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 f(hook) is a continuous self-map of σ having the branching point (0, 1) as a fixed point. We denote by Per(f(hook)) the set of periods of all periodic points of f(hook), and by N the set of positive integers. We prove that if f(hook) is a σ map and (2, 3, 4, 5, 7) ⊆ Per(f(hook)), then Per(f(hook)) = N. Conversely, if S ⊆ N is a set such that for every σ map f(hook) S ⊆ Per(f(hook)) implies Per(f(hook)) = N, then (2, 3, 4, 5, 7) ⊆ S. © 1994 Academic Press, Inc.