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On iterative roots of homeomorphisms of the circle. (English) Zbl 0996.39016
Summary: We deal with the problem of existence and uniqueness of continuous iterative roots of homeomorphisms of the circle. Let \(F:S^1\to S^1\) be a homeomorphism without periodic points. If the limit set of the orbit \(\{F^k (z)\), \(k\in \mathbb{Z}\}\) equals \(S^1\), then \(F\) has exactly \(n\) iterative roots of \(n\)-th order. Otherwise \(F\) either has no iterative roots of \(n\)-th order or \(F\) has infinitely many iterative roots depending on an arbitrary function. In this case we determined all iterative roots of \(n\)-th order of \(F\).

39B12 Iteration theory, iterative and composite equations
26A18 Iteration of real functions in one variable
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37C27 Periodic orbits of vector fields and flows