General construction of non-dense disjoint iteration groups on the circle

@article{Ciepliński2005,
abstract = {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 $\emptyset \ne L_\{\{\mathcal \{F\}\}\}\ne \{\mathbb \{S\}\}^\{1\}$.},
author = {Ciepliński, Krzysztof},
journal = {Czechoslovak Mathematical Journal},
keywords = {(disjoint; non-singular; singular; non-dense) iteration group; (strictly) increasing mapping; iteration group; (strictly) increasing mapping},
language = {eng},
number = {4},
pages = {1079-1088},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {General construction of non-dense disjoint iteration groups on the circle},
url = {http://eudml.org/doc/31011},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Ciepliński, Krzysztof
TI - General construction of non-dense disjoint iteration groups on the circle
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 4
SP - 1079
EP - 1088
AB - 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 $\emptyset \ne L_{{\mathcal {F}}}\ne {\mathbb {S}}^{1}$.
LA - eng
KW - (disjoint; non-singular; singular; non-dense) iteration group; (strictly) increasing mapping; iteration group; (strictly) increasing mapping
UR - http://eudml.org/doc/31011
ER -