General Construction of Non-Dense Disjoint Iteration Groups on the Circle

Let be a disjoint iteration group on the unit circle , that is a family of homeomorphisms such that F ^v1 ○ F ^v2 = F ^v1+v2 for v ₁, v ₂ ∈ V and each F ^v either is the identity mapping or has no fixed point ((V, +) is a 2-divisible nontrivial Abelian group). Denote by the set of all cluster points of {F ^v (z), v ∈ V} for . In this paper we give a general construction of disjoint iteration groups for which .