Limiting stochastic processes of shift-periodic dynamical systems. Issue 11 (27th November 2019)

• Record Type: Journal Article
• Authors: Stadlmann, Julia; Erban, Radek
• Is Part Of: Royal Society open science. Volume 6:Issue 11(2019)
• Publication Date: 2019-11-27
• Subjects: iterative sequences -- random walks -- Brownian motion -- one-dimensional maps -- conditionally invariant densities; Science -- Periodicals; 500
• DOI: 10.1098/rsos.191423
• Languages: English
• ISSNs: 2054-5703

Abstract

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 n +1 = F ( x n ) generated by such maps display rich dynamical behaviour. The integer parts ⌊ x n ⌋ 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.