The Full Periodicity Kernel for σ Maps

Let σ be the topological space formed by the points (x, y) of R2 such that either x2 + y2 = 1, or 0 ≤ x ≤ 2 and y = 1. A σ map ƒ is a continuous self-map of σ having the branching point (0, 1) as a fixed point. We denote by Per(ƒ) the set of periods of all periodic points of ƒ, and by N the set of positive integers. We prove that if ƒ is a σ map and {2, 3, 4, 5, 7} ⊆ Per(ƒ), then Per(ƒ) = N. Conversely, if S ⊆ N is a set such that for every σ map ƒ S ⊆ Per(ƒ) implies Per(ƒ) = N, then {2, 3, 4, 5, 7} ⊆ S.