We will focus on the mutual relations between singular values,
periodic rays and periodic orbits in transcendental dynamics.
It is classically known that the existence of non-repelling periodic
points is connected to the presence of a singular value 'near by'- for
example, attracting and parabolic basins need to contain at least one
singular value. For repelling periodic points, singular values come
into play when the periodic point in question is not the landing point
of any periodic ray.
In this talk we will show that under our assumptions, it is possible
to associate a specific singular orbit to every non-repelling cycle,
as well as to every repelling cycle whose points are not landing
points of periodic rays.
This gives a version of the Fatou-Shishikura inequality which takes
into account such repelling cycles.
This is joint work with N. Fagella.