On conjugacy of homeomorphisms of the circle possessing periodic points

We give a necessary and sufficient condition for topological conjugacy of homeomorphisms of the circle having periodic points. As an application we get the following theorem on the representation of homeomorphisms. The homeomorphism has a periodic point of period n iff there exist a positive integer q<n relatively prime to n and a homeomorphism such that the lift of Φ⁻¹○F○Φ restricted to 0,1] has the form