The set of rotation numbers of periodic points in certain nonseparating plane continua is a rational interval

Let F be an orientation-preserving homeomorphism of the plane that has a fixed point which is contained in an invariant, nonseparating continuum Lambda. If p/q is a reduced rational in the interior of the convex hull of the rotation set of Lambda about the fixed point, then there exists a q-periodic point in Lambda with rotation number p/q, provided that p/q is not the local rotation number about the fixed point and that Lambda satisfies certain technical requirements. We also show that the local rotation number is a point in the closure of the rotation set of Lambda and that if this rotation set is nondegenerate, then Lambda is indecomposable.