Dynamics of annulus maps. III: Periodic points and completeness (Q2821956)

Surjective continuous maps of an open plane annulus \(f:A\to A\) 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: {\parindent = 0.4 cm \begin{itemize}\item[1.] \(d>1\) and \(f\) has a completely invariant essential continuum; \item[2.] \(| d | >1\) and \(f\) has a forward invariant essential continuum which is locally connected; \item [3.] \(| d | >1\) and \(f\) extends continuously to the boundary such that both components of the boundary are attracting (or both are repelling). NEWLINENEWLINE\end{itemize}} Simple applications to maps of the 2-sphere are given.NEWLINENEWLINEFor 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)].