Initial powers of Sturmian sequences

Acta Arithmetica

Volumn 122, Issue 4, 2006, Pages 315-347

  Berthé, Valérie ^a,b   Holton, Charles ^a,d   Zamboni, Luca Q ^a,c  

^a LIRMM ^*

^b LIRMM   (France)

^c UNIVERSITY OF NORTH TEXAS   (United States)

^d UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33745674701     PISSN: 00651036     EISSN: None     Source Type: Journal    
DOI: 10.4064/aa122-4-1     Document Type: Article

References (44)

1. B. Adamczewski, Répétitions dans les codages de rotations, Adv. Appl. Math. 34 (2005), 1-29.
2. B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers II. Continued fractions, Acta Math., to appear.
3. J.-P. Allouche, J. P. Davison, M. Queffélec and L. Q. Zamboni, Transcendence of Sturmian or morphic continued fractions, J. Number Theory 91 (2001), 39-66.
4. P. Arnoux, S. Ferenczi and P. Hubert, Trajectories of rotations, Acta Arith. 87 (1999), 209-217.
5. P. Arnoux and A. M. Fisher, The scenery flow for geometric structures on the torus: the linear setting, Chinese Ann. Math. Ser. B 22 (2001), 427-470.
6. P. Arnoux et G. Rauzy, Représentation géométrique de suites de complexité 2n + 1, Bull. Soc. Math. France 119 (1991), 199-215.
7. G. Barat and P. Liardet, Dynamical systems originated in the Ostrowski alpha-expansion, Ann. Univ. Sci. Budapest. Sect. Comput. 24 (2004), 133-184.
8. J. Berstel, Recent results in Sturmian words, in: Developments in Language Theory II, J. Dassow, G. Rozenberg and A. Salomaa (eds.), World Scientific, Singapore, 1996, 13-24.
9. _, On the index of Sturmian words, in: Jewels are Forever, Springer, Berlin, 1999, 287-294.
10. V. Berthé, Autour du système de numération d'Ostrowski, Bull. Belg. Math. Soc. Simon Stevin 8 (2001), 209-239.
11. P. Billingsley, Ergodic Theory and Information, Wiley, New York, 1965.
12. J. Cassaigne, Special factors of sequences with linear subword complexity, in: Developments in Language Theory II (Magdeburg, 1995), World Scientific, Singapore, 1996, 25-34.
13. _, Limit values of the recurrence quotient in Sturmian sequences, Theoret. Comput. Sci. 218 (1999), 3-12.
14. E. M. Coven and G. A. Hedlund, Sequences with minimal block growth, Math. Systems Theory 7 (1973), 138-153.
15. D. Crisp, W. Moran, A. Pollington and P. Shiue, Substitution invariant cutting sequences, J. Théor. Nombres Bordeaux 5 (1993), 123-137.
16. D. Damanik, R. Killip and D. Lenz, Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity, Comm. Math. Phys. 212 (2000), 191-204.
17. D. Damanik and D. Lenz, The index of Sturmian sequences, Europ. J. Combin. 23 (2002), 23-29.
18. I. R. Descombes, Sur la répartition des sommets d'une ligne polygonale régulière non fermée, Ann. Sci. École Norm. Sup. 73 (1956), 283-355.
19. Y. Dupain and V. T. Sos, On the one-sided boundedness of discrepancy-function of the sequence {nα}, Acta Arith. 37 (1980), 363-374.
20. F. Durand, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998), 89-101.
21. _, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam. Systems 20 (2000), 1061-1078.
22. _, Combinatorial and dynamical study of substitutions around the theorem of Cobham, in: Dynamics and Randomness, Nonlinear Phenom. Complex Systems 7, Kluwer, 2002, 53-94.
23. S. Ferenczi, Rank and symbolic complexity, Ergodic Theory Dynam. Systems 16 (1996), 663-682.
24. P. J. Grabner, P. Liardet and R. F. Tichy, Odometers and systems of numeration, Acta Arith. 70 (1995), 103-123.
25. C. Holton and L. Q. Zamboni, Descendants of primitive substitutions, Theory Comput. Systems 32 (1999), 133-157.
26. S. Ito and H. Nakada, Approximations of real numbers by the sequence {nα} and their metrical theory, Acta Math. Hungar. 52 (1988), 91-100.
27. J. Justin and G. Pirillo, Fractional powers in Sturmian words, Theoret. Comput. Sci. 255 (2001), 363-376.
28. J. Lesca, Sur la répartition modulo 1 des suites (nα), Sém. Delange-Pisot-Poitou 8 (1966/67), Théorie des Nombres, Fasc. 1, Exp. 2, Secrétariat Mathématique, Paris, 1966/1967, 9pp.
29. _, Sur la répartition modulo 1 de la suite na, Acta Arith. 20 (1972), 345-352.
30. M. Lothaire, Algebraic Combinatorics on Words, Cambridge Univ. Press, Cambridge, 2002.
31. F. Mignosi, Infinite words with linear subword complexity, Theoret. Comput. Sci. 65 (1989), 221-242.
32. F. Mignosi and G. Pirillo, Repetitions in the Fibonacci infinite word, RAIRO Inform. Théor. Appl. 26 (1992), 199-204.
33. F. Mignosi, A. Restivo and S. Salemi, Periodicity and the golden ratio, Theoret. Comput. Sci. 204 (1988), 153-167.
34. M. Morse and G. A. Hedlund, Symbolic dynamics II: Sturmian trajectories, Amer. J. Math. 62 (1940), 1-42.
35. A. Ostrowski, Bemerkungen zur Theorie der diophantischen Approximationen, I, II, Abh. Math. Sem. Hamburg 1 (1921), 77-98 and 250-251.
36. B. Parvaix, Substitution invariant Sturmian bisequences, J. Théor. Nombres Bordeaux 11 (1999), 201-210.
37. N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel (eds.), Lecture Notes in Math. 1794, Springer, 2002.
38. R. N. Risley and L. Q. Zamboni, A generalization of Sturmian sequences: Combinatorial structure and transcendence, Acta Arith. 95 (2000), 167-184.
39. V. T. Sos, On the distribution mod 1 of the sequence nα, Ann. Univ. Sci. Budapest. Rolando Eötvös Sect. Math. 1 (1958), 127-134.
40. M. Stewart, Irregularities of uniform distribution, Acta Math. Acad. Sci. Hungar. 37 (1981), 185-221.
41. D. Vandeth, Sturmian words and words with a critical exponent, Theoret. Comput. Sci. 242 (2000), 283-300.
42. L. Vuillon, A characterization of Sturmian words by return words, Europ. J. Combin. 22 (2001), 263-275.
43. Z.-X. Wen and Z.-Y. Wen, Some properties of the singular words of the Fibonacci word, Europ. J. Combin. ibid. 15 (1994), 1-12.
44. S.-i. Yasutomi, On Sturmian sequences which are invariant under some substitutions, in: Number Theory and Its Applications, S. Kanemitsu and K. Gyo″ry (eds.), Kluwer, Dordrecht, 1999, 347-373.