Iterated contraction gives a summable orbit
= Iterated contraction gives a summable orbit
{title2=$\sum_{n\ge1}\|T^nx\|\le q\|x\|/(1-q)$}
If a map on a <normed vector space> satisfies $\|T(x)\|\le q\|x\|$ for every $x$, with $0\le q<1$, induction gives $\|T^n(x)\|\le q^n\|x\|$. The <geometric series> then proves <absolute convergence> of its orbit series whenever the ambient <normed vector space> is complete. The scalar map $T(x)=\sin(qx)$ satisfies the bound because $|\sin u|\le|u|$.