Rotation sets for graph maps of degree 1

Alsedà, Lluís; Ruette, Sylvie

doi:10.5802/aif.2384

Rotation sets for graph maps of degree 1
[Ensembles de rotation pour des transformations de graphes de degré 1]

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)

Annales de l'Institut Fourier, Tome 58 (2008) no. 4, pp. 1233-1294.

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.

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

Affiliations des auteurs :

Alsedà, Lluís ¹ ; Ruette, Sylvie ²

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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://www.numdam.org/articles/10.5802/aif.2384/

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