We survey several quantitative problems on infinite words related to repetitions, recurrence, and palindromes, for which the Fibonacci word often exhibits extremal behaviour.
@article{ITA_2008__42_4_701_0, author = {Cassaigne, Julien}, title = {On extremal properties of the Fibonacci word}, journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications}, volume = {42}, year = {2008}, pages = {701-715}, doi = {10.1051/ita:2008003}, mrnumber = {2458702}, zbl = {1155.68062}, language = {en}, url = {http://dml.mathdoc.fr/item/ITA_2008__42_4_701_0} }
Cassaigne, Julien. On extremal properties of the Fibonacci word. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) pp. 701-715. doi : 10.1051/ita:2008003. http://gdmltest.u-ga.fr/item/ITA_2008__42_4_701_0/
[1] Palindrome complexity. Theoret. Comput. Sci. 292 (2003) 9-31. | MR 1964623 | Zbl 1064.68074
, , and ,[2] Transcendence of Sturmian or morphic continued fractions. J. Number Theory 91 (2001) 39-66. | MR 1869317 | Zbl 0998.11036
, , and ,[3] On the index of Sturmian words, in Jewels are Forever. Springer, Berlin (1999) 287-294. | MR 1719097 | Zbl 0982.11010
,[4] Initial powers of Sturmian sequences. Acta Arith. 122 (2006) 315-347. | MR 2234421 | Zbl 1117.37005
, and ,[5] Enumeration of factors in the Thue-Morse word. Discrete Appl. Math. 24 (1989) 83-96. | MR 1011264 | Zbl 0683.20045
,[6] On Dejean's conjecture over large alphabets. Theoret. Comput. Sci. 385 (2007) 137-151. | MR 2356248 | Zbl 1124.68087
,[7] Special factors, periodicity, an application to Sturmian words. Acta Inform. 36 (2000) 983-1006. | MR 1776146 | Zbl 0956.68119
and ,[8] Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 67-88. | MR 1440670 | Zbl 0921.68065
,[9] On a conjecture of J. Shallit, in Automata, languages and programming (ICALP 1997), Springer, Berlin. Lect. Notes Comput. Sci. 1256 (1997) 693-704. | MR 1616227
,[10] Limit values of the recurrence quotient of Sturmian sequences. Theoret. Comput. Sci. 218 (1999) 3-12. | MR 1687748 | Zbl 0916.68115
,[11] Fonctions de récurrence des suites d'Arnoux-Rauzy et réponse à une question de Morse et Hedlund. Ann. Inst. Fourier (Grenoble) 56 (2006) 2249-2270. | Numdam | MR 2290780 | Zbl 1138.68045
and ,[12] Dejean's conjecture and Sturmian words. Eur. J. Combin. 28 (2007) 876-890. Also in Morteza Mohammad-Noori, PhD. thesis, Université Paris-Sud (2005). | MR 2300768 | Zbl 1111.68096
and ,[13] The index of Sturmian sequences. Eur. J. Combin. 23 (2002) 23-29. | MR 1878771 | Zbl 1002.11020
and ,[14] Sur un théorème de Thue. J. Comb. Theory A 13 (1972) 90-99. | MR 300959 | Zbl 0245.20052
,[15] Palindromes and Sturmian words. Theoret. Comput. Sci. 223 (1999) 73-85. | MR 1704637 | Zbl 0930.68116
and ,[16] On nonrepetitive sequences. J. Comb. Theory A 16 (1974) 159-164. | MR 332533 | Zbl 0279.05001
, and ,[17] Palindromic prefixes and episturmian words. J. Comb. Theory A 113 (2006) 1281-1304. | MR 2259061 | Zbl 1109.68082
,[18] Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications 90. Cambridge University Press, Cambridge (2002). | MR 1905123 | Zbl 1001.68093
,[19] Repetitions in the Fibonacci infinite word. RAIRO-Theor. Inf. Appl. 26 (1992) 199-204. | Numdam | MR 1170322 | Zbl 0761.68078
and ,[20] Periodicity and the golden ratio. Theoret. Comput. Sci. 204 (1998) 153-167. | MR 1637524 | Zbl 0913.68162
, and ,[21] Recurrent geodesics on a surface of negative curvature. Trans. Amer. Math. Soc. 22 (1921) 84-100. | JFM 48.0786.06 | MR 1501161
,[22] Symbolic dynamics. Amer. J. Math. 60 (1938) 815-866. | JFM 64.0798.04 | MR 1507944
and ,[23] Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940) 1-42. | JFM 66.0188.03 | MR 745
and ,[24] Proof of Dejean's conjecture for alphabets with 5, 6, 7, 8, 9, 10 and 11 letters. Theoret. Comput. Sci. 95 (1992) 187-205. | MR 1156042 | Zbl 0745.68085
,[25] À propos d'une conjecture de F. Dejean sur les répétitions dans les mots. Discrete Appl. Math. 7 (1984) 297-311. | MR 736893 | Zbl 0536.68072
,[26] Mémoire sur quelques relations entre les puissances des nombres. C. R. Acad. Sci. Paris Sér. I 33 (1851) 225.
,[27] Suites à termes dans un alphabet fini, in Seminaire de théorie des nombres 1982-1983, Univ. Bordeaux I, 1983. Exposé 25. | MR 750326 | Zbl 0547.10048
,[28] Über unendliche Zeichenreihen. Norske Vid. Selsk. Skr., I. Mat. Nat. Kl., Christiana 7 (1906) 1-22. | JFM 37.0066.17
,[29] Sturmian words and words with a critical exponent. Theoret. Comput. Sci. 242 (2000) 283-300. | MR 1769782 | Zbl 0944.68148
,