Bouquets of circles for lamination languages and complexities
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 48 (2014) no. 4, pp. 391-418.

Laminations are classic sets of disjoint and non-self-crossing curves on surfaces. Lamination languages are languages of two-way infinite words which code laminations by using associated labeled embedded graphs, and which are subshifts. Here, we characterize the possible exact affine factor complexities of these languages through bouquets of circles, i.e. graphs made of one vertex, as representative coding graphs. We also show how to build families of laminations together with corresponding lamination languages covering all the possible exact affine complexities.

DOI: 10.1051/ita/2014016
Classification: 14Q05, 37B10, 37F20, 57R30, 68R15, 68Q45, 68R10
Keywords: curves, laminations on surfaces, symbolic dynamics, shifts, factor complexity, embedded graphs, train-tracks, Rauzy graphs, substitutions, spirals
@article{ITA_2014__48_4_391_0,
     author = {Narbel, Philippe},
     title = {Bouquets of circles for lamination languages and complexities},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {391--418},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {4},
     year = {2014},
     doi = {10.1051/ita/2014016},
     mrnumber = {3302494},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita/2014016/}
}
TY  - JOUR
AU  - Narbel, Philippe
TI  - Bouquets of circles for lamination languages and complexities
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2014
SP  - 391
EP  - 418
VL  - 48
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita/2014016/
DO  - 10.1051/ita/2014016
LA  - en
ID  - ITA_2014__48_4_391_0
ER  - 
%0 Journal Article
%A Narbel, Philippe
%T Bouquets of circles for lamination languages and complexities
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2014
%P 391-418
%V 48
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ita/2014016/
%R 10.1051/ita/2014016
%G en
%F ITA_2014__48_4_391_0
Narbel, Philippe. Bouquets of circles for lamination languages and complexities. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 48 (2014) no. 4, pp. 391-418. doi : 10.1051/ita/2014016. http://www.numdam.org/articles/10.1051/ita/2014016/

[1] R.L. Adler, A.G. Konheim and M.H. Mcandrew, Topological entropy. Trans. Amer. Math. Soc. 114 (1965) 309-319. | MR | Zbl

[2] J-P. Allouche and J. Shallit, Automatic sequences. Cambridge University Press, Cambridge (2003). | MR | Zbl

[3] P. Arnoux and G. Rauzy, Représentation géométrique de suites de complexit*error*é2n + 1. Bull. Soc. Math. France 119 (1991) 199-215. | Numdam | MR | Zbl

[4] A. Ya. Belov and A.L. Chernyatiev, Words with low complexity and interval exchange transformations. Commun. Moscow Math. Soc. 63 (2008) 159-160. | Zbl

[5] F. Bonahon, Geodesic laminations on surfaces. In Laminations and foliations in dynamics, geometry and topology, vol. 269 of Contemp. Math. Amer. Math. Soc. (2001) 1-37. | MR | Zbl

[6] D. Calegari, Foliations and the geometry of 3-manifolds. Oxford Mathematical Monographs. Oxford University Press, Oxford (2007). | MR | Zbl

[7] J. Cassaigne, Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. 1 (1997) 67-88. | MR | Zbl

[8] J. Cassaigne and F. Nicolas, Factor complexity, Combinatorics, automata and number theory, vol. 135 of Encyclopedia Math. Appl. Cambridge University Press, Cambridge (2010) 163-247. | MR | Zbl

[9] A.J. Casson and S. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, vol. 9 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1988). | MR | Zbl

[10] V. Dujmović et al., A fixed-parameter approach to 2-layer planarization. Algorithmica 45 (2006) 159-182. | Zbl

[11] S. Ferenczi and L.Q. Zamboni Languages of k-interval exchange transformations. Bull. Lond. Math. Soc. 40 (2008) 705-714. | MR | Zbl

[12] N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics, vol. 1794 of Lect. Notes Math. Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. Springer, Verlag, Berlin (2002). | MR | Zbl

[13] R.K. Guy, Outerthickness and outercoarseness of graphs, Combinatorics in Proc. British Combinatorial Conf., Univ. Coll. Wales, Aberystwyth, 1973. London Math. Soc. Lect. Note Ser. Cambridge University Press, London (1974) 57-60. | MR | Zbl

[14] I. Hargittai and C.A. Pickover, Spiral Symmetry. World Scientific (1992). | MR

[15] A.E. Hatcher, Measured lamination spaces for surfaces, from the topological viewpoint. Topology Appl. 30 (198) 8 63-88. | MR | Zbl

[16] M. Keane, Interval exchange transformations. Math. Z. 141 (1975) 25-31. | MR | Zbl

[17] D. Lind and B. Marcus, Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995). | MR | Zbl

[18] L.-M. Lopez and Ph. Narbel, Languages, D0L-systems, sets of curves, and surface automorphisms. Inform. Comput. 180 (2003) 30-52. | MR | Zbl

[19] L.-M. Lopez and Ph. Narbel, Lamination languages. Ergodic Theory Dynam. Systems 33 (2013) 1813-1863. | MR | Zbl

[20] M. Lothaire, Combinatorics on Words, number 17 in Encyclopedia of Math. Appl. Cambridge University Press, Cambridge (1997). | MR | Zbl

[21] R. Mañé, Ergodic Theory and Differentiable Dynamics. Springer-Verlag, Berlin (1987). | MR | Zbl

[22] M. Morse and G.A. Hedlund, Symbolic dynamics I. Amer. J. Math. 60 (1938) 815-866. | JFM | MR

[23] M. Morse and G.A. Hedlund, Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940) 1-42. | JFM | MR

[24] R.C. Penner, A construction of pseudo-Anosov homeomorphisms. Trans. Amer. Math. Soc. 310 (1988) 179-197. | MR | Zbl

[25] R.C. Penner and J.L. Harer, Combinatorics of train tracks, vol. 125 of Annal. Math. Studies. Princeton University Press, Princeton, NJ (1992). | MR | Zbl

[26] D. Perrin and J.P. Pin, Infinite Words, number 141 in Pure Appl. Math. Elsevier (2004). | Zbl

[27] M. Quéffelec, Substitution dynamical systems-spectral analysis, 2nd Edition. Vol. 1294 of Lect. Notes Math. Springer-Verlag, Berlin (2010). | MR | Zbl

[28] W.P. Thurston, The geometry and topology of three-manifolds. Princeton University Lecture Notes (Electronic version 1.1, 2002). http://library.msri.org/books/gt3m (1980).

[29] W.P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. 19 (1988) 417-431. | MR | Zbl

Cited by Sources: