The structure of disjoint iteration groups on the circle

Krzysztof Ciepliński

Czechoslovak Mathematical Journal (2004)

  • Volume: 54, Issue: 1, page 131-153
  • ISSN: 0011-4642

Abstract

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The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle 𝕊 1 , that is, families ℱ = { F v 𝕊 1 ⟶ 𝕊 1 v ∈ V } of homeomorphisms such that F v 1 ∘ F v 2 = F v 1 + v 2 , v 1 , v 2 ∈ V , and each F v either is the identity mapping or has no fixed point ( ( V , + ) is an arbitrary 2 -divisible nontrivial (i.e., c a r d V > 1 ) abelian group).

How to cite

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Ciepliński, Krzysztof. "The structure of disjoint iteration groups on the circle." Czechoslovak Mathematical Journal 54.1 (2004): 131-153. <http://eudml.org/doc/30844>.

@article{Ciepliński2004,
abstract = {The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle $\{\mathbb \{S\}^1\}$, that is, families $\{\mathcal \{F\}\}=\lbrace F^\{v\}\:\{\mathbb \{S\}^1\}\longrightarrow \{\mathbb \{S\}^1\}\; v\in V\rbrace $ of homeomorphisms such that \[ F^\{v\_\{1\}\}\circ F^\{v\_\{2\}\}=F^\{v\_\{1\}+v\_\{2\}\},\quad v\_1, v\_2\in V, \] and each $F^\{v\}$ either is the identity mapping or has no fixed point ($(V, +)$ is an arbitrary $2$-divisible nontrivial (i.e., $\mathop \{\mathrm \{c\}ard\}V>1$) abelian group).},
author = {Ciepliński, Krzysztof},
journal = {Czechoslovak Mathematical Journal},
keywords = {(disjoint; non-singular; singular; non-dense; dense; discrete) iteration group; degree; periodic point; orientation-preserving homeomorphism; rotation number; limit set; orbit; system of functional equations; iteration group; degree; periodic point; orientation-preserving homeomorphism; rotation number; limit set; orbit; system of functional equations},
language = {eng},
number = {1},
pages = {131-153},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The structure of disjoint iteration groups on the circle},
url = {http://eudml.org/doc/30844},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Ciepliński, Krzysztof
TI - The structure of disjoint iteration groups on the circle
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 1
SP - 131
EP - 153
AB - The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle ${\mathbb {S}^1}$, that is, families ${\mathcal {F}}=\lbrace F^{v}\:{\mathbb {S}^1}\longrightarrow {\mathbb {S}^1}\; v\in V\rbrace $ of homeomorphisms such that \[ F^{v_{1}}\circ F^{v_{2}}=F^{v_{1}+v_{2}},\quad v_1, v_2\in V, \] and each $F^{v}$ either is the identity mapping or has no fixed point ($(V, +)$ is an arbitrary $2$-divisible nontrivial (i.e., $\mathop {\mathrm {c}ard}V>1$) abelian group).
LA - eng
KW - (disjoint; non-singular; singular; non-dense; dense; discrete) iteration group; degree; periodic point; orientation-preserving homeomorphism; rotation number; limit set; orbit; system of functional equations; iteration group; degree; periodic point; orientation-preserving homeomorphism; rotation number; limit set; orbit; system of functional equations
UR - http://eudml.org/doc/30844
ER -

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