= Fixed point
{wiki}
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.
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