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2 Bulletin of the Polish Academy of Sciences. Mathematics

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On iterative roots of homeomorphisms of the circle

Zdun M.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

203-213

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 belongs to 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.

Set-valued iterative square roots of bijections

Powierża T.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 4

377-383

There are different ideas how to generalize the notion of an iterative root, especially when a function does not have a "real" root. We consider a multifunction as a substitute for this notion.