General theorem for the existence of iterative roots of homeomorphisms with periodic points

We apply Zdun's factorization theorem (see Zdun (2008) [3]) to give the conditions for the existence and the form of continuous and orientation-preserving iterative roots of homeomorphisms of the circle with a rational rotation number. Our theorem generalizes the previous results given by Jarczyk (2003) in [2], Zdun (2008) in [3] and Solarz (2003, 2009) in [4] and [5].