On The Circumference Of The Four Periodic Orbits Of Continuous Self-mapping Of The Set Of Periods

We shall consider the period orbits of self-maps on the circle which have a period orbit with least period 4. Starting with the history and development of this problem, some pre-knowledge are introduced. We classify the maps on the circle with finite invariant subset into some homotopy conjugacy classes, then the set of periods of each class could be determined based on the use of f-graph, and some examples presented. Finally our main result is a classification to the relative homotopy minimal periods. When comparing our conclusion with the Sharkovskii theorem, some differences are pointed out, namely, the set of periods is N for almost all the self-maps on the circle having a period orbit with least period 4.