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Limiting stochastic processes of shift-periodic dynamical systems
Julia Stadlmann1, Radek Erban2
1Merton College, Merton Street, Oxford OX1 4JD, UK.
Abstract:
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 = F(x ) generated by such maps display rich dynamical behaviour. The integer parts 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.
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