# fixed points of normal functions

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

If $f$ is $\kappa$-normal (http://planetmath.org/KappaNormal) then $\operatorname{Fix}(f)$ 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 $\geq\omega$, 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