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Iterative roots of some homeomorphisms with a rational rotation number. (English) Zbl 1104.39018
The author considers the iterative functional equation \(G^m(z)= F(z)\), \(z\in S^1\subset\mathbb{C}\), \(m\geq 2\) integer, on the unit circle with positive orientation \(S^1\subset\mathbb{C}\), where \(F: S^1\to S^1\) is an orientation-preserving homeomorphism which has a nonempty and finite set of periodic points. The solutions of this functional equation (if such exist) are called iterative roots of \(m\)th-order of \(F\).
J. H. Mai [Acta Math. Sin. 30, No. 2, 280–283 (1987; Zbl 0634.39010)] has given conditions for existence of the iterative roots of a homeomorphism with a finite set of periodic points.
In the present paper the author gives different conditions which allow to find all possible rotation numbers of the solutions of the considered functional equation.

MSC:
39B12 Iteration theory, iterative and composite equations
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