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 $n=1,2,\mathrm{\dots}$, then $f$ is said to be a mapping of period $n$. Here, the notation ${f}^{n}$ means the $n$fold composition $f\circ \mathrm{\cdots}\circ f$.
0.0.1 Examples

1.
A mapping $f$ is of period $1$ if and only if $f$ is the 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 $n=1,2,\mathrm{\dots}$.

4.
Let us consider the function space^{} spanned by the trigonometric functions^{} $\mathrm{sin}$ and $\mathrm{cos}$. 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.
Title  period of mapping 

Canonical name  PeriodOfMapping 
Date of creation  20130322 13:48:53 
Last modified on  20130322 13:48:53 
Owner  bwebste (988) 
Last modified by  bwebste (988) 
Numerical id  12 
Author  bwebste (988) 
Entry type  Definition 
Classification  msc 03E20 
Related topic  Retract 
Related topic  Idempotency 