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 xn+1 = F(xn) generated by such maps display rich dynamical behaviour. The integer parts ⌊xn⌋ give a discrete...
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| Médium: | Journal article |
| Jazyk: | English |
| Vydáno: |
Royal Society
2019
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| Shrnutí: | 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 xn+1 = F(xn) generated by such maps display rich dynamical behaviour. The integer parts ⌊xn⌋ give a discrete-time random walk for a suitable initial distribution of x0 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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