Rotation sets for graph maps of degree 1

Alsedà, Lluís; Ruette, Sylvie

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 $\alpha$ dans cet intervalle il existe un point périodique de nombre de rotation $\alpha$.

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 $\alpha$ in this interval there exists a periodic point of rotation number $\alpha$.

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

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