Rotation sets for graph maps of degree 1

Alsedà, Lluís; Ruette, Sylvie

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), p. 1233-1294 / Harvested from Numdam

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Résumé
Abstract

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.

Publié le : 2008-01-01

MR 2427960 | Zbl pre05303675

DOI : https://doi.org/10.5802/aif.2384
Classification: 37E45, 37E25, 54H20, 37E15
Mots clés: nombres de rotation, transformations de graphes, ensembles de périodes

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     title = {Rotation sets for graph maps of degree~1},
     journal = {Annales de l'Institut Fourier},
     volume = {58},
     year = {2008},
     pages = {1233-1294},
     doi = {10.5802/aif.2384},
     zbl = {pre05303675},
     mrnumber = {2427960},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2008__58_4_1233_0}
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Alsedà, Lluís; Ruette, Sylvie. Rotation sets for graph maps of degree 1. Annales de l'Institut Fourier, Tome 58 (2008) pp. 1233-1294. doi : 10.5802/aif.2384. http://gdmltest.u-ga.fr/item/AIF_2008__58_4_1233_0/

Bibliographie

[1] Alsedà, Ll.; Juher, D.; Mumbrú, P. Sets of periods for piecewise monotone tree maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., Tome 13 (2003) no. 2, pp. 311-341 | Article | MR 1972155 | Zbl 1056.37049

[2] Alsedà, Ll.; Juher, D.; Mumbrú, P. On the preservation of combinatorial types for maps on trees, Ann. Inst. Fourier (Grenoble), Tome 55 (2005) no. 7, pp. 2375-2398 | Article | Numdam | MR 2207387 | Zbl 1085.37035

[3] Alsedà, Ll.; Juher, D.; Mumbrú, P. Periodic behavior on trees, Ergodic Theory Dynam. Systems, Tome 25 (2005) no. 5, pp. 1373-1400 | Article | MR 2173425 | Zbl 1077.37031

[4] Alsedà, Ll.; Juher, D.; Mumbrú, P. Minimal dynamics for tree maps, Discrete Contin. Dyn. Syst. Ser. A, Tome 3 (2006) no. 20, pp. 511-541 | MR 2373202 | Zbl 1146.37024

[5] Alsedà, Ll.; Llibre, J.; Misiurewicz, M. Periodic orbits of maps of $Y$ , Trans. Amer. Math. Soc., Tome 313 (1989) no. 2, pp. 475-538 | Article | MR 958882 | Zbl 0803.54032

[6] Alsedà, Ll.; Llibre, J.; Misiurewicz, M. Combinatorial dynamics and entropy in dimension one, World Scientific Publishing Co. Inc., River Edge, NJ, Advanced Series in Nonlinear Dynamics, 5 (1993) | MR 1255515 | Zbl 0843.58034

[7] Alsedà, Ll.; Mañosas, F.; Mumbrú, P. Minimizing topological entropy for continuous maps on graphs, Ergodic Theory Dynam. Systems, Tome 20 (2000) no. 6, pp. 1559-1576 | Article | MR 1804944 | Zbl 0992.37014

[8] Baldwin, S. An extension of Sharkovskiĭ’s theorem to the $n -od$ , Ergodic Theory Dynam. Systems, Tome 11 (1991) no. 2, pp. 249-271 | Article | Zbl 0741.58010

[9] Baldwin, S.; Llibre, J. Periods of maps on trees with all branching points fixed, Ergodic Theory Dynam. Systems, Tome 15 (1995) no. 2, pp. 239-246 | Article | MR 1332402 | Zbl 0831.58020

[10] Bernhardt, C. Vertex maps for trees: algebra and periods of periodic orbits, Discrete Contin. Dyn. Syst., Tome 14 (2006) no. 3, pp. 399-408 | Article | MR 2171718 | Zbl 1110.37033

[11] Block, L. Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc., Tome 72 (1978) no. 3, pp. 576-580 | Article | MR 509258 | Zbl 0365.58015

[12] Block, L.; Guckenheimer, J.; Misiurewicz, M.; Young, L. S. Periodic points and topological entropy of one dimensional maps, Global Theory of Dynamical Systems, Springer-Verlag (Lecture Notes in Mathematics, no. 819) (1980), pp. 18-34 | MR 591173 | Zbl 0447.58028

[13] Ito, R. Rotation sets are closed, Math. Proc. Cambridge Philos. Soc., Tome 89 (1981) no. 1, pp. 107-111 | Article | MR 591976 | Zbl 0484.58027

[14] Leseduarte, M. C.; Llibre, J. On the set of periods for $σ$ maps, Trans. Amer. Math. Soc., Tome 347 (1995) no. 12, pp. 4899-4942 | Article | MR 1316856 | Zbl 0868.54035

[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., Tome 13 (2003) no. 7, pp. 1743-1754 (Dynamical systems and functional equations (Murcia, 2000)) | Article | MR 2015625 | Zbl 1056.37051

[16] Misiurewicz, M. Periodic points of maps of degree one of a circle, Ergodic Theory Dynamical Systems, Tome 2 (1982) no. 2, p. 221-227 (1983) | MR 693977 | Zbl 0508.58038

[17] Rhodes, F.; Thompson, C. L. Rotation numbers for monotone functions on the circle, J. London Math. Soc. (2), Tome 34 (1986) no. 2, pp. 360-368 | Article | MR 856518 | Zbl 0623.58008

[18] Sharkovskiĭ, A. N. Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Ž., Tome 16 (1964), pp. 61-71 ((in Russian)) | MR 159905

[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. Publ., River Edge, NJ (World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc.) Tome 8 (1995), pp. 1-11 (Translated by J. Tolosa, Reprint of the paper reviewed in MR1361914 (96j:58058)) | Zbl 0890.58012

[20] Wall, C. T. C. A geometric introduction to topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1972) | MR 478128 | Zbl 0261.55001

[21] Zeng, F.; Mo, H.; Guo, W.; Gao, Q. $ω$ -limit set of a tree map, Northeast. Math. J., Tome 17 (2001) no. 3, pp. 333-339 | MR 2011841 | Zbl 1026.37032

[22] Ziemian, K. Rotation sets for subshifts of finite type, Fund. Math., Tome 146 (1995) no. 2, pp. 189-201 | MR 1314983 | Zbl 0821.58017