Fixed point

Fixed point $L$ of a functor $F$ is solution of equation

(1) $F[L]=L$

Simple examples

Elementary functions

In particular, functor can be elementaty function. For example, 0 and 1 are fixed points of function sqrt, because ${\sqrt {0}}=0$ and ${\sqrt {1}}=1$.

In similar way, 0 is fixed point of sine function, because $\sin(0)=0$ .

Operators

Functor in the equation (1) can be a linear operator. In this case, the fixed point of functor $F$ is its eigenfunction with eigenvalue equal to unity.

Exponential if fixed point or operator of differentiation D, because ${\rm {D}}~\exp(x)=\exp '(x)=\exp(x)$

The Gaussian exponential

(2) $L(x)=\exp(-x^{2}/2)$ , $~x\in$reals

is fixed point of the Fourier operator, defined with its action on a function $g$:

(3) $F(g)(p)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }g(x)\exp(-{\rm {i}}px){\rm {d}}x$

in general, functors have no need to be linear, so, there is no associativity at application of several functors in row, and parenthesis are necessary in the left hand side of eapression (3).

Fixed points of exponential and fixed points of logarithm

Graphical search for fixed points of logarithm. Function $f=|z-\ln(z)|$ is shown with levels $f=0.25$, $f=0.5$, $f=1$, $f=2$, $f=4$, $f=8$, $f=16$ in the complex $z$-plane