The full periodicity kernel for \(\sigma\) maps (Q1329301)

By \(\sigma\) the authors denote the topological space formed by the points \((x,y)\) of \(\mathbb{R}^ 2\) such that either \(x^ 2 + y^ 2 = 1\) or \(0 \leq x \leq 2\) and \(y = 1\). They consider such continuous self-maps of \(\sigma\) which have the branching point \((0,1)\) as a fixed point and call them \(\sigma\) maps. In the paper it is proved that the so-called full periodicity kernel for \(\sigma\) maps is the set \(K = \{2,3,4,5,7\}\). This means that (\(\text{Per}(f)\) denotes the set of periods of all periodic points of \(f\)): 1) \(\text{Per}(f) \supseteq K\) implies \(\text{Per}(r) = \mathbb{N}\) whenever \(f\) is a \(\sigma\) map, and 2) if \(S \subseteq \mathbb{N}\) is a set such that for every \(\sigma\) map \(f\), \(\text{Per}(f) \supseteq S\) implies \(\text{Per}(f) = \mathbb{N}\), then \(K \subseteq S\). In the paper one can also find a list of spaces for which the full periodicity kernel has been computed and corresponding references.