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(disjoint; non-singular; singular; non-dense) iteration group; (strictly) increasing mapping
Let ${\mathcal F}=\lbrace F^{v}\: {\mathbb{S}}^{1}\rightarrow {\mathbb{S}}^{1}, v\in V\rbrace $ be a disjoint iteration group on the unit circle ${\mathbb{S}}^{1}$, 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 $L_{{\mathcal F}}$ the set of all cluster points of $\lbrace F^{v}(z)$, $v\in V\rbrace $ for $z\in {\mathbb{S}}^{1}$. In this paper we give a general construction of disjoint iteration groups for which $\emptyset \ne L_{{\mathcal F}}\ne {\mathbb{S}}^{1}$.
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