Limiting stochastic processes of shift-periodic dynamical systems

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 [Formula: see text] 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 Lévy 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.