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On the embeddability of a homeomorphism of the unit circle in disjoint iteration groups. (English) Zbl 0935.39010
Let \(S^1\) denote a unite circle. A homeomorphism \(F:S^1\to S^1\) is said to be embeddable in a disjoint group if there exists a family of homeomorphisms \(F^t:S^1\to S^1\), \(t\in\mathbb{R}\) such that \(F^t \circ F^s=F^{t+s}\), \(t,s\in\mathbb{R}\), \(F^1=F\) and if \(F^t\) has a fixed point then \(F^t=\operatorname{id}\). In the paper the characterization of embeddable homeomorphisms is given. A homeomorphism with rational number is embeddable whenever one of its iterates equals identity. If the rotation number of homeomorphism \(F\) is irrational and the limit set \(L_F=\{F^n(x),\;n\in\mathbb{Z}\}^d\) is equal to the whole circle, then \(F\) is embeddable. In this case the general form of all disjoint iteration groups of \(F\) is obtained. The main result states that the embeddability of \(F\) with irrational rotation number and \(L_F\neq S^1\) is equivalent to the existence of a sequence of non-negative integers satisfying some conditions related to the iterative kernel of \(F\). The proof of the theorem contains the general construction of all disjoint iteration groups of \(F\).

MSC:
39B12 Iteration theory, iterative and composite equations
37C10 Dynamics induced by flows and semiflows
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