| Title:
|
General construction of non-dense disjoint iteration groups on the circle (English) |
| Author:
|
Ciepliński, Krzysztof |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
55 |
| Issue:
|
4 |
| Year:
|
2005 |
| Pages:
|
1079-1088 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\emptyset \ne L_{{\mathcal F}}\ne {\mathbb{S}}^{1}$. (English) |
| Keyword:
|
(disjoint |
| Keyword:
|
non-singular |
| Keyword:
|
singular |
| Keyword:
|
non-dense) iteration group |
| Keyword:
|
(strictly) increasing mapping |
| MSC:
|
20F38 |
| MSC:
|
37E10 |
| MSC:
|
39B12 |
| idZBL:
|
Zbl 1081.37024 |
| idMR:
|
MR2184385 |
| . |
| Date available:
|
2009-09-24T11:30:50Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128046 |
| . |
| Reference:
|
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| Reference:
|
[2] M. Bajger: On the structure of some flows on the unit circle.Aequationes Math. 55 (1998), 106–121. Zbl 0891.39017, MR 1600588, 10.1007/s000100050023 |
| Reference:
|
[3] K. Ciepliński: On the embeddability of a homeomorphism of the unit circle in disjoint iteration groups.Publ. Math. Debrecen 55 (1999), 363–383. MR 1721896 |
| Reference:
|
[4] K. Ciepliński: On conjugacy of disjoint iteration groups on the unit circle.European Conference on Iteration Theory (Muszyna-Złockie, 1998). Ann. Math. Sil. 13 (1999), 103–118. MR 1735195 |
| Reference:
|
[5] K. Ciepliński: The structure of disjoint iteration groups on the circle.Czechoslovak Math. J. 54 (2004), 131–153. MR 2040226, 10.1023/B:CMAJ.0000027254.04824.0c |
| Reference:
|
[6] K. Ciepliński: Topological conjugacy of disjoint flows on the circle.Bull. Korean Math. Soc. 39 (2002), 333–346. MR 1904668, 10.4134/BKMS.2002.39.2.333 |
| Reference:
|
[7] K. Ciepliński and M. C. Zdun: On a system of Schröder equations on the circle.Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 1883–1888. MR 2015635, 10.1142/S0218127403007709 |
| Reference:
|
[8] M. C. Zdun: The structure of iteration groups of continuous functions.Aequationes Math. 46 (1993), 19–37. Zbl 0801.39005, MR 1220719 |
| . |
