinin and E.V. Sukhanov, On combinatorial properties of the Arshon sequence. Discrete Appl. Math. 114 (2001) 155-169. | MR | Zbl

[13] Y. Kobayashi and F. Otto, Repetitiveness of languages generated by morphisms. Theoret. Comput. Sci. 240 (2000) 337-378. | MR | Zbl

[14] D. Krieger, On critical exponents in fixed points of binary k-uniform morphisms, in STACS 2006: 23rd Annual Symposium on Theoretical Aspects of Computer Science, edited by B. Durand and W. Thomas. Lect. Notes. Comput. Sci. 3884 (2006) 104-114. | MR | Zbl

[15] D. Krieger, On critical exponents in fixed points of non-erasing morphisms. Theoret. Comput. Sci. 376 (2007) 70-88. | MR | Zbl

[16] M. Lothaire, Algebraic Combinatorics on Words, Vol. 90 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press (2002). | MR | Zbl

[17] R.C. Lyndon and M.P. Schützenberger, The equation a M =b N c P in a free group. Michigan Math. J. 9 (1962) 289-298. | MR | Zbl

[18] F. Mignosi, Infinite words with linear subword complexity. Theoret. Comput. Sci. 65 (1989) 221-242. | MR | Zbl

[19] F. Mignosi and G. Pirillo, Repetitions in the Fibonacci infinite word. RAIRO-Theor. Inf. Appl. 26 (1992) 199-204. | EuDML | Numdam | MR | Zbl

[20] F. Mignosi and P. Séébold, If a D0L language is k-power free then it is circular. ICALP'93. Lect. Notes Comput. Sci. 700 (1993) 507-518. | MR

[21] B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution. Theoret. Comput. Sci. 99 (1992) 327-334. | MR | Zbl

[22] A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1-67. | JFM

[23] D. Vandeth, Sturmian Words and Words with Critical Exponent. Theoret. Comput. Sci. 242 (2000) 283-300. | MR | Zbl

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