Dynamics of annulus maps III: completeness

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.