  Consider a continuous surjective self map of the open annulus with degree d >
1. It is proved that the number of Nielsen classes of periodic points is
maximum possible whenever f has a completely invariant essential continuum. The
same result is obtained in negative degree |d| > 1 and for just forward
invariant essential continua, provided that the continuum is locally connected.
We also deal with the problem of wether there is a representative of each
Nielsen class in the filled set of the invariant continuum. Moreover, if the
map extends continuously to the boundary of the annulus and both boundary
components are either attracting or repelling, the hypothesis on the existence
of the invariant continuum is no longer needed for obtaining all the periodic
points in the interior of the annulus.
