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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