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

FIg.1. $f=|z-\ln(z)|$ is shown with levels $f=0.5$, $f=1$, $f=2$, $f=4$, $f=8$, $f=16$ in the complex $z$-plane

Fig.1. The same as FIg.1 for function $f=|z-\exp(z)|$

Fixed points can be searched graphically. Fig.1 shows the graphical search of fixed points of logarithm, i.e., soluitons of the equaiton

(10) $L=\ln(L)$

There are no real solutions fot this equation, but there are two complex-congjugated solutions $L_{\rm {e}}$ and $L_{\rm {e}}^{*}$. However, the value of $L_{\rm {e}}$ cannot be estimated well from the figure (1), but the straigtgorward iteration allows the precise estimate. Few hundreds of iterations are sufficient to get error of order of last significant figure in the approximation

(11) $L_{\rm {e}}\approx 0.318131505204764135312654+1.33723570143068940890116\!~{\rm {i}}$

Fixed points of logarithm should not be confised with fixed points of exponential, shown in FIgure 2. Therse fixed points are solutions of equaitons

(12) $L=\exp(L)$

They can be expressed also as solution of equation

(13) $L=\log(L)+2\pi \!~{\rm {i~m~}}$ for some integer $m$

For example,

(13)$L_{\rm {e,2}}\approx 2.062277729598284+7.5886311784725127\!~{\rm {i}}$

is fixed point of the exponential, but is not the fixed point of natural logarithm.