Fixed point

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

$F[L]=L$,

that is a point that is mapped to itself.

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

The functor in the equation defining a fixed point can be a linear operator. In this case, the fixed point of the 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

$L(x)=\exp(-x^{2}/2)$ , with $x\in \mathbb {R}$ and $\mathbb {R}$ the 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, and 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 equations

(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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=1}

(13)Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_b} the real fixed points exist at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1< b \le \exp(1/\rm e)} . For example, at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/ \rm e)>} , number e is fixed point of both, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_b} and $\exp _{b}$; and at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} , numbers 2 and 4 are their fixed points.

FIg.3. Example of graphic solution of equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\log_b(x)} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} (two real solutions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=4} ), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\rm e)} (one real solution $x={\rm {e}}$) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=2} (no real solutions).

Finding of real fixed points is presented graphically in Figure 3. The black curve represents the identical function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=f(x)=x} in the left hand side of equation

(14) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=\log_b(z)}

the colored curves represent the function in the right hand side for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\rm )e} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=2} . In the last case, there are no real solutions, but the complex fixed points are complex numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_2^*} . Within few hundred iterations of equation (14), they can be approcimated with many decimal digits;

(15)Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_b} . In some range of the complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -plane, the tetration can be approximated with asymptotic

(32) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  F(z)=L+\varepsilon(z) + o \big(\varepsilon(z)^{2}\big) }

where

(33) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \varepsilon(z)=\exp(Qz+r) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} are fixed complex numbers, dependent on $b$. Possible values of fixed points $L$ are shown with thin black curves versus logarithm on base $b$. At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} in (33), again, the dased curve shows the imaginary part, and real and imaginary parts of the asymptotic period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi{ \rm i}/Q} . See article tetration for details.

Projectors

Functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} on the space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} is called Projector, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^2 \phi=P \phi} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \in \Psi} .

Any function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } of subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi } of functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=P\phi} is fixed point of projector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} .

In common use are projectors of the 3-dimensional space to 2-dimensional space. Such projectors allow to make flat pictures of 2-dimensional objects. Any point at the plane of image is fixed point of such a projector.

Common fixed points

The same element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} may be fixed point of several projectors.

Sometimes, such common fixed points can be approximated using iterations. If there are two projectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_2} , then the sequence of elements, approximating the common fixed point of these projectors can be constructed in the following way:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n=P_1\Big(P_2(f_{n-1})\Big) }

Such a sequance is used to approximate the amplitude-phase behavior of ultra-short pulses in the FROG systems .

See also

• exponential
• Linear operators
• Eigenfunciton
• eigenvalue
• tetration
• projector