no. 4
Rotation sets for graph maps of degree 1
[Ensembles de rotation pour des transformations de graphes de degré 1]
Alsedà, Lluís1 ; Ruette, Sylvie2
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)
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) 
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2384/

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