  A shift-periodic map is a one-dimensional map from the real line to itself
which is periodic up to a linear translation and allowed to have singularities.
It is shown that iterative sequences $x_{n+1}=F(x_n)$ generated by such maps
display rich dynamical behaviour. The integer parts $\lfloor x_n \rfloor$ give
a discrete-time random walk for a suitable initial distribution of $x_0$ and
converge in certain limits to Brownian motion or more general L\'evy processes.
Furthermore, for certain shift-periodic maps with small holes on $[0,1]$,
convergence of trajectories to a continuous-time random walk is shown in a
limit.
