In this talk we will look at the mutual relations between singular values, periodic rays and periodic orbits for transcendental entire maps.

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.