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 $n=1,2,\\ldots$ , then $f$ is said to be a mapping of period $n$ .Here, the notation $f^{n}$ means the $n$ -foldcomposition $f\\circ\\cdots\\circ f$ .

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 $n=1,2,\\ldots$ .
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.

单词	PeriodOfMapping
释义	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 $n=1,2,\\ldots$ , then $f$ is said to be a mapping of period $n$ .Here, the notation $f^{n}$ means the $n$ -foldcomposition $f\\circ\\cdots\\circ f$ . 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 $n=1,2,\\ldots$ . 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.
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