On some iterative roots on the circle

Paweł Solarz
2003 Publicationes mathematicae (Debrecen)  
The aim of this paper is to investigate the problem of the existence of continuous iterative roots of a homeomorphism F : Let S 1 = {z ∈ C : |z| = 1} be the unit circle with the positive orientation. Let u, w, z ∈ S 1 , then there exist unique t 1 , t 2 ∈ [0, 1) such that we 2πit 1 = z, we 2πit 2 = u. Define −−−→ u, w := e 2πit : t ∈ t u , t w ). This set is said to be an open arc (resp. a closed arc). Let F : S 1 −→ S 1 be a continuous mapping, then there exist a continuous function f : R −→ R
more » ... called a lift of F and an integer k such that F e 2πix = e 2πif (x) and f We say that a homeomorphism F preserves orientation if f is increasing (reverses orientation if f is decreasing) (see for example [7]). Let u, w, z ∈ S 1 and w ∈ − −− → (u, z), then if F preserves orientation F (w) ∈ −−−−−−−−→ (F (u), F (z)). However, if F reverses orientation, then we have F (w) ∈ −−−−−−−−→ (F (z), F (u)).
doi:10.5486/pmd.2003.2823 fatcat:qbr6ne67jbfmvpymgxy4jt4rbm