period of mapping

Definition Suppose $X$ is a set and $f$ is a mapping $f:X\to X$ . If $f^n$ is the identity mapping on $X$ for some $n=1,2,\ldots$ , then $f$ is said to be a mapping of period $n$ . Here, the notation $f^n$ means the $n$ -fold composition $f\circ\cdots \circ f$ .

Examples

1. A mapping $f$ is of period $1$ if and only if $f$ is the identity mapping.
2. Suppose $V$ is a vector space. Then a linear involution $L:V\to V$ is a mapping of period $2$ . For example, the reflection mapping $x\mapsto -x$ is a mapping of period $2$ .
3. In the complex plane, the mapping $z\mapsto e^{-2\pi i/n}z $ is a mapping of period $n$ for $n=1,2,\ldots$ .
4. Let us consider the function space spanned by the trigonometric functions $\sin$ and $\cos$ . On this space, the derivative is a mapping of period $4$ .

Properties

1. Suppose $X$ is a set. Then a mapping $f:X\to X$ of period $n$ is a bijection. (proof.)
2. Suppose $X$ is a topological space. Then a continuous mapping $f:X\to X$ of period $n$ is a homeomorphism.