A converse result concerning the periodic structure of commuting affine circle maps

We analyze the set of periods of a class of maps φd,κ : Z∆ → Z∆ defined by φd,κ(x) = dx + κ, d, κ ∈ Z∆, where ∆ is an integer greater than 1. This study is important to characterize completely the period sets of alternated systems f, g, f, g, . . . , where f, g : S1 → S1 are affine circle maps that commute, and to solve the converse problem of constructing commuting affine circle maps having a prescribed set of periods. All rights reserved. Qc 2016 Keywords: Affine maps, alternated system, periods, circle maps, degree, combinatorial dynamics, ring of residues modulo m, Abelian multiplicative group of residues modulo m, Euler function, congruence, order, generator.