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[parent] period of mapping (Definition)

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$.

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,\ldots$.
  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$.

Properties

  1. Suppose $ X$ is a set. Then a mapping $ f:X\to X$ of period $ n$ is a bijection. (proof.)
  2. Suppose $ X$ is a topological space. Then a continuous mapping $ f:X\to X$ of period $ n$ is a homeomorphism.



"period of mapping" is owned by bwebste. [ owner history (1) ]
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See Also: retract, idempotency


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mapping of period $n$ is a bijection (Proof) by Koro
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Cross-references: homeomorphism, continuous mapping, topological space, bijection, derivative, trigonometric functions, spanned by, function space, complex plane, reflection, linear involution, vector space, composition, identity mapping, mapping
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This is version 9 of period of mapping, born on 2003-08-01, modified 2004-03-06.
Object id is 4539, canonical name is PeriodOfMapping.
Accessed 1714 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
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