Dynamics of annulus maps. III: Periodic points and completeness.

Surjective continuous maps of an open plane annulus f : A → A of degree |d| > 1 are studied. Such a map is complete if for every n the number of Nielsen classes of fixed points of an iteration f n is equal to |d n − 1|, i.e., it is maximal for a map of degree d. The authors give the following sufficient conditions for completeness: 1. d > 1 and f has a completely invariant essential continuum; 2. |d| > 1 and f has a forward invariant essential continuum which is locally connected; 3. |d| > 1 and f extends continuously to the boundary such that both components of the boundary are attracting (or both are repelling). Simple applications to maps of the 2-sphere are given. For Parts I and II, see [the authors, Math. Z. 284, No. 1–2, 209–229 (2016; Zbl 1367.37041); Fundam. Math. 235, No. 3, 257–276 (2016; Zbl 1375.37127)].