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.2. 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 exp are not the same as those of ln

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, at $m=1$

(13)$L_{\rm {e,1}}\approx 2.062277729598284+7.5886311784725127\!~{\rm {i}}$

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

In the case of exponential and natural logarithm, all fixed points are complex. However, $\log _{b}$ the real fixed points exist at $1<b\leq \exp(1/{\rm {e)}}$. For example, at $b=\exp(1/{\rm {e)>}}$, number e is fixed point of both, $\log _{b}$ and $\exp _{b}$; and at $b={\sqrt {2}}$, numbers 2 and 4 are their fixed points.

FIg.3. Example of graphic solution of equation $x=\log _{b}(x)$ for $b={\sqrt {2}}$ (two real solutions, $x=2$ and $x=4$), $b=\exp(1/{\rm {e)}}$ (one real solution $x={\rm {e}}$) $b=2$ (no real solutions).

Finding of real fixed points is presented graphically in Figure 3. The black curve represents the identical function $f=f(x)=x$ in the left hand side of equation

(14) $z=\log _{b}(z)$

the colored curves represent the function in the right hand side for $b={\sqrt {2}}$ $b=\exp(1/{\rm {e}}$ $b=2$. In the last case, there are no real solutions, but the complex fixed points are complex numbers $L_{2}$ and $L_{2}^{*}$. Within few hundred iterations of equation (14), they can be approcimated with many decimal digits;

(15)$L_{2}\approx 0.824678546142074222314065+1.56743212384964786105857\!~{\rm {i}}$.

Such evaluation can be berformed in real time with any high level algorithmic language – Maple, Mathematica, that allow to specify number of correct decimal digits desirable in the result.

Tetration

FIg.4. parameters of asymptotic of tetration versus logarithm of the base

The fixed points of logarithm determine the asymptotic properties of analytic extension of tetration $F_{b}$. In some range of the complex $x$-plane, the tetration can be approximated with asymptotic

(32) $F(z)=L+\varepsilon (z)+o{\big (}\varepsilon (z)^{2}{\big )}$

where

(33) $\varepsilon (z)=\exp(Qz+r)$,

$Q$ and $r$ are fixed complex numbers, dependent on $b<math>.Possiblevaluesoffixedpoints<math>L$ are shown with thin black curves versus logarithm on base </math>b</math>. At $\ln(b)>1/{\rm {e}}$, the fixed points are complex; the real part is shown with solid curve, the imaginary part is shown with dashed curve.

Green curves in FIgure 4 represent the parameter $Q$ in (33), again, the dased curve shows the imaginary part, and real and imaginary parts of the asymptotic period $2\pi {\rm {i}}/Q$. See article tetration for details.