# 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 $f\circ\cdots\circ f$.

## 0.0.1 Examples

1. 1.

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

2. 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. 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. 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. 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. 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 PeriodOfMapping 2013-03-22 13:48:53 2013-03-22 13:48:53 bwebste (988) bwebste (988) 12 bwebste (988) Definition msc 03E20 Retract Idempotency