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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  $$\emptyset \ne L_F \ne \mathbb{S}^1$$ .

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