Rotation sets for graph maps of degree 1

Abstract

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.