Fixed point

A fixed point of a self-map $f:X\to X$ is an element $x\in X$ with $f(x)=x$. Once an iteration reaches that element, every later iterate remains there.

For example, $f(x)=x^2$ on the real line has fixed points zero and one, because $x^2=x$ is equivalent to $x(x-1)=0$. The translation $f(x)=x+1$ has none. For real-valued self-maps, the fixed points are exactly the zeros of a function $h(x)=f(x)-x$.

Existence of a fixed point does not itself imply convergence of nearby iterates. For $f(x)=x^2$, inputs with $|x|<1$ tend to zero under repeated squaring, while the initial value two grows without bound. The same function can therefore have both fixed points and very different iterated behaviour elsewhere.

A fixed point of a function on a partially ordered set that is greater than or equal to every other fixed point. For the order-preserving function $H(X)=A\cap \{s:\mathrm{Post}(s)\subseteq X\}$ on the power set of a finite transition system with no terminal states, its greatest fixed point is the set of states from which every infinite path in a transition system remains in $A$. Iteration from the whole state space decreases to this set; every strict decrease removes at least one state.