Geodesic
General construction of non-dense disjoint iteration groups on the circle
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 1079-1088 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\emptyset \ne L_{{\mathcal F}}\ne {\mathbb{S}}^{1}$.
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 $\emptyset \ne L_{{\mathcal F}}\ne {\mathbb{S}}^{1}$.
Classification : 20F38, 37E10, 39B12
Keywords: (disjoint; non-singular; singular; non-dense) iteration group; (strictly) increasing mapping 
@article{CMJ_2005_55_4_a20,
     author = {Ciepli\'nski, Krzysztof},
     title = {General construction of non-dense disjoint iteration groups on the circle},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1079--1088},
     year = {2005},
     volume = {55},
     number = {4},
     mrnumber = {2184385},
     zbl = {1081.37024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2005_55_4_a20/}
}
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Ciepliński, Krzysztof. General construction of non-dense disjoint iteration groups on the circle. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 1079-1088. http://geodesic.mathdoc.fr/item/CMJ_2005_55_4_a20/