period of mapping

DefinitionSuppose $X$ is a set and $f$ is a mapping $f:X\to X$.If $f^n$ is the identity mapping on $X$ for some $n=1,2,\mathrm{\dots }$, then$f$ is said to be a mapping of period $n$.Here, the notation $f^n$ means the $n$-foldcomposition $f\circ \mathrm{\cdots }\circ f$.

0.0.1 Examples

1. A mapping $f$ is of period $1$ if and only if $f$ is the identitymapping.

2. Suppose $V$ is a vector space. Then a linear involution $L:V\to V$ isa 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 amapping of period $n$ for $n=1,2,\mathrm{\dots }$.

4. Let us consider the function space spanned by thetrigonometric functions $\sin$ and $\cos$. On this space,the derivative is a mapping of period $4$.

0.0.2 Properties

1. Suppose $X$ is a set.Then a mapping $f:X\to X$ of period $n$ is a bijection.(proof.) (http://planetmath.org/MappingOfDegreeNIsASurjection)

2. Suppose $X$ is a topological space.Then a continuous mapping $f:X\to X$ of period $n$ is a homeomorphism.