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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 $$F$$.

##### MSC:
 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