Substitutions on two letters, cutting segments and their projections

Sierk W. Rosema[1]

  • [1] Mathematical Institute Leiden University P.O. Box 9512, 2300 RA Leiden The Netherlands

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 2, page 523-545
  • ISSN: 1246-7405

Abstract

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In this paper we study the structure of the projections of the finite cutting segments corresponding to unimodular substitutions over a two-letter alphabet. We show that such a projection is a block of letters if and only if the substitution is Sturmian. Applying the procedure of projecting the cutting segments corresponding to a Christoffel substitution twice results in the original substitution. This induces a duality on the set of Christoffel substitutions.

How to cite

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Rosema, Sierk W.. "Substitutions on two letters, cutting segments and their projections." Journal de Théorie des Nombres de Bordeaux 19.2 (2007): 523-545. <http://eudml.org/doc/10810>.

@article{Rosema2007,
abstract = {In this paper we study the structure of the projections of the finite cutting segments corresponding to unimodular substitutions over a two-letter alphabet. We show that such a projection is a block of letters if and only if the substitution is Sturmian. Applying the procedure of projecting the cutting segments corresponding to a Christoffel substitution twice results in the original substitution. This induces a duality on the set of Christoffel substitutions.},
affiliation = {Mathematical Institute Leiden University P.O. Box 9512, 2300 RA Leiden The Netherlands},
author = {Rosema, Sierk W.},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {2},
pages = {523-545},
publisher = {Université Bordeaux 1},
title = {Substitutions on two letters, cutting segments and their projections},
url = {http://eudml.org/doc/10810},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Rosema, Sierk W.
TI - Substitutions on two letters, cutting segments and their projections
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 2
SP - 523
EP - 545
AB - In this paper we study the structure of the projections of the finite cutting segments corresponding to unimodular substitutions over a two-letter alphabet. We show that such a projection is a block of letters if and only if the substitution is Sturmian. Applying the procedure of projecting the cutting segments corresponding to a Christoffel substitution twice results in the original substitution. This induces a duality on the set of Christoffel substitutions.
LA - eng
UR - http://eudml.org/doc/10810
ER -

References

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  8. N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics. Springer, 2002. Zbl1014.11015MR1970385
  9. G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France 110 (1982), 147–178. Zbl0522.10032MR667748
  10. G. Richomme, Test-words for Sturmian morphisms. Bull. Belg. Math. Soc. 6 (1999), 481–489. MR1732884
  11. G. Richomme, Lyndon morphisms. Bull. Belg. Math. Soc. 10 (2003), 761–785. Zbl1101.68075MR2073025
  12. S. W. Rosema, R. Tijdeman, The tribonacci substitution. Integers: Electron. J. Combin. Number Th. 5(3) (2005), A13. Zbl1099.11004MR2191759
  13. P. Séébold, Fibonacci morphisms and Sturmian words. Theoret. Comput. Sci. 195 (1991), 91–109. Zbl0981.68104MR1131075
  14. C. Series, The geometry of Markoff numbers. Math. Intelligencer 7, no. 3 (1985), 20–29. Zbl0566.10024MR795536

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