Iterated contraction
= Iterated contraction
= Fixed point from a contracting iterate
{synonym}
If a self-map $T$ of a nonempty <complete metric space> has an iterate $T^n$ that is a <contraction mapping>, the <contraction mapping theorem> gives a unique <fixed point> $a$ of $T^n$. The maps commute, so $T^n(Ta)=T(T^na)=Ta$, and uniqueness forces $Ta=a$. Conversely every <fixed point> of $T$ is fixed by $T^n$. Thus $a$ is the unique <fixed point> of $T$.