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

Krzysztof Ciepliński

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 4, page 1079-1088
  • ISSN: 0011-4642

Abstract

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Let ℱ = { F v 𝕊 1 → 𝕊 1 , v ∈ V } be a disjoint iteration group on the unit circle 𝕊 1 , that is a family of homeomorphisms such that F v 1 ∘ F v 2 = F v 1 + v 2 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 ∈ 𝕊 1 . In this paper we give a general construction of disjoint iteration groups for which ∅ ≠ L ℱ ≠ 𝕊 1 .

How to cite

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Ciepliński, Krzysztof. "General construction of non-dense disjoint iteration groups on the circle." Czechoslovak Mathematical Journal 55.4 (2005): 1079-1088. <http://eudml.org/doc/31011>.

@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 -

References

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  8. The structure of iteration groups of continuous functions, Aequationes Math. 46 (1993), 19–37. (1993) Zbl0801.39005MR1220719

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