Fixed point
For a given function $f$, the fixed point is solution $L$ of equation
- $(1) ~ ~ ~ ~ ~ ~ L = f(L)$
In the simple case, $f$ is just holomorphic function of a single variable; then $L$ is assumed to be a complex number. [1]
In the theory of linear operators, for some operator $f$, the eigenfunction with eigenvalue unity is fixed point of operator $f$.
Examples
In some cases, the inverse function $g=f^{-1}$ exists, and often the fixed point of function $f$ is also fixed point of function $g$, id est,
- $(2) ~ ~ ~ ~ ~ ~ L=g(L)$
For example, the fixed point of the logarithm is also fixed point of the exponential, although not every fixed point of the exponential is also fixed point of logarithm.
In many cases, the fixed point can be approximated by the straightforward iteration of equation (1) or equation (2).
For example, the fixed point of exponential
- $L = -\mathrm{ProductLog}(-1)^* \approx 0.3181315052047641+ 1.3372357014306895 \mathrm{i}$
can be evaluated iterating expression
- $L=\ln(L)$
with initial approximation $L\approx\mathrm i$.
Fixed points of a transfer function $f$ are good choice as asymptotic s of the superfunction $F$; fixed point $L$ of the exponential in the example above is asymptotic value of tetration tet:
- $(3) ~ ~ ~ ~ ~ \displaystyle
\lim_{y\rightarrow +\infty} \mathrm{tet}(x\!+\mathrm i y) = L ~ ~ ~ \forall x \in \mathbb R$
The Wolfram (software) offers the table of evaluations of some fixed points for some elementary functions [2].
Hyperbolic fixpoint
The hyperops [3] uses the term fixpoint instead of fixed point. This section is borrowed from there.
A hyperbolic fixpoint is a fixpoint $a$ of $f$ such that $|f'(a)|\neq 0,1$.
For a locally analytic function with hyperbolic fixpoint at $0$, i.e. $f(z)=c_1 z + c_2z^2 + \dots$, $|c_1|\neq 0,1$, there is always a locally analytic and injective function $\sigma$ that satisfies the Schröder equation $$\sigma(f(z))=c_1 \sigma(z))$$ $\sigma$ is unique up to a multiplicative constant and is called the Schröder coordinates of $f$ at 0.
Hyperbolic fixed points are good for application of the regular iteration
Other meanings of the term
Sometimes, fixed point refers to the style of the numerical representation of real constants, that has maximum rounding error, uniformly distributed along the area of numbers allowed. Practically, in this case, any real number is approximated as ratio of some integer number to the integer constant that is the same for all numbers. To avoid confusions, the term fixed point arithmetics is recommended for such a case [4].
