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

Abstract

Let $F = \{ F^v :\mathbb{S}^1 \to \mathbb{S}^1 ,v \in V\}$ be a disjoint iteration group on the unit circle $\mathbb{S}^1$, 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 $L_F$ the set of all cluster points of {F ^v(z), v ∈ V} for $z \in \mathbb{S}^1$. In this paper we give a general construction of disjoint iteration groups for which $\emptyset \ne L_F \ne \mathbb{S}^1$.