Rotation sets for graph maps of degree 1
Annales de l'Institut Fourier, Volume 58 (2008) no. 4, p. 1233-1294

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.

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.

DOI : https://doi.org/10.5802/aif.2384
Classification:  37E45,  37E25,  54H20,  37E15
Keywords: Rotation numbers, graph maps, sets of periods
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {4},
     year = {2008},
     pages = {1233-1294},
     doi = {10.5802/aif.2384},
     mrnumber = {2427960},
     zbl = {pre05303675},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_4_1233_0}
}
Alsedà, Lluís; Ruette, Sylvie. Rotation sets for graph maps of degree 1. Annales de l'Institut Fourier, Volume 58 (2008) no. 4, pp. 1233-1294. doi : 10.5802/aif.2384. http://www.numdam.org/item/AIF_2008__58_4_1233_0/

[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