Abstract

We introduce a subclass of the class of continuous degree one functions on a topological graph containing a unique loop which displays a rotation theory analogous to the continuous degree one circle maps. We call these maps combed. We extend the notions of lower and upper lifting and water functions in the spirit of the circle case to the class of combed maps and, with the help of these tools, we show that the main results concerning rotation numbers and periods for continuous degree one circle maps extend nicely to the class of combed graph maps.