Directive words of episturmian words: equivalences and normalization

Amy Glen; Florence Levé; Gwénaël Richomme

RAIRO - Theoretical Informatics and Applications (2008)

  • Volume: 43, Issue: 2, page 299-319
  • ISSN: 0988-3754

Abstract

top
Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing the same episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.

How to cite

top

Glen, Amy, Levé, Florence, and Richomme, Gwénaël. "Directive words of episturmian words: equivalences and normalization." RAIRO - Theoretical Informatics and Applications 43.2 (2008): 299-319. <http://eudml.org/doc/92918>.

@article{Glen2008,
abstract = { Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing the same episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word. },
author = {Glen, Amy, Levé, Florence, Richomme, Gwénaël},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Episturmian word; Sturmian word; Arnoux-Rauzy sequence; episturmian morphism; directive word.; episturmian word; sturmian word; directive word},
language = {eng},
month = {11},
number = {2},
pages = {299-319},
publisher = {EDP Sciences},
title = {Directive words of episturmian words: equivalences and normalization},
url = {http://eudml.org/doc/92918},
volume = {43},
year = {2008},
}

TY - JOUR
AU - Glen, Amy
AU - Levé, Florence
AU - Richomme, Gwénaël
TI - Directive words of episturmian words: equivalences and normalization
JO - RAIRO - Theoretical Informatics and Applications
DA - 2008/11//
PB - EDP Sciences
VL - 43
IS - 2
SP - 299
EP - 319
AB - Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing the same episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.
LA - eng
KW - Episturmian word; Sturmian word; Arnoux-Rauzy sequence; episturmian morphism; directive word.; episturmian word; sturmian word; directive word
UR - http://eudml.org/doc/92918
ER -

References

top
  1. J.-P. Allouche and J. Shallit, Automatic sequences: Theory, Applications, Generalizations. Cambridge University Press (2003).  
  2. P. Arnoux and G. Rauzy, Représentation géométrique de suites de complexités 2 n + 1 . Bull. Soc. Math. France119 (1991) 199–215.  
  3. J. Berstel, Sturmian and episturmian words (a survey of some recent results), in Proceedings of CAI 2007. Lect. Notes Comput. Sci., Vol. 4728. Springer-Verlag (2007).  
  4. V. Berthé, C. Holton and L.Q. Zamboni, Initial powers of Sturmian sequences. Acta Arith.122 (2006) 315–347.  
  5. X. Droubay, J. Justin and G. Pirillo, Episturmian words and some constructions of de Luca and Rauzy. Theoret. Comput. Sci.255 (2001) 539–553.  
  6. S. Ferenczi, Complexity of sequences and dynamical systems. Discrete Math.206 (1999) 145–154.  
  7. A. Glen, On Sturmian and episturmian words, and related topics. Ph.D. thesis, The University of Adelaide, Australia (2006).  
  8. A. Glen, A characterization of fine words over a finite alphabet. Theoret. Comput. Sci.391 (2008) 51–60.  
  9. A. Glen and J. Justin, Episturmian words: a survey. RAIRO-Theor. Inf. Appl. (submitted). e-print arxiv:0801.1655 (2007).  
  10. A. Glen, J. Justin and G. Pirillo, Characterizations of finite and infinite episturmian words via lexicographic orderings. Eur. J. Combin.29 (2008) 45–58.  
  11. A. Glen, F. Levé and G. Richomme, Quasiperiodic and Lyndon episturmian words. Theoret. Comput. Sci. DOI: .  DOI10.1016/j.tcs.2008.09.056
  12. E. Godelle, Représentation par des transvections des groupes d'artin-tits. Group Geom. Dyn.1 (2007) 111–133.  
  13. J. Justin and G. Pirillo, Episturmian words and episturmian morphisms. Theoret. Comput. Sci.276 (2002) 281–313.  
  14. J. Justin and G. Pirillo, On a characteristic property of Arnoux-Rauzy sequences. RAIRO-Theor. Inf. Appl.36 (2003) 385–388.  
  15. J. Justin and G. Pirillo, Episturmian words: shifts, morphisms and numeration systems. Int. J. Found. Comput. Sci.15 (2004) 329–348.  
  16. F. Levé and G. Richomme, Quasiperiodic infinite words: some answers. Bull. Eur. Assoc. Theor. Comput. Sci.84 (2004) 128–138.  
  17. F. Levé and G. Richomme, Quasiperiodic episturmian words, in Proceedings of the 6th International Conference on Words, Marseille, France (2007).  
  18. F. Levé and G. Richomme, Quasiperiodic Sturmian words and morphisms. Theoret. Comput. Sci.372 (2007) 15–25.  
  19. M. Lothaire, Combinatorics on Words, Encyclopedia of Mathematics and its Applications, Vol. 17. Addison-Wesley (1983).  
  20. M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, Vol. 90, Cambridge University Press (2002).  
  21. M. Morse and G. Hedlund, Symbolic Dynamics II. Sturmian trajectories. Amer. J. Math.61 (1940) 1–42.  
  22. G. Paquin and L. Vuillon, A characterization of balanced episturmian sequences. Electron. J. Combin.14 (2007) 33.  
  23. N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics. Lect. Notes Math., Vol. 1794. Springer (2002).  
  24. G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France110 (1982) 147–178.  
  25. G. Rauzy, Mots infinis en arithmétique, in Automata on Infinite words, edited by M. Nivat, D. Perrin. Lect. Notes Comput. Sci., Vol. 192. Springer-Verlag, Berlin (1985).  
  26. G. Richomme, Conjugacy and episturmian morphisms. Theoret. Comput. Sci.302 (2003) 1–34.  
  27. G. Richomme, Lyndon morphisms. Bull. Belg. Math. Soc. Simon Stevin10 (2003) 761–785.  
  28. G. Richomme, Conjugacy of morphisms and Lyndon decomposition of standard Sturmian words. Theoret. Comput. Sci.380 (2007) 393–400.  
  29. G. Richomme, A local balance property of episturmian words, in Proc. DLT '07. Lect. Notes Comput. Sci., Vol. 4588. Springer, Berlin (2007) 371–381.  
  30. R. Risley and L. Zamboni, A generalization of Sturmian sequences: combinatorial structure and transcendence. Acta Arith.95 (2000) 167–184.  
  31. P. Séébold, Fibonacci morphisms and Sturmian words. Theoret. Comput. Sci.88 (1991) 365–384.  
  32. Z.-X. Wen and Y. Zhang, Some remarks on invertible substitutions on three letter alphabet. Chinese Sci. Bull.44 (1999) 1755–1760.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.