The structure of disjoint iteration groups on the circle

Krzysztof Ciepliński

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Let $ℱ = {F^{v} 𝕊^{1} \to 𝕊^{1}, v \in V}$ be a disjoint iteration group on the unit circle $𝕊^{1}$ , that is a family of homeomorphisms such that $F^{v_{1}} \circ F^{v_{2}} = F^{v_{1} + v_{2}}$ for $v_{1}$ , $v_{2} \in V$ 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 ${F^{v} (z)$ , $v \in V}$ for $z \in 𝕊^{1}$ . In this paper we give a general construction of disjoint iteration groups for which $\emptyset \neq L_{ℱ} \neq 𝕊^{1}$ .

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