... more information about the coexistence of periodic points on the circle, comparing with the results in 4, 5, 12, and so on. The relative homotopy-conjugacyclasses refine the homotopy classes, ... degree, on maps on the circle, because of the following.Proposition 4.6. The degree of fn0,n1,n2is n0 n1 n2 1.In fact, our improvement lies in the cases that the degree of a ... and active field in mathe-matics see 3.It is very natural to ask for an analogy of Sharkovskii’s theorem for maps on the circle. Unfortunately, the answer is negative if the period of a periodic...