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 = ( ) generated by such maps display rich dynamical behaviour. The integer parts give a discrete-time random...

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Bibliographic Details
Published in:Royal Society open science 2019-11, Vol.6 (11), p.191423-191423
Main Authors: Stadlmann, Julia, Erban, Radek
Format: Article
Language:English
Subjects:
brownian motion
conditionally invariant densities
iterative sequences
Mathematics
one-dimensional maps
random walks
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Summary: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.
ISSN:2054-5703
2054-5703
DOI:10.1098/rsos.191423