Abstract
We consider Billiard Words on alphabets with k = 3 letters: such words are associated to some 3-dimensional positive vector \({\overrightarrow{\alpha}=({\alpha_1},{\alpha_2},{\alpha_3})}\). The language of these words is already known in the usual case, i.e., when the α j are linearly independent over \({\rm \rule{.33em}{0ex}\rule{.08em}{1.52ex}\kern -0.33em Q}\), and so for the \({\alpha}_{j^{-1}}\)’s. Here we study the language of these words when there exist some linear relations. We give the complexity of Billiard Words in any case, which has asymptotically a ”polynomial-like” form, with degree less or equal to 2. These results are obtained by geometrical methods.
AMS Classification. 68R15.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Allouche, J.-P., Shallit, J.: Automatic sequences: Theory and Applications. Cambridge University Press, Cambridge (2003)
Arnoux, P., Mauduit, C., Shiokawa, I., Tamura, J.I.: Complexity of sequences defined by billiard in the cube. Bull. Soc. Math. France 122, 1–12 (1994)
Arnoux, P., Rauzy, G.: Représentation géomtrique de suites de complexité 2n + 1. Bull. Soc. Math. France 119, 199–215 (1991)
Baryshnikov, Y.: Complexity of trajectories in rectangular billiards. Comm. Math. Phys. 174, 43–56 (1995)
Bédaride, N.: Etude du billard dans un polyhédre, PhD Thesis, Univ. Aix-Marseille (2005)
Bédaride, N.: Billiard complexity in rational polyhedra. Regul Chaotic Dyn. 8, 97–104 (2003)
Berstel, J., de Luca, A.: Sturmian words, Lyndon words and trees. Theoret. Comput. Sci. 178, 171–203 (1997)
Berstel, J., Séébold, P.: Sturmian words. In: Lothaire, M. (ed.) Algebraic combinatorics on words. Cambridge University Press, Cambridge (2002)
Borel, J.-P., Reutenauer, C.: Palindromic factors of billiard words. Theoret. Comput. Sci. 340-2, 334–348 (2005)
Borel, J.-P., Reutenauer, C.: Some new results on palindromic factors of billiard words. In: De Felice, C., Restivo, A. (eds.) DLT 2005. LNCS, vol. 3572, pp. 180–188. Springer, Heidelberg (2005)
Borel, J.-P.: Complexity and palindromic complexity of billiard words. In: Brlek, Reutenauer (eds.) WORDS 2005 5th Intern. Conf. on Words, vol. 36, pp. 175–184. Publ. LACIM Montral (2005)
Bourbaki, N.: Topologie générale, ch. 5-10, Herman, Paris (1974)
Cassaigne, J.: Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. Simon Stevin 4, 67–88 (1997)
de Luca, A.: Sturmian words: structure, combinatorics, and their arithmetics. Theoret. Comput. Sci. 183, 45–82 (1997)
de Luca, A.: Combinatorics of standard Sturmian words, in Structures in Logic and Computer Science. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science. LNCS, vol. 1261, pp. 249–267. Springer, Heidelberg (1997)
Rauzy, G.: Des mots en arithmetique. In: Avignon Conf. on Language Theory and Algorithmic Complexity 1983, Lyon, pp. 103–113. Univ. Claude-Bernard (1984)
Rauzy, G.: Sequences with terms in a finite alphabet, Seminar on Number Theory Univ. Bordeaux I, vol. 25, p. 16 (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Borel, J.P. (2006). Complexity of Degenerated Three Dimensional Billiard Words. In: Ibarra, O.H., Dang, Z. (eds) Developments in Language Theory. DLT 2006. Lecture Notes in Computer Science, vol 4036. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11779148_35
Download citation
DOI: https://doi.org/10.1007/11779148_35
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35428-4
Online ISBN: 978-3-540-35430-7
eBook Packages: Computer ScienceComputer Science (R0)