Minimal periodic orbits of continuous mappings of the circle

Abstract

Let $f$ be a continuous map of the circle into itself and suppose that $n > 1$ is the least integer which occurs as a period of a periodic orbit of $f$. Then we say that a periodic orbit $\{ {p_1}, \ldots ,{p_n}\}$ is minimal if its period is $n$. We classify the minimal periodic orbits, that is, we describe how the map $f$ must act on the minimal periodic orbits. We show that there are $\varphi (n)$ types of minimal periodic orbits of period $n$, where $\varphi$ is the Euler phi-function.