Rotation set for maps of degree 1 on sun graphs

Abstract

For a continuous map on a topological graph containing a unique loop S, it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and for every rational r in this interval there exists a periodic point of rotation number r. The whole rotation set (i.e. the set of all rotation numbers) may not be connected and it is not known in general whether it is closed. A sun graph is the space consisting in finitely many segments attached by one of their endpoints to a circle. We show that, for a map of degree 1 on a sun graph, the rotation set is closed and has finitely many connected components. Moreover, for all but finitely many rational numbers r in the rotation set, there exists a periodic point of rotation number r.