The number of periodic points of a function depends on the context. The number of complex periodic points and rational periodic points have been shown to be infinite and finite, respectively, if f is a polynomial of degree at least 2. However, the number of real periodic points can be either finite or infinite. Sharkovskys Theorem states that if p is left of q in the “Sharkovsky ordering” and the continuous function f has a point of period p, then f also has a point of period q. This statement becomes very powerful when considering a function that has points of period 3, all the way to the left side of the Sharkovsky ordering, since having a point of period 3 implies the existence of points of all periods. We explore a continuous function with points of period 3 where the function can be restricted to an interval containing points of period all other natural numbers.
Seaton, Luke J., "Periodic Points and Sharkovsky’s Theorem" (2020). Mathematics Senior Capstone Papers. 15.