In this paper we deal with the problem of existence and uniqueness of continuous iterative roots of homeomorphisms of the circle. Let F:S~1 → S~1 be a homeomorphism without periodic points. If the limit set of the orbit {F~k(z), k ∈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.
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