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The structure of disjoint iteration groups on the circle. (English) Zbl 1047.37024

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 \(\mathcal F = \{F^v: \mathbb S^1 \rightarrow \mathbb S\), \(v\in V\}\) of homeomorphisms such that \(F^{v_{1}}\circ F^{v_2} = F^{v_1 + v_2}\), \(v_1, v_2 \in V\), and each \(F^v\) either is the identity mapping or has no fixed point (\((V, +)\) is an arbitrary \(2\)-divisible nontrivial (i.e., card \(V > 1\)) abelian group).

MSC:

37E10 Dynamical systems involving maps of the circle
20F38 Other groups related to topology or analysis
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37E45 Rotation numbers and vectors
39B12 Iteration theory, iterative and composite equations
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References:

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