Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 2 / 8D / c / i

Choose a bijection $h:\mathbb{N}\to \mathbb{Z}$, and conjugate the translation of the integers:Then $f$ is a bijection of $\mathbb{N}$. Its $k$th iterate is $h^{-1}(z\mapsto z+k)h$, which has no fixed point for every $k\ge 1$. Hence $f\in T$.