fixed points of normal functions
If f:M→𝐎𝐧 is a function from any set of ordinals to the class of ordinals then Fix(f)={x∈M∣f(x)=x} is the set of fixed points of f. f′, the derivative of f, is the enumerating function of Fix(f).
If f is κ-normal (http://planetmath.org/KappaNormal) then Fix(f) is κ-closed and κ-normal, and therefore f′ is also κ-normal.
For example, the function which takes an ordinal α to the ordinal 1+α has a fixed point at every ordinal ≥ω, so f′(α)=ω+α.
Title | fixed points of normal functions |
Canonical name | FixedPointsOfNormalFunctions |
Date of creation | 2013-03-22 13:28:59 |
Last modified on | 2013-03-22 13:28:59 |
Owner | Henry (455) |
Last modified by | Henry (455) |
Numerical id | 6 |
Author | Henry (455) |
Entry type | Definition |
Classification | msc 03E10 |
Related topic | ProofOfPowerRule |
Related topic | LeibnizNotation |
Related topic | ProofOfProductRule |
Related topic | ProofOfSumRule |
Related topic | SumRule |
Related topic | DirectionalDerivative |
Related topic | NewtonsMethod |
Defines | derivative |