no. 2
Infinite periodic points of endomorphisms over special confluent rewriting systems
[Points périodiques infinis d’endomorphismes sur des systèmes de réécriture spéciaux confluents]
Cassaigne, Julien1 ; Silva, Pedro V.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, Tome 59 (2009) no. 2, pp. 769-810.

On considère les endomorphismes d’un monoïde défini par un système de réécriture spécial confluent qui admettent une extension continue à sa complétion donnée par les mots infinis réduits, et on étudie d’un point de vue dynamique la nature de leurs points périodiques infinis. Pour les endomorphismes préfixe-convergents et les endomorphismes expansifs, on détermine la structure de l’ensemble de tous les points périodiques infinis en termes de valeurs d’adhérence, on borne les périodes et on prouve que tous les points périodiques réguliers sont des attracteurs.

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.

DOI : 10.5802/aif.2447
Classification : 68R15, 37B10, 20M35, 68Q70
Keywords: Periodic points, endomorphisms, rewriting systems, dynamics
Mot clés : points périodiques, endomorphismes, systèmes de réécriture, dynamique
Cassaigne, Julien 1 ; Silva, Pedro V. 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) 
@article{AIF_2009__59_2_769_0,
     author = {Cassaigne, Julien and Silva, Pedro V.},
     title = {Infinite periodic points  of endomorphisms over special confluent rewriting systems},
     journal = {Annales de l'Institut Fourier},
     pages = {769--810},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {2},
     year = {2009},
     doi = {10.5802/aif.2447},
     zbl = {1166.68021},
     mrnumber = {2521435},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2447/}
}
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
SP  - 769
EP  - 810
VL  - 59
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2447/
DO  - 10.5802/aif.2447
LA  - en
ID  - AIF_2009__59_2_769_0
ER  - 
%0 Journal Article
%A Cassaigne, Julien
%A Silva, Pedro V.
%T Infinite periodic points  of endomorphisms over special confluent rewriting systems
%J Annales de l'Institut Fourier
%D 2009
%P 769-810
%V 59
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2447/
%R 10.5802/aif.2447
%G en
%F AIF_2009__59_2_769_0
Cassaigne, Julien; Silva, Pedro V. Infinite periodic points  of endomorphisms over special confluent rewriting systems. Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 769-810. doi : 10.5802/aif.2447. http://www.numdam.org/articles/10.5802/aif.2447/

[1] Allouche, J.-P.; Shallit, J. Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, 2003 | MR | Zbl

[2] Benois, M.; LNCS Descendants of regular language in a class of rewriting systems: algorithm and complexity of an automata construction, Proceedings RTA 87, Volume 256, Springer-Verlag, Berlin (1987), pp. 121-132 | MR | Zbl

[3] Berstel, J. Transductions and Context-free Languages, Teubner, Stuttgart, 1979 | MR | Zbl

[4] Bestvina, M.; Feighn, M.; Handel, M. The Tits alternative for Out(F n ), I: Dynamics of exponentially-growing automorphisms, Ann. Math., Volume 151 (2000), pp. 517-623 | DOI | MR | Zbl

[5] Bestvina, M.; Handel, M. Train tracks and automorphisms of free groups, Ann. Math., Volume 135 (1992), pp. 1-51 | DOI | MR | Zbl

[6] Book, R. V. Confluent and other types of Thue systems, J. Assoc. Comput. Mach., Volume 29 (1982), pp. 171-182 | MR | Zbl

[7] Book, R. V.; Otto, F. String-Rewriting Systems, Springer-Verlag, New York, 1993 | MR | Zbl

[8] Cassaigne, J.; Silva, P. V. Infinite words and confluent rewriting systems: endomorphism extensions (2005) (preprint)

[9] Dugundji, J. Topology, Allyn and Bacon, Boston, 1966 | MR | Zbl

[10] Gaboriau, D.; Jaeger, A.; Levitt, G.; Lustig, M. An index for counting fixed points of automorphisms of free groups, Duke Math. J., Volume 93 (1998), pp. 425-452 | DOI | MR | Zbl

[11] Ghys, E.; de la Harpe, P. Sur les groupes hyperboliques d’après Mikhael Gromov, Birkhäuser, Boston, 1990 | MR | Zbl

[12] Hilion, A. Dynamique des automorphismes du groupe libre, Université Paul Sabatier, Toulouse III (2004) (Thèse de Doctorat)

[13] Levitt, G.; Lustig, M. Periodic ends, growth rates, Hölder dynamics for automorphisms of free groups, Comment. Math. Helv., Volume 75 (2000), pp. 415-429 | DOI | MR | Zbl

[14] Levitt, G.; Lustig, M. Automorphisms of free groups have asymptotically periodic dynamics, J. Reine Angew. Math., Volume 619 (2008), pp. 1-36 | DOI | MR

[15] Lothaire, M. Combinatorics on Words, Addison-Wesley, Reading, 1983 | MR | Zbl

[16] Lothaire, M. Algebraic Combinatorics on Words, Cambridge University Press, Cambridge, 2002 | MR | Zbl

[17] Lyndon, R. C.; Schupp, P. E. Combinatorial Group Theory, Springer-Verlag, Berlin, 1977 | MR | Zbl

[18] Perrin, D.; Pin, J.-E. Infinite Words: Automata, Semigroups, Logic and Games, Pure and Applied Mathematics Series, Volume 141, Elsevier Academic Press, Amsterdam, 2004 | Zbl

[19] Sénizergues, G. On the rational subsets of the free group, Acta Informatica, Volume 33 (1996), pp. 281-296 | DOI | MR | Zbl

[20] Silva, P. V. Rational subsets of partially reversible monoids, Theoret. Comp. Sci. (to appear)

[21] Silva, P. V. Free group languages: rational versus recognizable, Theoret. Inform. Appl., Volume 38 (2004), pp. 49-67 | DOI | Numdam | MR | Zbl

Cité par Sources :