Periodic point

Remarks

If $n$ is a period of $x$, so is every positive multiple of $n$, since $f^{kn}(x) = (f^n)^k(x) = x$. The orbit $\set{x, f(x), \dots, f^{n-1}(x)}$ of a periodic point is a finite set, called a periodic orbit or cycle. A point $x$ is eventually periodic, or preperiodic, if some $f^m(x)$ with $m \ge 0$ is periodic.

Example: for $f(x) = x^2 - 1$ on $\R$, the point $0$ is periodic with period $2$. Indeed $f(0) = -1$ and $f(-1) = 0$, so $f^2(0) = 0$.

Non-example: for the same $f$, the point $1$ is not periodic. One has $f(1) = 0$, after which the orbit alternates between $-1$ and $0$ and never returns to $1$. So $1$ is eventually periodic but not periodic.