Iterated contraction gives a summable orbit (${\sum }_{n\ge 1}\Vert T^{n}x\Vert \le q\Vert x\Vert /(1-q)$)

If a map on a normed vector space satisfies $\Vert T(x)\Vert \le q\Vert x\Vert$ for every $x$, with $0\le q<1$, induction gives $\Vert T^n(x)\Vert \le q^n\Vert x\Vert$. 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 $|\sin u|\le |u|$.