fixed points of normal functions

If $f:M\to \mathrm{𝐎𝐧}$ is a function from any set of ordinals to the class of ordinals then $\mathrm{Fix}(f)=\{x\in M\mid f(x)=x\}$ is the set of fixed points of $f$. $f^{\prime }$, the derivative of $f$, is the enumerating function of $\mathrm{Fix}(f)$.

If $f$ is $\kappa$-normal (http://planetmath.org/KappaNormal) then $\mathrm{Fix}(f)$ is $\kappa$-closed and $\kappa$-normal, and therefore $f^{\prime }$ is also $\kappa$-normal.

For example, the function which takes an ordinal $\alpha$ to the ordinal $1+\alpha$ has a fixed point at every ordinal $\ge \omega$, so $f^{\prime }(\alpha )=\omega +\alpha$.

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