Infinite periodic points of endomorphisms over special confluent rewriting systems

Julien Cassaigne[1]; Pedro V. Silva[2]

  • [1] Institut de Mathématiques de Luminy Case 907 13288 Marseille Cedex 9 (France)
  • [2] Universidade do Porto Centro de Matemática Faculdade de Ciências R. Campo Alegre 687 4169-007 Porto (Portugal)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 2, page 769-810
  • ISSN: 0373-0956

Abstract

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We consider endomorphisms of a monoid defined by a special confluent rewriting system that admit a continuous extension to the completion given by reduced infinite words, and study from a dynamical viewpoint the nature of their infinite periodic points. For prefix-convergent endomorphisms and expanding endomorphisms, we determine the structure of the set of all infinite periodic points in terms of adherence values, bound the periods and show that all regular periodic points are attractors.

How to cite

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Cassaigne, Julien, and Silva, Pedro V.. "Infinite periodic points of endomorphisms over special confluent rewriting systems." Annales de l’institut Fourier 59.2 (2009): 769-810. <http://eudml.org/doc/10411>.

@article{Cassaigne2009,
abstract = {We consider endomorphisms of a monoid defined by a special confluent rewriting system that admit a continuous extension to the completion given by reduced infinite words, and study from a dynamical viewpoint the nature of their infinite periodic points. For prefix-convergent endomorphisms and expanding endomorphisms, we determine the structure of the set of all infinite periodic points in terms of adherence values, bound the periods and show that all regular periodic points are attractors.},
affiliation = {Institut de Mathématiques de Luminy Case 907 13288 Marseille Cedex 9 (France); Universidade do Porto Centro de Matemática Faculdade de Ciências R. Campo Alegre 687 4169-007 Porto (Portugal)},
author = {Cassaigne, Julien, Silva, Pedro V.},
journal = {Annales de l’institut Fourier},
keywords = {Periodic points; endomorphisms; rewriting systems; dynamics; periodic points},
language = {eng},
number = {2},
pages = {769-810},
publisher = {Association des Annales de l’institut Fourier},
title = {Infinite periodic points of endomorphisms over special confluent rewriting systems},
url = {http://eudml.org/doc/10411},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Cassaigne, Julien
AU - Silva, Pedro V.
TI - Infinite periodic points of endomorphisms over special confluent rewriting systems
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 2
SP - 769
EP - 810
AB - We consider endomorphisms of a monoid defined by a special confluent rewriting system that admit a continuous extension to the completion given by reduced infinite words, and study from a dynamical viewpoint the nature of their infinite periodic points. For prefix-convergent endomorphisms and expanding endomorphisms, we determine the structure of the set of all infinite periodic points in terms of adherence values, bound the periods and show that all regular periodic points are attractors.
LA - eng
KW - Periodic points; endomorphisms; rewriting systems; dynamics; periodic points
UR - http://eudml.org/doc/10411
ER -

References

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  1. J.-P. Allouche, J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, (2003), Cambridge University Press Zbl1086.11015MR1997038
  2. M. Benois, Descendants of regular language in a class of rewriting systems: algorithm and complexity of an automata construction, Proceedings RTA 87 256 (1987), 121-132, Springer-Verlag, Berlin Zbl0643.68119MR903667
  3. J. Berstel, Transductions and Context-free Languages, (1979), Teubner, Stuttgart Zbl0424.68040MR549481
  4. M. Bestvina, M. Feighn, M. Handel, The Tits alternative for O u t ( F n ) , I: Dynamics of exponentially-growing automorphisms, Ann. Math. 151 (2000), 517-623 Zbl0984.20025MR1765705
  5. M. Bestvina, M. Handel, Train tracks and automorphisms of free groups, Ann. Math. 135 (1992), 1-51 Zbl0757.57004MR1147956
  6. R. V. Book, Confluent and other types of Thue systems, J. Assoc. Comput. Mach. 29 (1982), 171-182 Zbl0478.68032MR662617
  7. R. V. Book, F. Otto, String-Rewriting Systems, (1993), Springer-Verlag, New York Zbl0832.68061MR1215932
  8. J. Cassaigne, P. V. Silva, Infinite words and confluent rewriting systems: endomorphism extensions, (2005) Zbl1213.68477
  9. J. Dugundji, Topology, (1966), Allyn and Bacon, Boston Zbl0144.21501MR193606
  10. D. Gaboriau, A. Jaeger, G. Levitt, M. Lustig, An index for counting fixed points of automorphisms of free groups, Duke Math. J. 93 (1998), 425-452 Zbl0946.20010MR1626723
  11. E. Ghys, P. de la Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov, (1990), Birkhäuser, Boston Zbl0731.20025MR1086648
  12. A. Hilion, Dynamique des automorphismes du groupe libre, (2004) 
  13. G. Levitt, M. Lustig, Periodic ends, growth rates, Hölder dynamics for automorphisms of free groups, Comment. Math. Helv. 75 (2000), 415-429 Zbl0965.20026MR1793796
  14. G. Levitt, M. Lustig, Automorphisms of free groups have asymptotically periodic dynamics, J. Reine Angew. Math. 619 (2008), 1-36 Zbl1157.20017MR2414945
  15. M. Lothaire, Combinatorics on Words, (1983), Addison-Wesley, Reading Zbl0514.20045MR675953
  16. M. Lothaire, Algebraic Combinatorics on Words, (2002), Cambridge University Press, Cambridge Zbl1001.68093MR1905123
  17. R. C. Lyndon, P. E. Schupp, Combinatorial Group Theory, (1977), Springer-Verlag, Berlin Zbl0368.20023MR577064
  18. D. Perrin, J.-E. Pin, Infinite Words: Automata, Semigroups, Logic and Games, Pure and Applied Mathematics Series 141 (2004), Elsevier Academic Press, Amsterdam Zbl1094.68052
  19. G. Sénizergues, On the rational subsets of the free group, Acta Informatica 33 (1996), 281-296 Zbl0858.68044MR1393764
  20. P. V. Silva, Rational subsets of partially reversible monoids, Theoret. Comp. Sci. Zbl1155.68047
  21. P. V. Silva, Free group languages: rational versus recognizable, Theoret. Inform. Appl. 38 (2004), 49-67 Zbl1082.68071MR2059028

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