**Motivation.** I was wondering about the following when playing a card-shuffling game with my elder son.


If $\varphi: \omega \to \omega$ is a bijection, we define the *shuffling distance* of $\varphi$ by $$sh(\varphi) = \min\{|n-\varphi(n)| : n \in \omega\}.$$
Let $\varphi^{(1)}=\varphi$ and $\varphi^{(k+1)} = \varphi\circ\varphi^{(k)}$ for all integers $k \geq 1$.


What is an example of a bijection $\varphi: \omega \to \omega$ such that $sh(\varphi^{(k+1)}) > sh(\varphi^{(k)})$ for all integers $k\geq 1$?



Increasing the "shuffling distance" by iterating a permutation $\varphi: \omega \to \omega$
There's indeed an example of such a function.


Given a an infinite subset $I\subset\omega$, call indexation of $I$ the unique increasing bijection $\omega\to I$. If $(x\_n)$ is its indexation, define $f\_I$ the permutation of $I$, mapping $x\_1\mapsto x\_0$, $x\_{2n+1}\mapsto x\_{2n-1}$ ($n\ge 1$), $x\_{2n}\mapsto x\_{2n+1}$ ($n\ge 0$). Hence $f\_I^k$ maps $f\_{2n+1}\mapsto f\_{2n+1-2k}$ for $n\ge k$, maps $f\_{2n+1}\mapsto f\_{2k-2n-2}$ for $0\le n\le k-1$, and maps $f\_{2n}\mapsto f\_{2n+2k}$. 


Say that a subset $I$ of $\omega$ is "good" if it is infinite, and, $(x\_n)$ being its indexation, it satisfies $x\_n>2x\_{n-1}$ for all $n\ge 1$. If $I$ is good, then $\min\_n|x\_n-f\_I^k(x\_n)|=x\_{k+1}-x\_k$, and $(x\_{k+1}-x\_k)$ is strictly increasing.


Given a partition $\mathcal{P}$ of $\omega$ into good subsets, let $f$ be the permutation of $\omega$ whose restriction to $I$ is $f\_I$ for every $I\in\mathcal{P}$. Then $(\min\_n|x\_n-f^{k}(x\_n)|)\_k$ is strictly increasing. The existence of such a partition is obtained by a straightforward induction.


