# Rotation sets for graph maps of degree 1

## TL;DR

This paper investigates rotation sets for degree 1 maps on topological graphs with loops, revealing properties similar to circle maps, including the existence of periodic points for rational rotation numbers.

## Contribution

It characterizes the rotation set for degree 1 graph maps, especially for graphs with a single loop and for combed maps, showing they share properties with circle maps.

## Key findings

- Rotation set is a compact interval for graphs with a single loop.
- Existence of periodic points for every rational rotation number within the set.
- For combed maps, the rotation set exhibits properties akin to continuous degree one circle maps.

## 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.