Fixed point
Fixed point [math]\displaystyle{ L }[/math] of a functor [math]\displaystyle{ F }[/math] is solution of equation
- (1) [math]\displaystyle{ F[L]=L }[/math]
Simple examples
Elementary functions
In particular, functor can be elementaty function. For example, 0 and 1 are fixed points of function sqrt, because [math]\displaystyle{ \sqrt(0)=0 }[/math] and [math]\displaystyle{ \sqrt{1}=1 }[/math].
In similar way, 0 is fixed point of sine function, because [math]\displaystyle{ \sin(0)=0 }[/math] .
Operators
Functor in the equation (1) can be a linear operator. In this case, the fixed point of functor [math]\displaystyle{ F }[/math] is its eigenfunction with eigenvalue equal to unity.
Exponential if fixed point or operator of differentiation D, because [math]\displaystyle{ D \exp(x) =\exp'(x)=\exp(x) }[/math]
The Gaussian exponential
- (2) [math]\displaystyle{ L(x)=\exp(-x^2/2) }[/math], [math]\displaystyle{ x \in }[/math]reals
is fixed point of the Fourier operator, defined with its action on a function [math]\displaystyle{ g }[/math]:
- (3) [math]\displaystyle{ F(g)(p)=\frac{1}{\sqrt{2\pi}\int_{-\infty}^{\infty} g(x)\exp(-{\rm i}px) {\rm d}p }[/math]
in general, functors have no need to be linear, so, there is no associativity at application of several functiors in row, and parenthesis are necessary in the left hand side of eapression (3). [1]
==Fixed points of exponential and fixed points of logarithm
Notes
- ↑ Note that that there is certain ambiguity in commonly uused writing of mathematical formulas, omiting sign * of multiplication; in equaiton (3), expression [math]\displaystyle{ F(g)(p) }[/math] does not mean that [math]\displaystyle{ F(g)\lt math\gt is multiplied to value of \lt math\gt p }[/math]; it means that result [math]\displaystyle{ F(g)(p) }[/math] of action of operator [math]\displaystyle{ F }[/math] on function [math]\displaystyle{ g }[/math], whith is function, is evaluated at arcument [math]\displaystyle{ p }[/math].
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