Rotation sets for graph maps of degree 1
[Ensembles de rotation pour des transformations de graphes de degré 1]
Annales de l'Institut Fourier, Tome 58 (2008) no. 4, pp. 1233-1294.

Pour une transformation continue sur un graphe topologique contenant une boucle S, il est possible de définir le degré (par rapport à la boucle S) et, quand la transformation est de degré 1, des nombres de rotation. Nous étudions l’ensemble de rotation de ces transformations et les périodes des points périodiques ayant un nombre de rotation donné. Nous montrons que, si le graphe a une unique boucle S, alors l’ensemble des nombres de rotation des points de S a des propriétés similaires à celles de l’ensemble de rotation d’une transformation du cercle ; en particulier, c’est un intervalle compact et pour tout rationnel α dans cet intervalle il existe un point périodique de nombre de rotation α.

Pour une classe particulière de transformations appelées transformations peignées, l’ensemble de rotation possède les mêmes bonnes propriétés que celui des transformations continues de degré 1 sur le cercle.

For a continuous map on a topological graph containing a loop S it is possible to define the degree (with respect to the loop S) and, for a map of degree 1, rotation numbers. We study the rotation set of these maps and the periods of periodic points having a given rotation number. We show that, if the graph has a single loop S then the set of rotation numbers of points in S has some properties similar to the rotation set of a circle map; in particular it is a compact interval and for every rational α in this interval there exists a periodic point of rotation number α.

For a special class of maps called combed maps, the rotation set displays the same nice properties as the continuous degree one circle maps.

DOI : 10.5802/aif.2384
Classification : 37E45, 37E25, 54H20, 37E15
Keywords: Rotation numbers, graph maps, sets of periods
Mot clés : nombres de rotation, transformations de graphes, ensembles de périodes
Alsedà, Lluís 1 ; Ruette, Sylvie 2

1 Universitat Autònoma de Barcelona Departament de Matemàtiques 08913 Cerdanyola del Vallès, Barcelona (Spain)
2 Université Paris-Sud 11 Laboratoire de Mathématiques CNRS UMR 8628 Bâtiment 425 91405 Orsay cedex (France)
@article{AIF_2008__58_4_1233_0,
     author = {Alsed\`a, Llu{\'\i}s and Ruette, Sylvie},
     title = {Rotation sets for graph maps of degree~1},
     journal = {Annales de l'Institut Fourier},
     pages = {1233--1294},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {4},
     year = {2008},
     doi = {10.5802/aif.2384},
     mrnumber = {2427960},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2384/}
}
TY  - JOUR
AU  - Alsedà, Lluís
AU  - Ruette, Sylvie
TI  - Rotation sets for graph maps of degree 1
JO  - Annales de l'Institut Fourier
PY  - 2008
SP  - 1233
EP  - 1294
VL  - 58
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2384/
DO  - 10.5802/aif.2384
LA  - en
ID  - AIF_2008__58_4_1233_0
ER  - 
%0 Journal Article
%A Alsedà, Lluís
%A Ruette, Sylvie
%T Rotation sets for graph maps of degree 1
%J Annales de l'Institut Fourier
%D 2008
%P 1233-1294
%V 58
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2384/
%R 10.5802/aif.2384
%G en
%F AIF_2008__58_4_1233_0
Alsedà, Lluís; Ruette, Sylvie. Rotation sets for graph maps of degree 1. Annales de l'Institut Fourier, Tome 58 (2008) no. 4, pp. 1233-1294. doi : 10.5802/aif.2384. http://www.numdam.org/articles/10.5802/aif.2384/

[1] Alsedà, Ll.; Juher, D.; Mumbrú, P. Sets of periods for piecewise monotone tree maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., Volume 13 (2003) no. 2, pp. 311-341 | DOI | MR | Zbl

[2] Alsedà, Ll.; Juher, D.; Mumbrú, P. On the preservation of combinatorial types for maps on trees, Ann. Inst. Fourier (Grenoble), Volume 55 (2005) no. 7, pp. 2375-2398 | DOI | Numdam | MR | Zbl

[3] Alsedà, Ll.; Juher, D.; Mumbrú, P. Periodic behavior on trees, Ergodic Theory Dynam. Systems, Volume 25 (2005) no. 5, pp. 1373-1400 | DOI | MR | Zbl

[4] Alsedà, Ll.; Juher, D.; Mumbrú, P. Minimal dynamics for tree maps, Discrete Contin. Dyn. Syst. Ser. A, Volume 3 (2006) no. 20, pp. 511-541 | MR | Zbl

[5] Alsedà, Ll.; Llibre, J.; Misiurewicz, M. Periodic orbits of maps of Y, Trans. Amer. Math. Soc., Volume 313 (1989) no. 2, pp. 475-538 | DOI | MR | Zbl

[6] Alsedà, Ll.; Llibre, J.; Misiurewicz, M. Combinatorial dynamics and entropy in dimension one, Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co. Inc., River Edge, NJ, 1993 | MR | Zbl

[7] Alsedà, Ll.; Mañosas, F.; Mumbrú, P. Minimizing topological entropy for continuous maps on graphs, Ergodic Theory Dynam. Systems, Volume 20 (2000) no. 6, pp. 1559-1576 | DOI | MR | Zbl

[8] Baldwin, S. An extension of Sharkovskiĭ’s theorem to the n-od, Ergodic Theory Dynam. Systems, Volume 11 (1991) no. 2, pp. 249-271 | DOI | Zbl

[9] Baldwin, S.; Llibre, J. Periods of maps on trees with all branching points fixed, Ergodic Theory Dynam. Systems, Volume 15 (1995) no. 2, pp. 239-246 | DOI | MR | Zbl

[10] Bernhardt, C. Vertex maps for trees: algebra and periods of periodic orbits, Discrete Contin. Dyn. Syst., Volume 14 (2006) no. 3, pp. 399-408 | DOI | MR | Zbl

[11] Block, L. Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc., Volume 72 (1978) no. 3, pp. 576-580 | DOI | MR | Zbl

[12] Block, L.; Guckenheimer, J.; Misiurewicz, M.; Young, L. S. Periodic points and topological entropy of one dimensional maps, Global Theory of Dynamical Systems (Lecture Notes in Mathematics, no. 819), Springer-Verlag, 1980, pp. 18-34 | MR | Zbl

[13] Ito, R. Rotation sets are closed, Math. Proc. Cambridge Philos. Soc., Volume 89 (1981) no. 1, pp. 107-111 | DOI | MR | Zbl

[14] Leseduarte, M. C.; Llibre, J. On the set of periods for σ maps, Trans. Amer. Math. Soc., Volume 347 (1995) no. 12, pp. 4899-4942 | DOI | MR | Zbl

[15] Llibre, J.; Paraños, J.; Rodríguez, J. A. Periods for continuous self-maps of the figure-eight space, Internat. J. Bifur. Chaos Appl. Sci. Engrg., Volume 13 (2003) no. 7, pp. 1743-1754 Dynamical systems and functional equations (Murcia, 2000) | DOI | MR | Zbl

[16] Misiurewicz, M. Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, Volume 2 (1982) no. 2, p. 221-227 (1983) | MR | Zbl

[17] Rhodes, F.; Thompson, C. L. Rotation numbers for monotone functions on the circle, J. London Math. Soc. (2), Volume 34 (1986) no. 2, pp. 360-368 | DOI | MR | Zbl

[18] Sharkovskiĭ, A. N. Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Ž., Volume 16 (1964), pp. 61-71 (in Russian) | MR

[19] Sharkovskiĭ, A. N. Coexistence of cycles of a continuous map of the line into itself, Thirty years after Sharkovskiĭ’s theorem: new perspectives (Murcia, 1994) (World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc.), Volume 8, World Sci. Publ., River Edge, NJ, 1995, pp. 1-11 Translated by J. Tolosa, Reprint of the paper reviewed in MR1361914 (96j:58058) | Zbl

[20] Wall, C. T. C. A geometric introduction to topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972 | MR | Zbl

[21] Zeng, F.; Mo, H.; Guo, W.; Gao, Q. ω-limit set of a tree map, Northeast. Math. J., Volume 17 (2001) no. 3, pp. 333-339 | MR | Zbl

[22] Ziemian, K. Rotation sets for subshifts of finite type, Fund. Math., Volume 146 (1995) no. 2, pp. 189-201 | EuDML | MR | Zbl

Cité par Sources :