Let $f:\mathbb{Z}_p\to\mathbb{Z}_p$ be a "nice" map on the $p$-adic integers (or a map on some more general space with a $p$-adic topology). People who study $p$-adic dynamcis investigate what the iterates of $f$ do to points of the space. So if we fix a point $\alpha\in\mathbb{Z}_p$, we can define an iteration map
`$$
I : \mathbb{N} \longrightarrow \mathbb{Z}_p,\qquad
I(n) = f^n(\alpha).
$$
The map $I$ is naturally defined on $\mathbb{N}$, and if $f$ is invertible, then it clearly extends to $\mathbb{Z}$. But for various applications, one would like to evaluate $I(n)$ for $n\in\mathbb{Z}_p$. So the example is
- iteration an integral number of times $\to$ iteration a $p$-adic number of times.
A very pretty application of this idea is in the paper:
Bell, J. P. ; Ghioca, D. ; Tucker, T. J. The dynamical Mordell-Lang problem for étale maps. Amer. J. Math. 132 (2010), no. 6, 1655--1675.
