Rotation sets for maps of degree 1 on sun graphs
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.