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 $\sin$ 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.

Title	period of mapping
Canonical name	PeriodOfMapping
Date of creation	2013-03-22 13:48:53
Last modified on	2013-03-22 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