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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
(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
Summary:
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).
References:
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