Geodesic
The structure of disjoint iteration groups on the circle
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 1, pp. 131-153.

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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., $\mathop {\mathrm card}V>1$) abelian group).
Classification : 20F28, 20F38, 30D05, 37B99, 37E10, 37E45, 39B12, 39B32, 39B72
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 
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     author = {Ciepli\'nski, Krzysztof},
     title = {The structure of disjoint iteration groups on the circle},
     journal = {Czechoslovak Mathematical Journal},
     pages = {131--153},
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     volume = {54},
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     zbl = {1047.37024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2004__54_1_a11/}
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Ciepliński, Krzysztof. The structure of disjoint iteration groups on the circle. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 1, pp. 131-153. http://geodesic.mathdoc.fr/item/CMJ_2004__54_1_a11/