Limiting stochastic processes of shift-periodic dynamical systems

摘要

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 = ( ) generated by such maps display rich dynamical behaviour. The integer parts give a discrete-time random walk for a suitable initial distribution of 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.