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BALANCE PROPERTIES OF ARNOUX–RAUZY WORDS

VALÉRIE BERTHÉ

LIAFA, CNRS, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France

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JULIEN CASSAIGNE

IML, CNRS, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France

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, and

WOLFGANG STEINER

LIAFA, CNRS, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France

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https://doi.org/10.1142/S0218196713400043Cited by:12 (Source: Crossref)

Abstract

This paper deals with balances and imbalances in Arnoux–Rauzy words. We provide sufficient conditions for C-balancedness, but our results indicate that even a characterization of 2-balanced Arnoux–Rauzy words on a 3-letter alphabet is not immediate.

Dedicated to Christophe Reutenauer on the occasion of his 60th birthday

Keywords:

AMSC: 68R15, 37B10

References

P. Arnoux and S. Ito , Bull. Belg. Math. Soc. Simon Stevin 8 , 181 ( 2001 ) . Crossref, Web of Science, Google Scholar
P. Arnoux and G. Rauzy, Bull. Soc. Math. France 119(2), 199 (1991). Crossref, Web of Science, Google Scholar
J. Berstel et al. , J. Algebra 369 , 146 ( 2012 ) . Crossref, Web of Science, Google Scholar
J. Berstel et al. , Combinatorics on Words. Christoffel Words and Repetitions in Words , CRM Monograph Series 27 ( American Mathematical Society , Providence, RI , 2009 ) . Google Scholar
V. Berthé, A. de Luca and C. Reutenauer, European J. Combin. 29(2), 535 (2008). Crossref, Web of Science, Google Scholar
V. Berthé, S. Ferenczi and L. Q. Zamboni, Algebraic and Topological Dynamics, Contempory Mathematics 385 (American Mathematical Society, Providence, RI, 2005) pp. 333–364. Crossref, Google Scholar
V. Berthé , T. Jolivet and A. Siegel , Uniform Distribution Theory 7 , 173 ( 2012 ) . Google Scholar
J.-P. Borel and C. Reutenauer , Theor. Inform. Appl. 40 , 15 ( 2006 ) . Crossref, Web of Science, Google Scholar
M. G. Castelli , F. Mignosi and A. Restivo , Theoret. Comput. Sci. 218 , 83 ( 1999 ) . Crossref, Web of Science, Google Scholar
J. Cassaigne and N. Chekhova, Ann. Inst. Fourier (Grenoble) 56(7), 2249 (2006). Crossref, Web of Science, Google Scholar
J. Cassaigne, S. Ferenczi and A. Messaoudi, Ann. Inst. Fourier (Grenoble) 58(6), 1983 (2008). Crossref, Web of Science, Google Scholar
J. Cassaigne, S. Ferenczi and L. Q. Zamboni, Ann. Inst. Fourier (Grenoble) 50(4), 1265 (2000). Crossref, Web of Science, Google Scholar
F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam. Syst. 20(4) (2000) 1061–1078, Corrigendum and addendum, Ergodic Theory Dynam. Syst. 23(2) (2003) 663–669 . Google Scholar
A. Glen and J. Justin, Theor. Inform. Appl. 43(3), 403 (2009). Crossref, Web of Science, Google Scholar
P. Hubert , Theoret. Comput. Sci. 242 , 91 ( 2000 ) . Crossref, Web of Science, Google Scholar
C. Kassel and C. Reutenauer , Ann. Mat. Pura Appl. (4) 186 , 317 ( 2007 ) . Crossref, Web of Science, Google Scholar
M. Lothaire , Algebraic Combinatorics on Words ( Cambridge University Press , 2002 ) . Crossref, Google Scholar
G. Paquin , Theoret. Comput. Sci. 410 , 3782 ( 2009 ) . Crossref, Web of Science, Google Scholar
G. Paquin and C. Reutenauer, Theoret. Comput. Sci. 412(5), 402 (2011). Crossref, Web of Science, Google Scholar
G. Paquin and L. Vuillon , Electron. J. Combin. 14 , ( 2007 ) . Crossref, Google Scholar
N. Pytheas Fogg , Substitutions in Dynamics, Arithmetics and Combinatorics , Lecture Notes in Mathematics 1794 ( Springer-Verlag , 2002 ) . Crossref, Google Scholar
G. Richomme , K. Saari and L. Q. Zamboni , Adv. Appl. Math. 45 , 212 ( 2010 ) . Crossref, Web of Science, Google Scholar
L. Vuillon , Bull. Belg. Math. Soc. Simon Stevin 10 , 787 ( 2003 ) . Crossref, Web of Science, Google Scholar