period of mapping

Definition Suppose X is a set and f is a mapping f:XX. If fn is the identity mapping on X for some n=1,2,, then f is said to be a mapping of period n. Here, the notation fn means the n-fold composition ff.

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 spaceMathworldPlanetmath. Then a linear involution L:VV is a mapping of period 2. For example, the reflectionMathworldPlanetmath mapping x-x is a mapping of period 2.

  3. 3.

    In the complex plane, the mapping ze-2πi/nz is a mapping of period n for n=1,2,.

  4. 4.

    Let us consider the function spaceMathworldPlanetmath spanned by the trigonometric functionsDlmfMathworldPlanetmath sin and cos. On this space, the derivativeMathworldPlanetmathPlanetmath is a mapping of period 4.

0.0.2 Properties

  1. 1.

    Suppose X is a set. Then a mapping f:XX of period n is a bijection. (proof.) (

  2. 2.

    Suppose X is a topological spaceMathworldPlanetmath. Then a continuous mapping f:XX 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